m LIBRARY 2- Michigan State 2008 University This is to certify that the dissertation entitled ON SOME INFERENCE PROBLEMS FOR CURRENT STATUS DATA presented by- DEEPA AGGARWAL has been accepted towards fulfillment of the requirements for the Ph.D. degree in STATISTICS AND PROBABILITY flag/'— Major Professor's Signature ”“27 9,1 M? Date MSU is an Affirmative Action/Equal Opportunity Institution -.-—.—.—p-.~._-—.-.~.-.~.-.—.-—.—-—u--.—.—.-—.--—---—-—.--—.-.—---._.—.—.—‘a.-.-—-_I-.--—.—.—-—-—.-—o—.—.—-—u—.-n- PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 5/08 KrlProflAccS-Pres/CIRC/DateDue.mdd On Some Inference Problems For Current Status Data By Deepa Aggarwal A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 2008 ABSTRACT On Some Inference Problems For Current Status Data By Deepa Aggarwal In the current status or interval censored case 1 data, one does not observe the event occurrence time but only the inspection time and whether the event has occurred prior to the inspection time or not. This thesis consists of two parts. The first part pertains to fitting a parametric model to the distribution function of the event occurrence time in the one sample set up with current status data. In this part, we first discuss two analogous minimum distance inference procedures for fitting a regression function in the classical regression set up. These distances are based on squared deviations of a nonparametric regression function estimator and the model being fitted. In the one distance the integrating measure is a— finite and in the second, it is data dependent. The thesis establishes asymptotic normality of the proposed empirical minimum distance statistic and that of the corresponding estimator under the fitted model in a general regression set up. Then, these results for empirical minimum distance test are adapted to fit a parametric model to the distribution of the event occurrence time based on current status data. It also contains a finite sample comparison of the proposed test with Koul and Yi test and the one sample Cramér-von Mises test based on nonparametric maximum likelihood estimator of the distribution function of the event occurrence time. The second part of the thesis pertains to testing for the equality of the two event occurrence time distribution functions in the two sample setting when the data is in- terval censored case 1 from both samples. It derives the asymptotic distribution of the underlying test statistic both under the null hypothesis and under local alternatives. It also contains a finite sample comparison of the proposed test with the two sample Cramér-von Mises test based on nonparametric maximum likelihood estimators of the time to event distribution functions. ACKNOWLEDGMENTS I wish to express my deepest regard to my advisor Professor Hira L. Koul for his invaluable guidance, generous support. I have enjoyed our interactions tremendously. I thank Dr Koul for providing me with countless opportunities to grow both personally and professionally. This dissertation could not have been completed without his help and support. I would also like to thank Professors Sarat Dass, Dennis Gilliland and Habib Salehi for serving on my guidance committee. My special thanks go to Professors Connie Page, Dennis Gilliland and Sandra Herman for their advice when I was at the consulting service. I would like to thank my family especially my mother, my husband and my daugh- ter for all the support and encouragement provided by them during my graduate education. This research was partly supported by the NSF grant DMS 0704130 with Professor Koul as P.I. iv TABLE OF CONTENTS 1 Introduction 1 2 Empirical Minimum Distance Lack-Of—Fit Testing In Regression Model 8 2.1 Introduction ................................ 8 2.2 Assumptions ................................ 12 2.3 Consistency of 0;; and fin ......................... 14 2.4 Asymptotic distribution of «in ...................... 28 2.5 Asymptotic normality of Mn (0n) .................... 41 3 Minimum Distance Goodness-Of—Fit Tests For Current Status Data 60 3.1 Introduction ................................ 60 3.2 Minimum Distance Statistics and Tests ................ 63 3.3 Empirical Minimum Distance Statistic ................. 67 3.4 Simulations ................................ 68 4 Testing the equality of two distributions with Current Status Data 75 4.1 Introduction ................................ 75 4.2 Asymptotic behavior under the null hypothesis and local alternatives 78 4.3 Simulations ................................ 88 BIBLIOGRAPHY 93 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 LIST OF TABLES Mean and MSE of tin, X, T N exp(1), 00 = 1 ............. 72 Empirical sizes of Mn(én), X, T ~ exp(1) ............... 72 Power of Mn(én), T ~ exp(1), (c1,c2) = (.9, 1) ............ 73 Simulated percentiles of CV1, X, T N exp(1) ............... 73 Empirical sizes of CV1, X, T ~ exp(1) ................. 73 Mean and MSE of (An, X, T ~ exp(1), 00 = 1 ............. 74 Power of CV1, T = exp(1) ......................... 74 Empirical sizes, X,T ~ exp(1), (01,02) = (.9, .8) ............ 74 Empirical sizes of T, X, Y N exp(1), S,T ~ exp(1) .......... 90 Empirical sizes of T, X, Y ~ exp(1), (n1,n2) = (180, 200) ...... 90 Power of T, S,T ~ exp(1), X ~ exp(1), n1 = 77.2 = 50 ......... 91 Simulated percentiles of CV2, X, Y ~ exp(1), S,T ~ exp(1.5) ..... 91 Empirical sizes of CV2, X, Y ~ exp(1), S, T N exp(1.5) ........ 91 Simulated 95th percentile of CV2, X, Y ~ exp(1). ........... 92 Empirical sizes, X, Y N exp(1), (c1,c2) = (25,1). ........... 92 Power, S,T ~ exp(1), X ~ exp(1), (01,02) = (.2, .9). ......... 92 vi CHAPTER 1 Introduction In recent years there has been a considerable research on the analysis of interval- censored data. In the case I interval censored data, an event occurrence time X is unobservable, but one observes an inspection time T and whether an event has occurred prior to this time or not. This type of data is also known as current status data. It is different from the right censored data where one observes true life time in the case of no censoring and only censoring time when life time is censored. Current status data often arises in epidemiology, demography and economics. For example, as mentioned in Jewell and Van der Laan (2004), in the study of infectious disease Human Immunodeficiency Virus (HIV), in particular, the partner studies of HIV infection. These partnerships are assumed to include an index case who has been infected via some external source, and a susceptible partner who has no other means of infection except the contact with the index case. Suppose X denotes the time from infection of the index case to infection of the susceptible partner and T is the time the susceptible partner is examined after infection of the index case. Then the infection status of the susceptible partner provides current status data. For some more applications for current status data, see Hoel and Walburg (1972), Finkelstein and Wolfe (1985), Finkelstein (1986), Diamond, McDonald and Shah (1986), Diamond and McDonald (1991), Keiding (1991) and Jewell and Van der Laan (2004). This thesis is concerned with the following two problems. The first problem per- tains to fitting a parametric model to the distribution function of the event occurrence time in the one sample set up with current status data. The second problem is con- cerned with testing for the equality of the two event occurrence times distribution functions in the two sample setting when the data is interval censored case 1 from both samples. We shall now focus on the first problem for the moment. To describe this problem a bit more precisely, let F denote the distribution function (d.f.) of the event occurrence time X, G be a subset of q-dimensional Euclidean space IN, and (Fe, 0 E G} be a known parametric family of d.f.’s on [0,00). Let T be the inspection time and 6 := I (X S T) and I be a compact interval of [0, 00), where I [A] denotes the indicator function of the event A. We assume X to be independent of T. The problem of interest here is to test the hypothesis H01 : F(t) = F90“), for all t E I, for some 90 E 6, against the alternative H11 : H01 is not true. It is natural to base tests of H01 on a distance between the nonparametric max- imum likelihood estimates F of F and (Fe, 0 E 6}. One such test statistic is the 2 Cramér—von-Mises statistic cvl = 1an / (F(x) —F0(z))2dfi(x). But unfortunately neither the finite sample nor the asymptotic null distribution of this statistic is known because of the complicated nature of the distribution of F‘. Even asymptotic distribution of a suitably standardized E is intractable, cf., Groeneboom and Wellner (1992). An alternative way to proceed is to use the well known regression relationship between 5 and F(T), i.e., E(6|T) = F (T), and the fact that this regression is het- eroscedastic. In this context then the problem of testing H01 is equivalent to testing the lack—of—fit of the parametric regression model {F9, 6 E 6}. There is a vast literature on the problem of testing for the lack—of-fit of a para- metric regression model. The monograph of Hart (1997) provides a nice overview on the subjet till 1997. Using the ideas of Khmaladze (1979), Stute, Thies and Zhu (1998) proposed an asymptotically distribution free test for this problem based on a martingale transform of a certain marked empirical process of the residuals. Koul and Ni (2005) used the minimum distance methodology to propose a class of tests for the same problem. In all this literature the data is completely observable. Using the above mentioned equivalence between testing H01 and the correspond- ing lack-of-fit testing of a regression model, Koul and Yi (2006) adapted the Stute— Thies-Zhu test to test for H01. They provide sufficient conditions for consistency of their test at a fixed alternative and derive an expression for its asymptotic power against local alternatives. Koul and Ni (2004) used the integrated square distance between a kernel type nonparametric estimator of the regression function and the model being fitted, where the integrating measure is a a—finite measure. A practical problem that arises in using these statistics is the choice of the integrating measure. Although one may choose this by using some optimality criteria, such a measure will invariably depend on the model being fitted and the design distribution. In this thesis we first discuss two analogous minimum distance inference proce- dures in the classical regression set up, first when the integrating measure is a—finite and second when the integrating measure is data dependent-viz, the empirical d.f. of the design variable. We prove asymptotic normality of the proposed empirical mini- mum distance statistic and that of the corresponding estimator under the fitted model in a general regression set up. Then, these results for empirical minimum distance test are adapted to fit a parametric model to the distribution of the event occurrence times based on current status data. We also show consistency of the proposed mini- mum distance tests against a fixed alternative and obtain asymptotic power against a class of local alternatives for current status data. We now describe the second problem of this thesis. To describe it more precisely, let F1 (F2) denote the d.f. of event occurrence time X (Y) from the first (second) population, and let S (T) be the corresponding inspection time. In the two sample current status data set up, one observes (6, S) and (n, T), where 6 = I [X S S] and 17 = I [Y S T]. The problem of interest here is to test the null hypothesis that the two event occurrence distributions are the same, i.e. H02: F1(a:) = F2(rc), for all :1: E I, against the alternative H12: F1(:r) 79 172(1), for some :2: E I. Similar to the one sample set up discussed above, it is natural to base tests of H02 on nonparametric maximum likelihood estimates F1 and F2 of F1 and F2. One such test is based on the Cramér-von—Mises statistic CV2 : 7117-2712 ‘/J:' (131(1) - F2(33))2d13‘1 (at) 1217:2112 /1_(fi‘1(:r)— fi2($))2dfig($). Again for the same reasons given above asymptotic null distribution of such a statistic is not currently tractable. An alternative way to proceed is to use the well known regression relationship between 6 and 171(5), and 77 and F2(T), i.e., E((5|S) = F1(S) and E(77IT) = F2(T), and the fact that these two regressions are heteroscedastic. In this context then the problem of testing H02 is equivalent to testing the equality of the two regression functions under heteroscedasticity. The problem of comparing the two regression functions has been discussed by several authors. In general, see, e.g., Hall and Hart (1990), King, Hart and Wehrly (1991), Carroll and Hall (1992), Delgado (1993), Kulasekera (1995), Koul and Schick (1997, 2003), Neumeyer and Dette (2003), among others. The data is completely observable in the above mentioned literature. 5 Koul and Schick (2003) proposed a test using covariate matching for the same problem in a general regression set up. In this thesis, we adapt this test to the two sample current status data and discuss its asymptotic normality under a general set of assumptions. This thesis is organized as follows. Chapter 2 studies empirical minimum dis- tance tests of lack of fit in classical regression set up. Corollary 2.3.1 and Theorem 2.3.1 state and prove consistency of empirical minimum distance estimates of the underlying parameters of the model being fitted. Theorem 2.4.1 and Theorem 2.5.1 give asymptotic distribution of the parameter estimator and the empirical minimum distance statistic under the null hypothesis. In chapter 3, section 2, we apply the results of Koul and Ni (2004) for minimum distance tests of goodness of fit hypothesis based on current status data. After that, we discuss consistency of these tests against a fixed alternative and obtain asymptotic power against a class of local alternatives. Section 3.3 uses the results of Chapter 2 for empirical minimum distance test to fit a parametric model to the distribution of the event occurrence times based on current status data. Section 3.4 reports the numerical results of the three simulation studies in the one sample set up. The first one assesses the finite sample level and power behavior of the empirical minimum distance test. The simulation results of empirical minimum distance statistic are consistent with asymptotic theory. Also, simulation results show little bias in the estimator of the best fitted parameter for all the chosen sample sizes. The second simulation study investigates Monte Carlo size and power behavior of the Cramér-Von-Mises test CV1. The finite sample level of this test approximates 6 the nominal level well for all the chosen sample sizes. The third simulation study investigates Monte Carlo size comparison of empirical minimum distance test, CV1, and Koul and Yi (2006) test. Simulation results show that empirical sizes are better for C V1 and Koul and Yi (2006) test as compared to empirical minimum distance test, when sample size is less than 200. But when sample size is 200 or large, empirical sizes are comparable in all the three tests. In our simulations, F is obtained by the one step procedure for the calculation of the nonparametric maximum likelihood estimator, based on isotonic regression, cf. Groeneboom and Wellner (1992). Chapter 4 deals with the problem of testing the equality of two distribution func- tions against the two sided alternative based on the current status data. Proposition 4.2.1 discuss asymptotic normality of the underlying test statistic under a general set of assumptions. Section 4.3 reports the numerical results of the two simulation stud- ies. The first one assesses the finite sample level and power behavior of the proposed test statistic. The simulation results of the proposed test statistic are consistent with asymptotic theory. In the second simulation study, the finite sample comparison of the proposed test statistic with the two sample Cramér-Von Mises test is made. Simulation results show that for all the chosen alternatives, bandwidths and sample sizes, significance level and power of the proposed and CV2 tests are comparable. Again, in our simulations, F1 and F2 are obtained by the one step procedure for the calculation of the nonparametric maximum likelihood estimator, based on isotonic regression. CHAPTER 2 Empirical Minimum Distance Lack-Of-Fit Testing In Regression Model 2.1 Introduction This chapter discusses an empirical minimum distance method for fitting a parametric model to the regression function “(3) :2 E(Y|X = :13), :r 6 Rd, d 2 1, assuming it exists, where Y is the one dimensional response variable and X is a d dimensional design variable. Let {m0(:z:) : a: 6 Rd, 6 E G C 189, q 2 1} be a given parametric family of regression functions and let I be a compact subset of Rd. The problem of interest is that of model checking, i.e., to test the hypothesis H0 : p(x) = m90(:r), for some 60 E 8 and for all :1: E I; H1 : H0 is not true, based on a random sample (Xi, Y,;), 1 S 2' g n, from the distribution of (X, Y). Several authors have addressed the problem of regression model checking: see, Hart (1997) and references there in. The recent paper of Koul and Ni (2004) (K—N) uses the minimum distance ideas of Wolfowitz (1953, 1954, 1957) and Beran (1977, 1978) to propose tests of lack-of-fit for the regression model with heteroscedastic errors. In a finite sample comparison of these tests with some other existing tests, they noted that a member of this class preserves the asymptotic level and has very high power against some alternatives when compared to some other existing lack-of- fit tests. The distance used in their paper is the integrated square deviation between a nonparametric estimator of the regression function and the parametric model being fitted with respect to a general integrating measure. To be specific, K—N considered the following tests of H0 where the design is random and observable, and the errors are heteroscedastic. Let (I) be a sigma finite measure on Rd, G denote the d.f of the design variable X, and Gn be the empirical d.f. based on Xi, 1 g i S n. For any density kernel K, let Kh(:r) 2: K(a:/h)/hd, h > 0, .r 6 Rd. Define, as in K-N, the) = $2: Km — x», i=1 i=1 1 n 2 d(:c) 77L 6 3: — K 27 — Xi Y2- — m X2- , < > [I In; M >( 9( >>] 9122(1)) and tin := argminfleeTnW), where K, K * are kernel density functions, possibly different, h = hn and w = 2117; are the window widths, depending on the sam- ple size n. K—N gave some sufficient conditions on the underlying entities for con- sistency and asymptotic normality of tin under HO, and asymptotic normality of Dn := nhg/2(Tn(1§n) — ISM/6132 under H0 , where Rn I: '—— dq) 2.1 "2 .221/1 gaze) ( ) I ) 372' = Yi'mén (X,), .. _ Kh($-X,)Kh($-X')E,§' 2 9n := 712/143; (/ 9h“) 3 J d(:c)) . A practical problem that arises in using these statistics is the choice of the inte- grating measure . Although one may choose (I) using some optimality criteria, such a (I) will invariably depend on the model being fitted and the design distribution. One way to simplify the choice of (I) is to use the empirical d.f. of design in the above entities. We are thus motivated to propose empirical minimum distance tests of lack-of-fit in the classical regression model. Accordingly, let 1']- : 1(Xj E I) and define K},(Xj— —X ,)(Y,-—— m9(X,- )) n 2 Mn“) 72-12:" I: 9111(le Ij , and (in := argminoeean). We also need the following entities: . K x—-—X - ,)Y,' mn(:c) := — n: n*h(92 *llx , a: 6 Rd, w M519) :2 n—l ”1(mn(x —m9(x- n21 -, gang, J 6,"; := argminjgleeMnW). 10 In this thesis we prove the consistency of 6;; and 877,. We also prove asymptotic normality of Jaw”, — 60), and nhd/ 2(1I71n(00) — Cn) under H0, where Cn is given below at (2.2). Then, similar to K-N, sequences of estimators C’n and I‘n are provided such that Cn is nhd/z- consistent for Cu and En is consistent for I‘, and under some sufficient conditions on the underlying entities, asymptotic null distribution of nhd/ 2F; 1/ 2(anfin) — C77,), is shown to be standard normal. These results are similar in nature to Theorems 4.1 and 5.1 of K—N. Here, Tl. . __ . “ ._ —3 h J z' , ll .5 I 3 ~ . 2 ,, K X——X'K X—X’e-e- n, .= win-4: X h” 2) h“ J) 2 J: , . . 9 (X1) 2%] l h where 02(3) := E[(Y — m60 (112))2IX = 1r] , x 6 Rd, and g is Lebesgue density of G. This chapter is organized as follows. Section 2.2 states the needed assumptions. In the beginning of section 2.3, we summarize some of the results of K-N and Koul and Song (2006) (K-S) for the sake of completeness. Section 2.3 contains the proofs of consistency of 6;; and 0n, while section 2.4 and 2.5 contains the proofs of asymptotic normality of fin and that of the proposed empirical minimum distance test statistic, respectively. 11 2.2 Assumptions Here we shall state the assumptions that are in K-N for reference where theorems and lemmas are proved. Throughout the thesis 00 denotes the true parameter value under H0 assumed to be in the interior of 8. About the errors, the underlying design and the 0— finite measure (P on Rd we assume the following: (e1) The random variables {(X,,Y,-) : X,- E Rd,Y, E R1; = 1,2,--- ,n} are i.i.d., EIYI < 00, and the conditional expectation 11(17) := E(Y|X = :r) satisfying f p2(:1:)d(x) < 00. (e2) E(Y — ,u(X))2 < co, and the function 0%) = E[(Y — ,u(X))2|X = I] is as. ((1)) continuous on I and 030 (.13) is continuous on Rd. (e3) EIY — n(X)|2+6 < 00, for some 6 > 0. (e4) EIY — [.l.(X)|4 < 00 and 760(zr) := E[(Y -— m90(X))4|X = x] is continuous on Rd. (g1) The d.f. G of the design variable X has a uniformly continuous Lebesgue density 9 that is bounded from below on I. (g2) The density 9 of the d.f. G is twice continuously differentiable. (p) (I) has a continuous Lebesgue density qS. About the kernel functions K and K *, we shall assume the following: (k) The kernels K, K * are positive symmetric density functions on [—-1,1]d with finite variances. In addition, K is a bounded kernel and K * satisfies a Lipschitz condition. 12 About the parametric family of functions to be fitted we need to assume the following: (m1) For each 6, m9(a:) is a.e. continuous in 51:, w.r.t. the integrating measure on Rd. (m2) The function ing (1:) is identifiable w.r.t. 6. i.e., if mgl (11:) = "1.92 (x) for almost all :r(), then 61 = 62. (m3) For some positive continuous function E on Rd with E€(X) < co and some fl>0, |m92(:r) — m61(:r)| g "02 -— olnfiea), val, 92 6 6,3: 6 I. m4 For every 3;, m .1: is differentiable in 6 in a neighborhood of 6 with the vector 6 0 of derivatives #1903), which is continuous on Rd such that for every 0 < k < co, m X- —m X- — 6—6 ’r'n X- I 6( 1,) 00( 7,) ( O) 60( z)I 20131), "9-90" sup where C := {1 g 1' S n, Vnhg|l6 - 60” _<_ k}. (m5) For every 0 < k < oo, —d/2 . . Slép hn ||m0(X,) — m60(X,)|| 2 019(1), Vn > NE. . 2 .___ . . I . . . . (m6) f ||m90|| d‘I) < co, and 20 .— fmgomgodq) 1S posmve definite. About the bandwidth hn we shall make the following assumptions: (hl) fin—103511400. 13 (h2) nhgd —> 00 as n——> oo. (h3) hn ~ 11—“, where a < min(1/2d,4/(d(d + 4))). Let 9,, and 9.2:) denote the kernel density estimators of g with bandwidth h and 112, respectively. From Mack and Silverman (1982), we obtain that under (g1), (k), (hl) and (112), :23 léhtv) - g($)| = 019(1), £1611; lgiiitv) - 9(m)| = 012(1), (2-3) 9(56) _ =0 :22 game) 1| 10’- These conclusions are often used in the proofs below. In the sequel, 5 := Y — m90(X). The integrals with respect to <1) and G measures are understood to be over the compact set I. The inequality ((1 + b)2 S 2(a2 + b2), for any real numbers a, b, is often used without mention in the proofs below. The convergence in distribution is denoted by —+ d and Np(a, B) denotes the p—dimensional normal distribution with mean vector or and covariance matrix B, p 2 1. 2.3 Consistency of 6,”; and 6,, This section proves the consistency of 6;; and 6”. To state and prove these results we need some more notation. For a a-finite measure Or on d—dimensional Borel space (Rd,Bd), let L2 ((1) denote a class of square integrable real valued functions on Rd with respect to a. Define pm. V2; e) == [Io/1m — WWW), pow/2) := /I(V1(x)-V2(x))2dG(w), l4 Pn(V1iV2) :-—- /I-u2(x)>2dcn(x) Tl. = n—12(ui(X,)—u2)21j. Viz/2613(0). i=1 and the maps T(I/, Q) := argmingee p(1/, me; Q), T(V) :2 argmingee p(u,m6), Tn(1/) := argmingee pn(u,m6), u E L2(G), n 2 1. The following lemma has its roots in Beran (1977) and is proved in Ni (2002). Lemma 2.3.1 Let m satisfy the conditions (m1), (m2), and (m3). Then the follow- ing hold. (a) VV E L2(Q), T(V; Q) always exists. (b) If T(u; Q) is unique, then T (V; Q) is continuous at 1/ in the following sense: For any sequence of {un}, V E L2(Q), p(z/n, V; Q) -—> 0 implies T(l/n; Q) —-> T(u; Q). (c) V 6 E O, T(m6) = 6, uniquely. We need an analog of this lemma for the random distance pn and the correspond- ing Tn given as follows. Lemma 2.3.2 Let m satisfy the conditions (m1), (m2), and (m3) with Q replaced by G. Then the following hold. (a) VV E L2(G), T(z/) always exists, and Tn(1/) exists Vn 2 1, mp]. (b) If T(u) is unique, then the following holds. For any sequence of {Un},V E L2(G). pn(un, V) ——>p 0, implies Tn(un) -—->p T(l/), as n ——+ 00. 15 (c) V 6 E 9, T(m6(-)) = 6, uniquely, and Tn(m6(o)) = 6 uniquely, for all n 2 1, w.p.1. Proof. The following proof is a suitable modification of the proof that appears in Ni. Proof of Part (a). The existence of T(V) follows from (a) of Lemma 2.3.1. We shall prove that the family of random functions 6 H pn(u,m0), n 2 1, is almost surely equi-continuous. Then the claim (a) pertaining to Tn follows from the compactness of 8. By the Cauchy-Schwarz inequality, for any 19, 6 E 8, 1 2 1 2 Ipn(Vi 771,9) _' pn(Vim6)I S Pn(m19im0) + 2107/ (Vim0)pn/ (mflimg) But, by (m3), pn(m,,,m,) = [Ilma($)-ma(x))2d0n(x)sll19-9ll2n_lz€2(xi)1i- i=1 Since 6 is continuous on Rd and I is compact then I? is bounded on I and hence sup pn(m,9,m6) S C||19 — 6H2, w.p.1. n21 Similarly, under (m3), p(m,9,m9) 3 one — 9H2 van 6 9. Because of the SLLN’s and because my, V E L2(G), pn(i/,m,9) —+ p(u,m,9), as. for each 19 E 8. Also in view of the above bounds, both functions 19 I—) pn(-,m,9) and 19 H p(-, my) are Lipschitz(2) uniformly in n and with probability 1. These facts together with the compactness of 9 imply sup |pn(u,m6) — p(u, m9)| —+ 0, as, as n —+ 00, (2.4) Slfip‘pn(u’ "‘19) —— pn(u,m9)| -—> 0, as, as “19 — 6|] —> 0, 16 thereby completing the proof of equi-continuity of the map 6 H pn(u,m6), and of part (a). Proof of part (b). Let {Vn}, V in L2(G) be such that pawn, u) = ope). (2.5) Let 6 = T(l/), 6n = Tn(un). For an e > 0, let Ame == {PnO/niV) S 6’ IMO/me) - p(V.m.9)| S 6}- By (2.4) and (2.5), there is an N6 such that F(Am) 2 1 — c, vn > NE. (2.6) Now, by the definition of Tn, pn(1/n,m,9n)§ pn(un,m0), Vn Z 1,w.p.1. By subtracting and adding 11 inside of the square of the integrand, expanding the quadratic and using the Cauchy—Schwarz inequality on the cross product term, 1 2 1 2 men, m9) s me, me) + man, u) + 2M (un. m/ (me). On A7116, we thus obtain man, man) 3 pa, mg) + e + 2.1/ 2(. + pox, m9>)1/2. (2.7) On the other hand, again by the definition of T, Tn, 6, and 1%, pn(u,m6) S pn(u,m,9n), for all n 2 1, as. This, together with an argument like the above, 17 implies Pn(Vnim19n) _ PRU/img) 2 Pn(Vn,mgn) — pn(V.m.9.,,) 1 2 1 2 2 pn—2pn/ (um/)4 (V.m.9,,)Vn.>_1, w.p.1. But, pn(1/,m,9n) S 6pn(un,u)+4pn(u,m6). Hence, on A7115, Pn(Vn,m,9n) 2 pn(u,m9) + [mo/"1”) “ 2671/2971, V){6Pn(Vni V) + 41071.01, m6)}1/2 Z IMP/imp) — e — 261/2(26 + p(1/,m9))1/2 > p(u,m6) — 26 — 261/2(2€ + p(V,m9))1/2. Thus, in view of (2.6), (2.7), and the arbitrariness of e, we obtain pn(Vn.m.9,,) = puma + ope). From these facts it follows that 197,, —+p 6. For, suppose 19n 4+ 6, in probability. Then, by the compactness of 6, there are subsequences 197% of {1977,} such that 197;, k -+ 19 7Q 6, and by (2.8), pnk(1/nk,m,9nk) —-+ p(u, my), in probability. Hence, p(u, m,9) p(1/,m9), implying, in view of the uniqueness of T(I/), a contradiction, unless 19 = 6. Proof of part (c). The claim here follows from the identifiability condition (m2) withQ=G. 18 As in K-N, for any average L. = n"1 E? _X1(7( j)/g,"2,(XJ-)), the replacement of gw by g is reflected by the notation L := n‘12?=1(7(Xj)/9(Xj)). We also need to define for x E Rd, 6 E R9, maze) == - :2: Kh( z — "1909) (2.9) ,[in(x, 6) := — n: Kh(x — X,-)m6l(X,), (171(1), 6) I: Kh(£lI — X2)” — [171(2), 6) £1101: — X’l)[Y'i — m0(X'i)l7 (171(3) 3: Un(.’L‘,00) i}; 5. M0?) 2: Emma): EXha-Xmgou). Zn(x,6) :2 un(x, 6) — an(x, 60) = -ZKh<.x—X ilma< Xa— m90(X)l Un($)llh($) 1n Un(X] ____)_Xj]IJ Sn 2: -—-———— -:_ , / 92(56) (mm) %n;[g,,2( (Xj) n 2 Cn2(6) := ij: I[fl—W—ngg) (jX) —m9(Xj)] Ij, 21 2 .2 [0,0 0m90(x)’ «We 9 (x) dx. Many of these entities are the empirical analogues of the entities defined at (3.1) in K-N. Now, we will summarize the results of K-N and K—S for the sake of completeness. The next two results state the consistency of 19;; and 6n. Result 2-3-1. Suppose H0, (e1), (e2), (k), (g1), (h1), (h2), and (m1)-(m3) hold. 19 Then, 19,”; —+ 60, in probability. Result 2-3—2. Suppose H0, (e1), (e2), (k), (g1), (hl), (h2), and (m1)-(m3) hold. Then, in ——» 60, in probability. The following result gives asymptotic normality of 197,. Result 2-3-3. Suppose H0, (e1), (e2), (e3), (g1), (g2), (k), (p) , (h3), and (m1)—(m5) hold. Then, Consequently, 711/2097, - 60) —->d Nq(0, 26123261), where 8;; and 2 are as in (2.9) and 230 is as in (m6). The following result states asymptotic normality of the minimized distance min). Result 2-3—4. Suppose H0, (e1), (e2), (e4), (k), (p), (g1), (g2), (h3), and (m1)- (m5) hold. Then, nhd/2(Tn(19n)—Rn) T’d N1(0, 1‘). Moreover, IQnFTl —1| = 0p(1), where fin, Rn, and F are as in (2.1) and (2.2). The following result from K-S gives consistency of 19:, and 197,, for T(m), where m is a given regression function, different from the model being fitted. Result 2-3-5. Suppose (k), (g1), (m3) hold, and m is a given regression function such that m E {mm 6 E O}, m E L2(Q), and T(m) is unique. (a) In addition, suppose m is a.e.(Q) continuous. Then 19:, = T(m) + 013(1). (b) In addition, suppose m is continuous on I. Then 6n = T(m) + 019(1). Next, we shall prove consistency of empirical minimum distance estimates of the underlying parameter vectors under H0. 20 Corollary 2.3.] Assume H0, (e1), (e2), (91), (k), (m1)-(m3), (hl), and (h2) hold, with Q replaced by G. Then 6:, —+ 60, in probability. Proof. Note that Mfi(60) 2: pn(mn, m00)’ 6;“, = Tn(mn), and by the identifiability condition m2 , T m = 6 . It thus suffices to prove 60 0 Pn(mn,m90) = 0p(1)- (2-10) To prove this, substitute m60 (X,) + e,- for Y,- inside the ith summand of M7”: (60) and expand the quadratic summand to obtain that mean, mgo) is bounded above by the sum QICnl + 0112 (60)], where Cnlv 07,2 are as in (2.9). It thus suffices to show that both of these terms are op(1). 2 Since 6, is conditionally centered, given X,, and by continuity of g and 060’ assured by (e2), (g1), (k) and (h2), we obtain ”—1 n U’n(Xj) . 2 E( 'Jélgwfllj (2.11) _ " K(X X,)e 2 3:12:14 h M) l j 1 2 =n‘3h—2dK2(0)ZE 9,), 1,2] +n‘3ZE [7‘09 (X176 02 1,] Xi K2(0) 000031),” 090 (x— uh)-g(x uh)K2 (u) :(nhd)2/I 9(37) +nhd/fo 9(55) dxdu =O(1/nhd). Hence {1}”: M12 —- o ((nhd)_1) (212) i=1 909') J — p ’ ' and, by (2.3), Cnl = 039(1). Next, we shall show Chemo) = 0p(1)- (2.13) By taking the summations for i = j and i 7e j, and by using the inequality (a + b)2 g 2(a2 + b2), for any real numbers a, b, Cn2(9) S 210n21(0)+5‘n22(9)l. 668. (2-14) where . _ “ “(Xh(0)—Xt(0))mg(x~) 2 __ ,, 3 J . . 0.21(0)— )2 gwj, I, . (215) . _ n l (X(X-—X.-)m(X.-)—Xt(X-—-X.-)m(X-)) 2 Cn22(0) : n 3: 2: h J 6 g*(X~) J 6 J 2.9 i=1»i¢j w J By the compactness of 8, every open cover of 9 has a finite subcover {6); 1 g j S k}, say. For any 6 > 0 such that ||6 —— 6j|| g 6, and by (m3), supoeeE(én21(9)) 2K2(0) (h—2d _ w—2d) S ———:—,——— sup sup 11 193k “9"9jIIS5 (172.9(1)-—mg,.(a:))2 (ninja)? X +—-————— dx H he) at») l 2CK2(0) —2d___ —2d 25 k g n, (h w )(5 +152 [I 9,5,) dx. Thus by (g1), (k), and by continuity of m9,V6 E O, we obtain supeeeE(C‘n21(6)) = 0(nhd)’1. (216) Hence Cn21(60) :2 op(1) follows from (h2), and (2.3).. 22 To deal with C7722, let, for j 74 i 6},(.’L‘, 0) = E[I(h(Xj -- X,)m9(X,)lXj = :13], efu(n,6) = E[K,";,(X,- — X,)m9(XJ-)|Xj = .13]. By adding and subtracting eh(X]-,6) and e;,(XJ-,6) in the quadratic term of the summand of Cn22, one obtains én22(9) S 3Cn221(6)+30n222(6)+3Cn223(9)i 668, (217) where n ' .- 'X(X-—-X.-)m(X.-)-e (X-,6)) 2 07.2216) = n22 :22 h 2 .(X2, " 2 1)] (2.18) j=1 _ 911) J _ " (2.2-[Xi1X-—X.-)mn(X-)—et(X-.e)) 2 an (6) ___ n 3 J .7 * .7 .7 I] , 222 3;] 910(le J __ (n—l)2 " €h(Xj 9)-661(Xja9) _2 Cn223(6) _ n3 32::1I 961(le I] By the fact that the variance is bounded above by the second moment, one obtains \7’6EO, ~ 1 ECn221(9) S 7 X E n . . 15‘] Again proceeding as for (2.16), for any 6 > 0 such that “6 — 6le _<_ 5, we obtain 9(X ') J [Kh(Xj - Xi)m9(Xi)1.] 2 J 68111,)3 E(Cn221(0)) (219) e 2 Kh(Xj ~ X,)(m6(X,-) - may-(X0) 2 S sup sup —3 2: E' (X) I]- 133516 IIQ-lelSJ 7‘ 2,4]- 9 J _ Kh(Xj-Xi)msj(Xi) . 2 -+- sup 32E I] lgjgkn #3. 909') 23 _<_ sup sup ;E—Zne 0II22E 1e ,2 l M > 1,] 909') J 2622 K2(u)e2m§ (y - uh>g(y - uh) sup // dydu +nhd1 90, in probability. Proof. Arguing as in K-N, we shall again use part (b) of Lemma 2.3.2 with V(:r) E m90(;r), z/n(;r) E mén(:c). Then by (m2), (in = Tn(un), 60 = T(V), uniquely. It thus suffices to show that supganm — MW): = 0pm. (2.25) For, (2.25) implies that Mmén) = Mn(én) + 010(1), M3092“): Mn(9n )+ 010(1) M,*,(én) — M;;(6;‘,) = Mn(én) — Mn(6;';) + 033(1). (2.26) By the definitions of én and 6;, for every n, the left hand side of (2.26) is non negative, while the first term on the right hand side is non-positive. Hence, Mflén) Mn(9n )= 012(1) This together with the fact that Mflafi) .<_. M;(00) and (2.10) then proves the required result. 26 Arguing as in the proof of Theorem 3.1 of K-N, it thus suffices to Show that sngnZUJ) = 019(1), sgpllln(6)=0p(1). (2.27) Using the same argument as in (2.14) - (2.23), one obtains, (77,209) = 017(1), for each 0 E G. This and (2.3) in turn imply that V6 6 e. (2.28) C 2(9) < sup 92(27) 5 n2(9)=0 (1) " ‘ 3619* 2m" p ’ By the Cauchy-Schwarz inequality, for any 61, 02 E 9 ICn2(92) — 03,2be _<_ 2091 + E2) + 403320122 + E211/2, where, by (m3), " K1109 — Xi)(m62(Xi) — m61(Xi))I n 2 ._ -3 . E1 '_ n 2 [2 9221(Xj) 3] i=1 S ”92"61ll2’68uP161'29( H )"[ 3 j: 1(2 h(J 9X( le) ( )Ij) l 9322 (9:) E2 :— 22—1:[lm62(Xj)—m61(xj)lzj2] I-12 j=1 j: —1 Hence (2.24) applied with O E (f, (2.3) together with the compactness of G and (2.28) completes the proof of the first part of (2.27). To prove the second part of (2.27), note that by adding and subtracting m00(X,-) to the ithsummand in Mn(6), we obtain 2 2 < gl —::—2 ——-—2J 27 But, by a similar argument as for (2.19), and by (2.24) applied with a E Z , Zn 391- 2 sup9[n '1j21[ (XJ J)J]] (2.29) 9( X3) 2 " Kh(Xj - Xi)€(Xi)Ij n 5 ll9 — 60n22n‘3 Z [Z . j=1 i=1 9(X]) This together with (2.12) complete the proof of the second part of (2.27), and hence that of Theorem 2.3.1. CI 2.4 Asymptotic distribution of én In this section we shall prove the asymptotic normality of n1/2((§n — 60). Let [23(2) := Etna, 60) = Eth: — X)m30(X), (2.30) .. .:_1jzl Un(Xj)(/I;Y]:()Xj)1j . We shall prove the following Theorem 2.4.1 Assume that (61), (e2), (e3), (91), (92), (k), (m1)-(m5), and (h3) hold, and (I) is replaced by G. Then under H0, n1/2(9n _ (90) = 251,3”an + 033(1), (2.31) Consequently, n1/2(én — 00) _’d Nq(0, 2612261), where 20 and E are as in (m6) and (2.9), respectively. Proof. The proof consists of several steps. The first is to show that nhdué‘n—90u2 = 03(1). (2.32) 28 Let ._ ”—1 n Zn(X"9)Ij 2 We claim nthn(én) = 033(1). (2.33) To see this observe that ” n U (X-)I- 2‘ d d —1 n J J - j=1[ 9212(XJ) ] J n '1 dl -—1 U"(XJ)IJ 2 S nh n 2[ J _ 3:1 9(le ’2 U X- I; 2 2 +nhd[n_1 Z [M] J sup 9 2(13) -1| j=1 9(XJ) x61 92):) (1') by (2.12) and (2.3). But, by definition, M33023) 3 M72190), implying that nhdMn(én,) = Op(1). These facts together with the inequality Dn(6) _<_ 2[Mn(60) + Mn(6n)] proves (2.33). To complete the proof of (2.32), arguing as in K-N, it suffices to show for any 0 < a < 1, there exists an Na such that D - P(—.—nfifl—2 _>_ a+ inf bTZOb) > 1 — a, Vn > Na, (2-34) "on — 90H Hbll=1 where 20 as in (m6). To prove (2.34), let an := (in — 00, (2.35) I . . dni 3: "297109) — "2900(2) — unm00(Xz'), 1 _<_. z _<_ n, 29 1 n - l n d ' 2 on, = n- Z - 219,09- -X.-> A 2,- /g(Xj) , . n . Nun” 3:1 - 2221 n 'u ’p X-,0 1-2 0722 I: "-1: n n( ]X(.)) J A .. 1 2 1 2 Then, we have Dn(6n)/||6n — 00||2 _>_ Dnl + Dn2 — 2Dn/l D122 . By assumption (m4), consistency of én, and by using (2.24) with a = 1, one verifies that Dnl = 019(1). For the term D712, note that ||b||=1 where n b" X-,6 I- 2 271(0) := 72-1: M“ J 0) J , beRQ_ j=1 91(le Now, we will prove that for each I) 6 Kg, ”1)“ = 1, 277(1)) ——> b'EOb, in probability. For this it suffices to show that E[En(b) — #201212 = 0(1), Vb e 13‘]. (2.37) Rewrite I . b Kh(Xj - Xi)m90(Xi)Ij>2 1 n n 272(1)) = -—;,ZZ( M.) n . J=1 2:1 .7 I . - I +_1_ i Z (J Kh(Xj - Xi)m90(Xi)Kh(Xj — Xk)m90(xk) bIj 3 2 . where b’KMXJ- — Xamaom) 2 2122(1)) : n13ZZ[ 9(Xj) Ij] I z jyéz K 0 b’Kh,(X- 43mg (xamg (X->’b 27‘3“» = n3);clz:§[ J 9208].) O J 11') z z ] 1 (”(1109 - Xilm60(xi)Kh(Xj — Xklm90(Xk)' b 21240)) : "7:; . 92(X') Ij - 276],k .7 The left hand side of (2.37) is less than or equal to 6[E2%1(b ) + 1.3222(1)) + 13233, 3b( )1 )+ 2E[2,,4(b) — b’ 20312 Thus to prove (2.37), it is enough to show for each b 6 Kg, 1323,13) = 0(1), 3232(3) = 0(1), (2.33) 0(1), E[Zn4(b) — b’EOb]2 = 0(1). 323,3(3) Now, we shall prove the first part of (2.38). By the Cauchy-Schwarz inequality, 4 llm mu? 232(1)) )_K(O)]le92( 60 1]]. n1 5h4d Xj) Therefore, by (g1), (k) and (h2), one obtains 4 IIm (on2 g 5:492 / _29__d n4h4d I 9(3) = 0(nhd)—1 = 0(1). (17 Esup 272110)) b Similarly, K 0 b’Kh(Xj- lemo (Xi)ThI9(X j)’b 2 E53233“) < 71543532? 9203]) O 11' z j 2 K2“) Kiflr-y)llm90(x)ll2llm90(y)ll2g(y) < —— dxd _ n2h4d//I 93(3) 31 Km) K4(u)llm90( z>II2IImgO(z-hu>u29 (cc—1m) S ”Tth// g3(:z:) dxdu = 0(n2h3dr1 20(1). 31 by (g1), (k), (h2), and continuity of 75290. This proves the third claim in (2.38). Now we shall prove the fourth claim in (2.38). Since, E2724“) _ E[b'KMX2 - X1)m90(X1)Kh(X2 - X3)m90(X3)’12 b 92(X2) :_— bI/1_//K(u)K(v)T'n90(x — hu)7i290(x — vh)’ x g($ _ hu)g(x — Uh) du d1) d2: b 9(23) —+ b’ZOb. Thus, to prove E[Zn4(b) —b’20b]2 = 0(1), it is enough to show E23403 = (b’EO (2)2. Now, 2 23n4(b) : Z: Z zijkzlmn i¢j¢kl¢m¢n S 0 Z Zijk(zlmi+zlmj+zljm)+ Z zijkzlmn i¢j¢k¢m751 i7éj7ék7ém¢l7én = 272,410?) + Z3n42(b) + 371430)) + 212.44%), say, where . - I _ ._ b’ _3 Kh(Xj—Xi)m90(Xz')Kh(Xj-Xk)m90(Xk) I- b Ziflt" ” 2X. J 9( J) By independence of Xi’s, (k), (m1), (g1), and (h2), one obtain for each b 6 KW, Ilbll = 1, E27241“) g 5:73 [f [ [1' K(u)K(v)b’r'n90(x—uh)1'n90(a:—vh)’b X (gee — uh);((:)— vim”? dz] 2,, d, = 0(nh2d)_1 = 0(1). 32 Similarly, one can obtain that E2n42(b) = 0(1), for each (2 E W such that “b” = 1. Again by independence of Xi’s, (k), (m1), and (g1), one obtain for each b E Rq, Ilbll =1, E3n43(b) sn—f [ f /( K(u K(v)llmgo ( —uh)llllv‘n90(x — vh)” we: — hu)g(x — hv)du (11))29-3 (3)33 = 0(n—1)= 0(1). Also by independence of Xi’s, E2n44(b) = (EZn4(b))2 ——> (b’ZOb)2, for each b 6 EU, “b“ = 1. This also completes the proof of (2.37). Also note that for any A > O, and any two unit vectors ()1, b2 6 Rq, ”bl“ = 1 = H52”, “()2 - b1“ S A, we have l2n1(b2)— Z37"L1(b1)l ,1 ” Kh(Xj -Xz')"n00(Xz') )2 j n—1 b—b) — ,.>::,[<2 .,2 IX.) i=1 _1 n 1 " Kh(Xj_Xi)m60(Xi) n J;[(b2—b1)lgz 909) 1']- X[_l_>_’1_ " Kh(:j‘X:)7h00(Xi)I] J n i=1 9(X ') +2 n Kh(Xj ’Xi)m60(X i) 2 909') Ij S (/\+2n But similar to the argument as for Zn, 1 n _1 n . 2 E[gzjun :Kh,mg,/g>zj1, j: —1 .= "_1 " Un(Xj)[/~."n(Xj>90)_flh(Xj)lI.] 9n]. ' j;1[ 92(Xj) J 1 W == n*lz[ Un(Xj>Ipn(Xj.eo>~12h—g"2(Xj))I:rj]. j=1 34 Un( (Xj)[l1n(Xj an) " fln(Xj» 90)] —1 . {:1 92(Xj) 1’ ’ 9724 := n‘lZIUn(X,->tzn(X,-,én)-pn(Xj.6o)I(g.;2(X,->~g“2(X,-)>I)I. 971.3 Note that these gnj’s are the empirical analogs of the similar entities in K—N. Anal- ogous to the proof there, we need the following lemmas. Lemma 2.4.1 Suppose (61), (e2), (91), (k), (m1)-(m5), (h1), (h?) and Hg with (I) replaced by G hold. (i) If, additionally, (e3) hold, then \fliSAn —’d Nq(0, 2), where E is as in (2.9). (ii) If, additionally, (g2) and (h3) holds, then VA 6 R9 x/FLIIX’Snlll = 0pm). (2.40) Lemma 2.4.2 Suppose (e1), (e2), (g1), (k), (m1), (m2), (m4), (m5), (M), and (h?) with (D replaced by G hold. Then, under ’HO, VA 6 Rq n1/2llAignkH = 0pm), k = 1,234. (2.41) The proof of (2.40) is facilitated by the following lemma, which along with its proof appears as Theorem 2.2 part (2) in Bosq (1998). Lemma 2.4.3 Let 9:0(13), :1: 6 Rd, d 2 1, be the kernel estimator associated with a kernel K * which satisfies a Lipschitz condition. If (92) holds and wn = an(log n/n)1/(d+4), where an ——> a0 > 0, then for any positive integer 1:, (log;. n)-1(../ logo” 52p l9w*(:v) — g(sv)| —> o 35 Proof of Lemma 2.4.1. Let Sn denote 8n of (2.9) with (I) 2: Gn. To prove the first part of Lemma 2.4.1, by Slutsky’s Theorem, it suffices to show that [”571 ”*d Nq(0a 2); fillén - 5n“ = Op“)- (242) First part of (2.42) follows from Lemma 4.1 of K-N. To prove the second part of (2.42), it suffices to show that, VA 6 1R", lam/Von — sun? = 0(1). (2.43) —3/2 Let 1),,(23) = A'flh(:r) and aij := n 8,; Cija where dG'(:r) . 6.. 2 Kh(XJ' " Xilfhwjll. _/ Khlx — Xithl-T) 2] 122(le J I 92”) Now, the left hand side of equation (2.43) can be rewritten as the following sum: 6: E(a?Z-) + Z E(a12j) + 4 Z E(“ij“ii) + Z E(aijaiml- (2.44) .- #i #2‘ man" To prove (2.43), it suffices to show that each of the four terms of (2.44) are 0(1). By continuity of 77290, (k), and (g1), VA E Rq and V2: 6 I, one obtains supIIz>h(z)II s EKh(x—X)IIX’m9 (X>II (2.45) n O = /K(u)||A’7n60 (a: — hu)||g(x — hu)du = 0(1). Now, we shall show that 23-75,- E(a22j) = 0(1). By (2.45), (k), (g1), and continuity of 030, forj 7é i, and VA 6 IR", 36 K2 x— {/2 :1; = ”—2 [0303) [I ’2 9(3)) ’2 29(y)d:vdy 2 _<_ Cn —3h—d/1260 02 (x— hu )/K (2:26:34, u>dudx = 0(n_3h-d) Hence, by (h2), 2.7-5.5, E(a2 j_) — 0(nh d)_1=o,(1) VA 6 KW. Similarly, we can show that Z, E(a2 i) = 0(1), VA 6 HM. Note that, VA 6 KW, Vi 75 j , E[cz-lez- = y] = 0. Thus, by the independence of Xi’s, 23-79,- E(a,;ja,-,-|X,-)= 0- — Zm¢i3£j Ea(,- jaileiv) for all n 2 1. This completes the proof of the second part of (2.42), and hence that of the first part of Lemma 2.4.1. I To prove (2.40), by the Cauchy-Schwarz inequality, (2.45) , (2.12), by Lemma 2.4.3, and by (h3), we obtain VA 6 Kg an'5n112 21 n Un(Xj)l'/h(Xj) Jnr 92(1):) 2 < n — I- su ——1 ‘ _ngl 9209-) 3 $612 922,2(15) i_1 n Un(Xj)Ij 2 _1 WWW 2 92(2) 2 22 J; 909') Z 909') Ij i229 *2(a:)-1 .. ,_1 W W- ,_1 2... 92cc) 2 < C Elly 909) 53'— XXX .3} 93(22) 1 d 1 2 4 = n0p((nh )— )0p(1)0p((logkn) (hemmm) 2 .4— Gal—32- : 0p((logkn) (logn) +4n +4) =op(1). This completes the proof of Lemma 2.4.1. [:1 37 Proof of Lemma 2.4.2. Let I'ln 2: A’fin, 19h := A’ph, and 1) := A’rn. By the Cauchy-Schwarz inequality, VA 6 Rq UnX-I- 2 n1/2IIX’9n1ll s (n‘1/22[——g-((—,{—))i] ) (2.46) n - 2 ”_1/2 Ill'/n(X 60)—uh(X,-)II , x< ,‘El 909) z, . By (2.11) and (h2), En—l/Z Z The second factor is bounded by 2bn1 + 2bn2, where IX 0)); X K X yf/ d0 2 ||h() 60f -) f h( —y)00(y)ll (3.0%], __ n5——/2 Z 1[ 909 ) I[Un__;_Xi) ‘22)]: -——-—J2 = 0(n_1/2h—d) = 0(1). bn2 = "221/2 2% j=1 IIKh(Xj — X.)1>90(X.-) -— f Xh(X,- - 2290(2))” dam/)1 2 "37:3 g(X,—) 1' Now, by using (g1), (k), (h2), and continuity of mo, we obtain that the expected value of bnl is bounded above by _2___K2(0) / I___Iveo(x)II2d n3/2h2d 9(3) 2 (f K(u)llz>oo(x-hu)llg(x — hu)du)2 J'W/z go) 22 = “(2' To handle bn2: first note that conditional on X j, the inner term of bn2 is (n— 1) / n times the average of centered i.i.d. r.v.’s. Using the fact that the variance is bounded above by the second moment, we obtain that the expected value of bn2 is bounded 38 above by K X —X z; X 2 "-1/2E[1 h( 2 g()1()2|;(90( 1)” 2] X2 (u) Mow — mum — uh) _ =n1—_/12hd// g($) du d1: — 0(1). This completes the proof of (2.41) for k = 1. This together with (2.1) implies (2.41) for k = 2. To prove (2.41) for k = 3, similarly by the Cauchy-Schwarz inequality, VA 6 Rq 1/29 n3”2 Un(X ” IIVn(X -,én>—z>nu 2 ( 1Zl__-)_Ijl2"_)( 2i 909-) J Ijl) lln But the second summation is bounded above by )2 _321th(X—X,-j)z 2 saw > mm film: .( J) = 0p(hd) X 012(1), by (2.32) and the assumption (m5), and by (2.24) applied with a E 1. This together with (2.12) proves (2.41) for k = 3. The proof of (2.41) for k = 4 uses (2.41) for k = 3 and (2.3), thereby, completing the proof of Lemma 2.4.2. Next, we shall show that the right hand side of (2.39) equals Qn(én — 60), where Qn = 20 + 019(1). Recall the notation at (2.9) and (2.35). Let n l- A _ . . 9h(X') d - Vn 1= n 1: Vn(Xj,9n)";§‘J—Ij'”7nl"'], nlD(ngX9)l°/’(jé) ._ _1 ”X *0217, 1 n . Ln ._ n El 102)“ j) I], VAEIR‘]. i=1 39 So, the right hand side of (2.39) can be written as the sum [Vnugz + Ln]un. But, Id "V75“ S max1ur snvnun+uvn12|t *2 J j=1 9w (Xj) where, lan11|| == maxlgin(XJ-,00)]’I. n1 '— TL 2 *2(X) J ’ j=l _ 9w J L ._ _1" 'vnvn’1. j=1 _ 9w J 40 But, by (2.3), (m5), by the Cauchy-Schwarz inequality and (2.24) , VA 6 Rq, “Lnl II 2 019(1), while _ n-1Xj:60)9h(Xja00)’Ij "2 ,2?!“ 912%,) {171,0 -—l'/ 330 2 ”.42: ||( (X3 0) :12)“ 2;ng 0))” Ij] n ||(Vn(XJ :ol-Vh( Xj,90))||||Vh(Xj,90)|| . nlj; [ 9132(Xj) 1]] But, by using same argument as in the second factor of the right hand side of (2.46), and by (2.3), this upper bound is op(1). Moreover, similar to the argument used for Zn in (2.37) and using (2.3), one obtains VA 6 KW, lth(X j902) Vh(X 90)'|| —1 ; 9w 2(Xj) This proves Qn = 20 + op( 1), thereby also completing the proof of Theorem 2.4.1. 2.5 Asymptotic normality of Mn(én) This section contains a proof of the asymptotic normality of the minimized distance Mn(én). The replacement of 91"}, by g in Mn and Tn is reflected by notation Mn and Tn. The main result proved in this section is the following Theorem 2.5.1 Suppose (CI), (62), (e4), (91), (92), (k), {m1)-(m5) and (h3) with<1> replaced by G hold. Then under H0, nhd/2(Mn(dn) - C'n) —+d N1(0, F). Moreover, If‘nP—l — 1| = 013(1). 41 Consequently, the test that rejects H0 whenever nhd/Zf‘;1/2|Mn(én) —- C'nl > Za/2 is of the asymptotic size a, where 20 is the 100(1 — a)% percentile of the standard normal distribution. Analogous to the proof in K—N, the proof of this theorem is facilitated by the following five lemmas. Lemma 2.5.1 Suppose (e1), (e2), (e4), (91), (k), (hl), and (h2) with <1) replaced by G hold. Then, under ’HO, nhd/2(A7In(60) — a.) 2d N1(0, r). Lemma 2.5.2 Suppose (61), (e2), (91), (k), (m3)-(m5), (M), and (h2) with (I) re— placed by G hold. Then under H0, nhd/2an(én) — ano): = 0100). Lemma 2.5.3 Suppose (e1), (e2), (91), (92), (k), (m3)-(m5), and (h?) hold with (I) replaced by G. Then under H0, nhd/2|Mn(90) -— anon = 0pm). Lemma 2.5.4 Under the same conditions as in Lemma 2. 5. 3, nhd/ZIC‘n —- on) = 0,,(1). Lemma 2.5.5 Under the same conditions as in Lemma 2.5.2, f‘n — F = 019(1), Consequently, the positive definiteness of F implies [fur-1 — 1| = 0p(1). The proof of Lemma 2.5.1 is facilitated by Theorem 1 of Hall (1984) which is reproduced here for the sake of completeness. 42 Theorem 2.5.2 Let fit-,1 S i S n, be i.i.d. random vectors, and let Un := Z Hn()?2in)i GTl($ay) I: EHn(X1,$)Hn(X1a3/), l_<_i(:c) Tn 6 := — K a: — Xi Y2- — m Xi —, ( ) (1b,; h< >( 9( ))] 92a) - ._ _1_ " Kg(x—X,)é,2 Rn ._ ”2221/1 9201:) d(:r). To prove Lemma 2.5.1, by Slutsky’s Theorem, it suffices to show that nhd/ 2(21(60) — fin) —+d N1(0, r), (2.47) nltd/2|1V4n(6’o)- rm): = ops), mid/2mm — on) = 0,,(1). The first claim in (2.47) is proved in Lemma 5.1 of K-N. For the second claim, it suffices to show that E [nhd/ 2|Mn(00) — Tn (60)”2 = 0(1). Let dG(a:) , f.. ._ K1109 - XilKh(Xj - X191 _/ Kh(a: — X,)K,,(x — Xk) iJk '— 9209.) .7 I 92(x) 43 and eijk = flgflfijh Hence, [nhd/2(Mn(90) Tn(90))l = 2:26 eijk- i Expanding the quadratic and using the fact E [ f,- j kIX'l’ X k] = O, \7’ j at i, k, we obtain E[nhd/2(Mn — Tn(90))]2 (2.48) 2 2 -<- 04282 eiii + Zleiiie ejjj +ez‘jie J'iJ +ei 2J2 ie+eijj iii “we Jii +6 m i J7“ 2 +eljjejji + eiijejji + eiij + emjejn- + eijiejjj + ej’ljeiiil] . To prove the required claim, it suffices to show that all terms on the right hand side of inequality (2.48) are 0(1). By (e4), (g1), (k), and (h2), one obtains E [ Z 62221] i _ hd K%(0)11 Kg (:1: — X1) 2 _ 3E 790(x1)[ 920(1) —/1 9(3) dx] ] 2K4(O) T__60 (13)“ K4(u)g( (a: — hu)r90 (:c — hu) 3 (nhd)3 /;r— g3(x)d +n3h2d/z / 92(1') dim = 0(nhd)-1= 0(1). Similarly, using the independence of Xi’s, by (e4), (g1), (k), and (h2), one shows El2j#ileiiiejjjl = 0(1). Next, hd 2 2 E[éezjiefij] = ;E[090(X1)090(X2)f121f212l- J 2 44 By the independence of Xi’s, (g1), and (k) and continuity of 050, Eiago(X1)0§0(X2)f121f212] 4(u)o2 (:c)o2 (x— hu) = h3d1/I [K g:()g(wf0hU) dadx “2353 f [/1 K2(u)K2(v)060(x — hu)ogo(x —— hu — hv) g(a: — hu —- vh) g($)9($ - hu) 1: — mag (a: — hu) K2 (u)g( 2 O _ d —3 +h_2_d[.//I g(:c) dxdu -O(h ) . Hence, El2j¢i eijiejij] = 0(nh2d)“1 = 0(1). Similarly, one can show that other dxdudv terms of (2.48) are 0(1), thereby completing the proof of the second claim in (2.47). To prove the third claim in (2.47), it suffices to Show that Let 2 ._ . 2 __ . (in: Kh(XJ X1)I._/ Mdga) .3 9209-) 3 2: 92(2) ’ hd/22 ~ ~ and bij =—2—Ld,-j Then, [rind/2w” — 07.)] = ZiZJ-bZ-j. Expanding the quadratic and using the fact that Eldileil = 0, V j # i, we obtain Elnhd/2(Rn — Grill2 < C1512?“ + Z [bzibmm + bimalmm + bim + bmill] i mséi To prove the required claim, it suffices to show that all the terms on the right hand side of above inequality are 0( 1). 45 Thus by (e4), (g1), (k), and (h2) and continuity of r60, one obtains 4 r (x) E[Zifil 5 ill??? [I get-fwd K2(u)g1/2(x — hu)r1/2((x — hu) 2 2 90 +n3h2d / [/1 9(3) d2: d“ = 0(nhd)’1 = 0(1). For the fourth term, since hd El: bimbmil - 7Elago (X1)090 (X2)d12d21] maéi 2 thus, by independence of Xi’s, (g1), and (k) and continuity of 000, one obtains E[0§0(X1)0§0 (X2ld12d21l K4(u)000 (at)000(1:—hu) —2;§8—1///1_K2(u)1{2(v)000(x — uh)ogO (2c — vh — uh) g(:r — vh — hu) game—hm a: — hu)060 (x — uh) 2 +h_2—d1[//1'K2(u)g( 9(5'3) dxdu =O(hd)"3 Hence, E l2m¢i bimbmil = 0(nh2al)"1 = 0(1). By similar arguments, one can show dxdudv that other terms are also 0(1). Hence we are done with the third part of (2.47) and also with the proof of Lemma 2.5.1. [:1 Proof of Lemma 2.5.2. Recall the definitions of Un and Zn from (2.9). Add and subtract mgo (Xi) to the ith summand inside the squared integrand of Mn(én), 46 to obtain that Mn(90) — Mn(én) =n_1 :1[Un(X]*2MZn( 9nllj J] 9w 2(Xj) _ . X19“ )1". 1 2 [22? W212?) fl = 2Q1] - —Q12, say. It thus suffices to show that nhd/2Q1 = op(1), nhd/2Q2 = 013(1) (2.49) Add and subtract (én — 60)’7'n00(X,-) to the ith summand of Zn(Xj,én,), we can rewrite X 621 = WIZFW fig“ jd") mfg] 9w 2( XjU)n( _1X.7')HTL(X]60) +(0n _ 60), n JZU 9* 10% Xj) Ij = Q11+Q12, say, where dm- are as in (2.35). By using (2.3) and (2.24) with a E 1, one obtains Xj) 2 lzlifiX 1].] = 0,,(1) (2.50) By the Cauchy-Schwarz inequality, (2.3), (2.12) and (2.50), one obtain that (nhd/2 lQlll) is bounded above by ldni l W Zuén — 00||(nhd)1/2~Op((nhd)—1/2) -maxim. n _ 47 But, by (m4) and (2.32), this entire bound in turn is 019(1). Hence, to prove the first part of (2.49), it remains to prove that nhd/ZIQmI = 013(1). But Q12 can be rewritten as n U77(X )#n(X] 6in) (an _6),n—11(g[ J I] _(én _ 60) ”_1 Z": [UM XXj)lfln(J ' ,29n) — Iln(Xj,90)le] j=1 9u12X(j) = Q121 - Q122, SEW- But, by the Cauchy-Schwarz inequality, (2.3), (2.12), (2.50), one obtains (nhd/2 “0122“)? is bounded above by thduén - 60u20p(nhd>—1maxinménm.) — Th60(X7;)ll- By assumptions (m5) and (h2), and consistency of an for 90, this entire bound is 0p(1). Next, note that the average in (2121 is the same as the expression in the left hand side of (2.39). Thus it is equal to . " zn(x.,én) pn(x (in) (an — 0 )ln-1 [ J I] =(9n —00)n ), n_1 j: 1:7:UO([J jaJo‘n) MAX] j60)l_j:l (2.51) 920 209') _ Zn(XJ 9n )l#n(X 671—) MM 90)] 6n _ 60)”, I: ll 9w2(Xj) Ij] =D1 + D2, say, 48 But, by the Cauchy-Schwarz inequality, (2.3), (2.29) and (2.24) with a E moo, VA 6 1M, Emir” 2"01”) _<- nhd/2l|(én——6 0)”le _1 Z 1|:Z'n(XX jion)l, 132% If [Ill/7“)? Xgn)lle:|2 9112(le j=l (XJ') z nhd/2“(én — 60)II2O(1).= 0(1), by Theorem 2.4.1 and the assumption (m5) and (h2). Hence, nhd/ZIIDIH = op(1). Similarly, one obtains (nhd/2IID2II) is bounded above by nhd/2ll(én — 60)||2op(1) = 019(1). This completes the proof of the first part of (2.49). The proof of the second part of (2.49) is similar. [:1 Proof of Lemma 2.5.3. By (2.11) and Lemma 2.4.3. nhd/gerzWo) — Numb): U2(*“2(1>’2 x 0p((nhd)_l) = 0p(. Hence, |nhd/2An21| 3 mild/2 sup lAw(a:)|n _3 2: 2K 161 2=1j— 1 = Mid/20p(logkn(logn/n)2/d+4) ng((nhd)(_1) Xiz.)e x-> 19] = Op(h_d/glogkn(logn/n)2/d+4) = 0p(1), by Lemma 2.4.3 and (2.3). Similarly, one obtains that Inhd/QAnZZI n n K2(X- —X-) __ Z S mild/2 sup IAw($ $)|max1_<_i—1)= 0pm, and lnhd/gAn23l 192m- - Mia-II] J 92(Xj) n n g 2nhd/2 sup IAw($)lmaxlgignltiln—3 Z Z [ xEI 2' lj—l 52 = nhd/Zop(Iogkn(logn/n>2/d+4)op(W)*1/2)0p((nhd)—1> = cam/125*”) = 0pm), thereby completing the proof of the claim in (2.52), and hence that of the Lemma 2.5.4. E] Proof of Lemma 2.5.5. Recall the notation from (2.1), (2.2). Let n 2w To prove the first part of Lemma 2.5.5, we need to prove the following steps: EKh(a: — X)Kh(y —- X)o§O(X) 2 9(95)9(y) dG(a:) dG (y) lf‘n —— m = 0pm), lfn —- in = 0,,(1) (2.54) Ign — 972' = 010(1), 9n —> I‘. Now, we shall prove the first part of (2.54). For the sake of convenience, write Kh(Xj — X2") by Ki(XJ-) and Ah(.r) :2 g2(.2:)(§;2(:1:) — g"2(:r)). Now, rewrite f‘n as the sum of the following three terms: .__. d _2 -1 19(le >-K (szez— tixcj-tj) 2 Bl ._ 2h 12 2;}; _ 2 920(1) I, , 32 .___ 2hdn_22n —1: K2(Xl) Kj(Xl)(€22— tij)(€ “’9 j)Ah(Xl)I 1,21] , #J’ - 920(1) _ XlK (X )(67, t¢)(€'—t') B3 := Zhdn 2}: (n1: ’ 912W) 9 91,) i752 - Ki(Xz)K°(Xz)(€i- tz'jXC -t jrlAMXz) 11)] X ”-1 J ( Z 92(Xl) l In order to prove the first part of (2.54), it suffices to prove that Bl — Ln = 0p(1), B2 = 0p(1), and B3 = 019(1). (2.55) 53 For this, we shall show that K°X K-X e; c,- B :2 hdn—ZZ ”—1: 2( l) _7( I” 2”le 92(X) l iaéj l 1 This expression is bounded by the sum of the following two terms: '2 2K2(0) e2e2[K9(X9) _+K2(Xj) . n4hd #j 9 9 920(2) 2 92(le J, K'(X)K-(X) ' ._ d —4 2 2 2 l J l #1 1m 9209’) , By using (g1), (k), and continuity of 030, we obtain [K%(X2)€%€% J 94(X2) —d ‘30 (:1: — uh)a§0 (x)g(:r -— uh) = h [I] 93(33) dudx 2 H1 ’ 2 II ,9 a". a. V Hence, EH1 = 0(nhd)_2 = 0(1), by (h2), and H1 2 op(1). Next, rewrite H2 as the sum of the following two terms: K2 x K2 X 5262. H21 = hdn—4 Z 2( z)4J( z), 112: 179279] 9 (X1) H22 = h dn_ 4 Z Ki(Xl)Kj(X1)K,-(Xm)Kj(Xm)ezze%IlIm. m752°5£j¢l 92(X1)92(Xm) By (92), (81): (k), (112), and independence of Xi’s, K2(X3)K§030“ — uh — vh)ago(x - Uh — wh) X 9(2: — vh -- wh)g(a: — ’Uh — uh) 9(15 - vh)g(~’c) dw du dv dz] = 0(1). Hence, H2 = Op(1), and (2.56) is proved. By a similar argument, under (2.56), (e2), (g1), (k), and (h2) one obtains Min—22L, 4: K2 2(X;)2Kj (Xz)|€2lzl]2 =Op(1), (2.57) #J’ - (X1) __ Xz'()Xl )K j(Xl) dn 2§""_1Z 92(X2) l2 =0p(1). (2.58) 2 J - Furthermore, sup lAh(x)| = 019(1), by (2.3), (2.59) 2:61 maxlgignltil = 019(1), by (m4) and (2.32). (2.60) Note that by expanding (c,- — ti)(ej — tj) and the quadratic terms, '31 — in] is bounded above by the sum of 812 and B13, where 2 . d _2K2(X1)Kj(Xz)(|t2tj|+|6212|+lt2€'l) 312 .= 2h ’n. E[n— 1}: 92(Xl) '7 It , 2763' 1 313 := 4hdn_2}:(n— 1: K2(Xl)Kj g2((X)l)l€2 9'11) 272]: z X n_1 Ki(Xz)Kj(Xl)l(ltitjl + l€2til + Itz'éjl) ( Z 92(Xz) II . l 55 But 812 = op(1) by (2.57), (2.58),(2.60), and the fact that ti’s are free of X1. Sim- ilarly, by applying the Cauchy-Schwarz inequality to the double sum and by (2.56) 813 : 019(1). Hence lBl — f‘nl = 017(1). Next, consider 82. By using the inequality léz' - tilléj - tjl S Iéz'éjl + ”£th + lfz‘tz‘l + lti€jl, and by (2.59), 132 S 2 sup IAh(fr)| [312 + B] = 0p(1). :rEI Similarly, an application of the Cauchy-Schwarz inequality to the double sum yields B3 = 013(1). This completes the proof of (2.55), and hence that of the first part of (2.54). To prove the second part of (2.54), it suffices to Show that E[f‘n — fin]? = 0(1). Let S. 'kl [Ki(Xk)Kj:Xk)K:(Xz)Kj(X1)Ik11 2] 2(ch) 2(Xz) _K/fz z($)Kj ($)Kz' (30K 3'30 y>dxdy , 9((x)9 y) d .. .. and uz’jkl: [772 62 62’ J'Szjkl Hence, [P71 - 9n] = ijéz' 2k 21 uijkl' Expanding the quadratic, we obtain lf‘n - 5n]2 = 22222 X ZZuz-jszmnpq ijaéz'k l mnyémp q Since we have four kernel terms in each Sijkl term, thus by using (g1) and (k) h—Sd) maximum order of Es,- j kl smnpq = O( , and hence that of E[uz’jkl umnpq] = 56 2d h—g— x 0(h—8d) = 0(n—8h‘6d). Also, we have eight summations, if there are at n most five different subscripts in the summations, then the expected value of those terms which has at most five different subscripts is at most n5 X 0(n—8h-6d) = 0(n‘3h’6d), and hence by (h2), it is 0(1). So, we will only discuss those terms, which are involved with more than five different subscripts in the summations. According to that criteria, [f‘n — 67212 is bounded by the following terms: 2 Z Z Z Z Z “ijklumnpq, (2.61) j#i k l ”#m p#i’j7k3l1m,n q?éi,j7k)l,m3n)p ”2 = Z Z Z Z Z uz‘jkl j¢2 n#m k#zij3m)n l#z)j1k)m)n p¢23j3k3l3min X lumnpi + umnpj + umnpm + umnpnl, U3 = Z: Z Z ”mnpk is“ 7175771 kfldflfi P¢i,j,k,m,n Xluijii + “ijz'j + “ijim + “ijin + uz’jjz’ + uz’jjj + '“ijjm + uz‘jjn U1 +uz'jmz' + “ijmj + uijmm + uz‘jmn + uijnz' + “z'jnj + uijnm + uz‘jnn], U4 = Z “2' j kl umnpl’ p¢i¢j¢k¢m¢n¢z U5 2 Z luz'jkn(umipk + umjpk + “mnpk) ##jaékaémaén +umnpk(uijki + uz’jkj + ”ijkk) + 7‘2’jkm(uinpk + “jnpk + umnpk) +(umnpi + umnpj + umnpm + "mnzmxuijki + uz’jkj + “ijkk + uz’jkm +uz‘jkn) + (“ijz'k + uz'jjk)umnpk + (Umpm + ujnpm + uknprn)uijkm' To prove the claim, if suffices to show that the expected value of all of these terms is 0(1). 57 Note that, for all k, 1% i,j and k 75 l, ElsijkllXi’le = 0. Now, by using this fact, Vzmsé 233', k,m,n, 1, and p # q, Hence, E(U1) = 0. Similarly, for all k,l 75 i,j, m, 71,}? and k 74 l, E(uijl€l umnMIXi,Xj,Xm,Xn,Xp) = 0 and, expected value of other terms of U2 is zero. Hence, EU2 = 0. Similarly, for all km 74 2333mm and k 25 p, E(U3) = 0- Again by the above fact, hzd 2 2 2 E(U4) = 753(616365868123485674] h2d = —,—,-E[{E[0303123429514191 W 2 2 2 = TEl090(X1)090(lefEl31234lX3aX4l} l=0- By independence of Xi’s, expected value of the first term of U5 is equal to 2d 6 h 2 2 n Eluz’jknumz'pkl = n2 E[700(X1)090(X2)090(X5)8123485163]- By independence of Xi’s, (g1), (k) and continuity of 060 and 790, one obtains 2 2 E [T90(X1)090(X2)090(X5)8123435123l = Elmo(X1){E[0§0(X2)81234|X1,X3:X4]}2] 2&3[/;'// [/TOIJZCC—hv—hw)ogo(:c—hu—hw)K(U)K(v) 2 g(:r «- hv - hw) 3 3 dvdwdx g (av-hwy (x) xK(w + v)K(w + u)g(:r — hu — hw)du] 58 + f [/1/ [7,13% .. hv —— hw)a§0K(v> _ _ 1/2 _ _ 2 g(a: hu hw)g (a: hv hw)dudwdx] dv] xK(w + v)K('w + 1‘) g(:z:)g(:17 - wh) _—. 0(h_4d). Hence, the expected value of the first term of U5 is equal to 0(72.hal)“2 = 0(1), by (h2). Similarly, by using (g1), (k) and (h2), expected value of the other terms of U5 is 0(1). Hence, the second part of (2.54) is proved. The proof of the third and fourth part of (2.54) is given in Lemma 5.5 of K-N. Hence (2.54) is proved, and so is the Lemma 2.5.5. Cl 59 CHAPTER 3 Minimum Distance Goodness-Of-Fit Tests For Current Status Data 3. 1 Introduction This chapter discusses a minimum distance method for fitting a parametric model to the distribution function of the event occurrence time in the one sample set up with current status data. Let X and T denote the event occurrence and inspection times, respectively. Let F (G) denote the d.f. of X (T). Assume X and T are independent. In the current status data set up, one observes 6 = I [X S T] and T, where I [A] denotes the indicator function of the event A. Let .A := {F0(t) : t E R+, 0 E 8 C Rq, q 2 1} be a given parametric family of d.f.’s. Let I be a compact sub-interval of [0, 00). The 60 problem of interest here is to test the hypothesis H01 : F(t) = F000), for all t E I, for some 00 E 9, against the alternative H11 : H01 is not true. based on the random sample {(Ti, 67;) : 1 S i S n} from the distribution of (T, 6). In this chapter, we adapt the inference procedures discussed in chapter 2 to the current status data. More precisely, let 03(Ti) 2 F9 (Ti)(1 — F9(T,-)), and consider the regression model 6i = F6(Ti) + 06(Tz')Ci, I S ’i S n. Here {(2} are i.i.d. r.v’s such that E((,-|T,-) = O and E(§2-2|T,;) = 1, , 1 g i g n. we shall be using the notation of chapter 2 with X, Y, p(:c) and me replaced by T, (5, F (t) and F9, respectively, where now (1 = 1. Thus e.g., now l Tl K t—T- 6-—F T- 2 W, .= fl [nazl Myself“ an» m), (3.1) 1 n 1 n Kh(Tj-Ti)(5i_F6(Ti)) 2 Mn(9) 3: - [”2 * ' , nj=1 ”i=1 9w“) ‘7 Dn I: nhl/Z égl/2(Tn(1§n) '- Rn), 6n := argminoeeTn(0), 611:: argmingeeMnW), -- n K2 t—T- é? Rn :: n—ZZ/ h(*2 1.) 1d i=1 I 9w (0 . K (t—‘T°)K (t-fl-)é.-é' 2 ._ -2 d h i h _7 a 9n ._ 2n h E j (/ §h(t) J d(t)) ’ (t), 52-2: i—F~ (Ti), lsisn, 61 2 .7" j=1i 1 9i}; (Tj) - . 2 . K T—T-K T-T'E'E' Fn := thn—4Z (2: h( l 2)“ h( l J) z 311) . #j z 9h(Tl) This chapter is organized as follows. Section 2 adapts the results discussed in chapter 2 based on minimum distance statistic to the current status data. First, we discuss consistency of 19;; and 127; for T(F*), where F * E L2() is a d.f., different from the null model A. Then, we discuss the consistency of 19;, 1972 and asymptotic normality of 1% and Dn, under H01. Similar to Koul and Song (2006) (K-S), we also obtain consistency of Dn against a fixed alternative, under some regularity conditions. Additionally, we obtain asymptotic power of the proposed minimum distance tests under a class of local alternatives Hln : F(t) = F6005) + gb(t)/nh1/2, where 2/1 is a continuously differentiable function such that f w2d < 00 and f F9¢d = O, for all 6 E 9. Section 4 adapts the results of chapter 2 based on empirical minimum distance statistic to the current status data and discusses consistency of 6}“, and 6n and asymp- totic normality of 6n and nhl/zf‘;l/2(Mn(én) — Cm), under H01. Section 5 reports results of the three simulation studies. The first simulatiOn study investigates Monte Carlo size and power of empirical minimum distance test. The finite sample level approximates the nominal level well for large sample sizes. Simulation results also show little bias in the estimator 6n, for all the chosen sample sizes. The second simulation study investigates the empirical size and power behavior 62 of the Cramér-Von—Mises test CV1, where CV1 is defined in chapter 1. Since the asymptotic distribution of C V1 is not known, so in order to find the Monte Carlo levels and powers of this test, we need to estimate its out off points. Estimated cut off points are obtained by first getting 10,000 values of C V1 and then by finding percentiles from the distribution of 10,000 values. The finite sample level approximates the nominal level well for all the chosen sample sizes. In our simulations, F is computed by the one step procedure for calculating the nonparametric maximum likelihood estimator, based on isotonic regression, cf. Groeneboom and Wellner (1992). The third simulation study investigates Monte Carlo size comparison of the em- pirical minimum distance test with the tests of Koul and Yi (2006) (KY) and CV1. Simulation results show that the empirical levels of CV1 and KY tests are better than Mn(én), when sample size is less than 200. But when the sample size is 200, the significance levels of all the three tests are comparable to each other. 3.2 Minimum Distance Statistics and Tests In this section we adapt the results discussed in section 2 based on a class of minimum distance statistics to the current status data. Here we shall be using the same assump- tions discussed in chapter 2 with X, Y, ”(22) and ma replaced by T, 6, F(t) and F9, respectively, where now d = 1 and I is a bounded interval in [0, 00). Also under the current status data set up, assumptions (e1), (e2), (e3), and (e4) are automatically satisfied. First, we discuss the consistency of 19}: and 1§n for T(F*), where F * E L2() is a 63 d.f., different from the null model A. Let H11 : F(t) = F*(t), t E I. Lemma 3.2.1 Suppose assumptions (k), (91), and (m3) of chapter 2 hold with ma replaced by F6, (1 = 1 and I is a bounded sub-interval in [0, 00). Let F * be a given d.f. such that F* ¢ A, F* E L2(), and T(F*) is unique. (a) In addition, suppose F * is a.e. ((1)) continuous. Then, under H11, 19;: = T(F*) + op(1). (b) In addition, suppose F * is continuous on I. Then, under H11, dn = T(F'*) + Upon taking F * = F 00 in the above result one immediately obtains the following: Corollary 3.2.1 Suppose assumptions (91), (k), (m1)-(m3), (hl), and (h2) of chap- ter 2 hold with me replaced by F9, where now d = 1 and I is a bounded sub-interval in [0, 00). Then, under H01, 19,"; ——> 60, and dn -—> 6’0 in probability. Next result gives the asymptotic normality of n1/2(1§n — 60) under H01. Let Fn(t, 6) := 11-1 1': Kh( (1— 7",), (3.2) n Fn(t,6) :2 n—12Kh(r,)1—)F9(T,-,) Fh(1) := EFn(t,60)= EKh(t—T)F,90(T), _ mow) sn _ f1 92(t) d(t), 2 __ F90(t)(1“F00(t))F00(t)F90(0,4520)dt °_ /l' 900 64 Corollary 3.2.2 Suppose assumptions (g1), (g2), (p), (k), (m1)-(m5), and (h3) of chapter 2 hold with my replaced by F6, where now (1 = 1 and I is a bounded sub- interval in [0, 00). Then, under H01, n1/2(i§n — 60) = 261n1/2Sn + op(1). Conse- quently, n1/2(1§n — 00) —»d Nq(0, 23512251). Next, we state the asymptotic normality result about Dn under H01. Let F (t)(1F (t))¢(t)2 r := 2/;r{ 00 9(th }2d 1(/ /K(u) K()v+udu)2dv. Corollary 3.2.3 Suppose assumptions (91), (92), (p), (k), (m1)-(m5) and (h3) of chapter 2 hold with me replaced by F6, where now d = 1 and I is a bounded sub- interval of [0,00). Then under H01, Dn -—+d N1(O,F) and [STEP—1 — 1| = 019(1), where Q", is as in (3.1). Consequently, the test that rejects H01 whenever anI > z a /2 is of the asymptotic size a, where za is the 100(1 — a)% percentile of the standard normal distribution. The following corollary provides a set of sufficient conditions under which IDnI —> oo, in probability, for any sequence of consistent estimators tin of T(F*) under the fixed alternative H11. Corollary 3.2.4 Suppose assumptions (9]), (g2), (p), (k), (m3), (h3) of chapter 2 hold with ma replaced by F9, where now d = 1 and I is a bounded sub-interval of [0,00). Assume the alternative hypothesis H11 hold with the additional assumption that infg p(F*, F0) > 0. Then, for any sequence of consistent estimator 157;, of T(F *), IDnl —> oo, in probalility. Consequently, the test that rejects whenever |’Dn| > za is consistent against the fixed alternative H11. 65 Its proof is similar to that of Theorem 5.1 in KS adapted to the current status data. Next, let 1,!) be a known continuously differentiable real valued function. In addi- tion, assume i/J E L2() and fF9¢d = 0, ‘v’ 0 E 8. (3.3) Consider the sequence of local alternatives 111,, = F(t) = F10(t)+ 1.112(1). 11. = 1/(nh1/2)1/2- (3.4) The following corollary gives asymptotic power of the minimum distance test against the local alternative H In- Its proof is similar to that of Theorem 5.3 in K-S adapted to the current status data. Corollary 3.2.5 Suppose assumptions (91), (92), (p), (k), (m4), and (h3) of chap- ter 2 hold with m9 replaced by F9, where now d = 1 and I is a bounded sub- interval of [0, 00), then under the local alternative hypothesis (3.3) and (3.4), 'Dn —> d N(F’1/2fw2d<1>,1). The following corollary gives the asymptotic distribution of Sn under H 1n- Its proof is similar to that of Theorem 5.2 in K-S adapted to the current status data. Corollary 3.2.6 Suppose assumptions (91), (92), (p), (k), (m1)-(m6), (h3) of chap- ter 2 hold with m9 replaced by F9, where now at = 1 and I is a bounded sub- interval of [0, 00), then under the local alternative (3.3) and (3.4), ill/2(3); —00) "’d Nq(o,2512251). 66 3.3 Empirical Minimum Distance Statistic In this section, we adapt the results of chapter 2 based on empirical minimum distance statistic to the current status data. First, we discuss the consistency of 9;“, and Sn. The consistency of these estimators for 60 under H01 follows from Lemma 2.3.2 of chapter 2. Applying Corollary 2.3.1 to the current status set up, we have Corollary 3.3.1 Suppose assumptions (91), (k), (m1)-(m3), (hl), and (h2) of chap- ter 2 hold with m9 and (I) replaced by F9 and C, respectively, where now d = 1 and I is a bounded sub«interval of [0, 00). Then under H01, 0;; —1 60, in probability. Applying Theorem 2.3.1 to the current status set up, we have Corollary 3.3.2 Suppose assumptions (g1), (k), (m1)-(m3), (hl), and (h2) of chap- ter 2 hold with m9 and (I) replaced by F9 and C, respectively, where now (1 = 1 and I is a bounded sub-interval in [0, 00). Then, under H01, (in —1 60, in probability. Now, we discuss asymptotic normality of n1/2(én — 60). Let n .' . . a. z: 11-1 U”(TJ)Fh(TJ)IJ. 9(7)“) i=1 Applying Theorem 2.4.1 to the current status set up, we obtain Corollary 3.3.3 Suppose (61), (g1), (92), (k), (m1)-(m5) and (h3) of chapter 2 hold with m9 and (I) replaced by F9 and G, respectively, where now d = 1 and I is a bounded sub-interval in [0, 00). Then under H01, Til/2(én—00) = 261n1/23n+0p(1), 67 Consequently, n1/2(dn —60) _’d Nq(0, 2612261), where 20 and Z are as in (m6) and (3.2), respectively. Next, we discuss asymptotic distribution of the empirical minimized distance Mn (Sn). It follows from Theorem 2.5.1 adapted to current status data. Corollary 3.3.4 Suppose (e1), (g1), (92), (k), (m1)-(m5) and (h3) of chapter 2 hold with m9 and (I) replaced by F9 and C, respectively, where now d = 1 and I is a bounded sub-interval in [0, 00). Then under H01, nh1/2(Mn(dn) —C'n) —’d N1(0, F). Moreover, If‘nI‘_1 —— II = op(1), where f‘n and Ch are as in (3.1). Consequently, the test that rejects H0 whenever nh1/2f‘; 1/ 2IMn(l9An) — Cnl > Za/g is of the asymptotic size a. 3.4 Simulations This section contains the results of three simulation studies. The first one assesses finite sample level and power behavior of the empirical minimum distance test statistic Mn(0). The second simulation study investigates finite sample level behavior of the Cramér-Von-Mises test CV1. The third simulation study investigates a Monte Carlo size comparison of Mn (6), CV1 , and KY test. The simulations are done using Matlab. The kernel functions and the bandwidths used in the simulations are Ka) = 11*(1): 2i“ — x2>mx1 s 1) h = c1n_1/3, w =c2n_1/5(l0gn)1/5, 68 with some choices for CI and c2. In the tables below, exp(d), 6’ > 0, de- notes the exponential distribution with parameter 6 under the null hypothesis H01 :F = exp(0), for somed > 0. The Weibull distribution with density w(t) :2 ba—btb“1exp(—t/a)b is denoted by W(a, b) and G(a,b) represents the Gamma distribution with density g(t) := mta_13$P(—t/b)1 a > 0, b > 0. The asymp- totic level is taken to be 0.05 in all the cases. The sample sizes chosen are 50, 100, 200, each repeated 1,000 times. Table 3.1 reports the Monte Carlo mean and the MSE(én) under H01 which are obtained by minimizing Mn(0) and employing the Newton-Raphson algorithm. The sample sizes chosen are 50, 100, 200, 500, each repeated 1,000 times. One can see there appears to be little bias in én for all the chosen sample sizes and MSE decreases as the sample size increases. To assess the effect of the choice of (c1, eg) that appears in the bandwidths on the level and power, we ran the simulations for various choices of (c1, c2), ranging from 0.1 to 1. Table 3.2 reports the simulation results for those (c1, c2) which gave the best results. The entries in the tables for Mn(én) are obtained by computing the number of times (lnhl/Qfgl/2(Mn(dn) — Cn)| 2 1.96)/1,000. Table 3.2 summarizes the empirical levels for test statistic Mn(én). It shows that as the sample size increases the simulated levels are getting closer to the asymptotic level 0.05. Table 3.3 represents the power for test statistic Mn(dn) for four different alterna- tives, when (c1,c2) = (.9, 1). It shows that the power is getting better as the sample size increases. The second simulation study investigates the behavior of the Cramér—Von—Mises 69 test CV1. Since the asymptotic distribution of CV1 is not known, so in order to find the Monte Carlo levels and powers of this test, we need to estimate its out off points. Estimated cut off points are obtained by first generating 10,000 values of CV1 and then by finding percentiles from these 10,000 values. After that, for CV1, the empirical level and power are obtained by computing the number of (CV1 2 estimated cut off point) / 1,000. In our simulations, F is obtained by the one step procedure for the calculation of the nonparametric maximum likelihood estimator, based on isotonic regression, cf. Groeneboom and Wellner (1992). Table 3.4 contains the simulated 90th, 95th, 97.5th, 99th, and 99.5th percentiles of CV1 when distributions of X, T are exp(1). Table 3.5 represents simulated signifi- cance levels by using the corresponding simulated percentiles given in Table 3.4 when testing F = exp(1) and the distribution of T is exp(1). It shows that the simulated significance levels of CV1 for different chosen sample sizes are very close to the true nominal sizes. Let A Cn := argminCEe CV1((). I Table 3.6 reports the Monte Carlo mean and the MSE of (in) under F = exp(1) which are obtained by minimizing CV1 and employing the Newton-Raphson algorithm. One can see there appears to be little bias in Sn for all the chosen sample sizes and MSE decreases as the sample size increases. Table 3.7 represents the power of CV1 for five different alternatives, when distri- bution of T is exp(1). It shows that the power is getting better as the sample size 70 increases. In the third simulation study, we make a comparison of the Monte Carlo level of the proposed empirical minimum distance test Mn (Sn) with the other two tests C V1 and KY. KY test. Let S M L E denote the maximum likelihood estimator, obtained by using the the following score statistics Sn(6) given in (3.3) of KY: SW) := Z [——5_:6_T, — 1]T,-. 1:1 1“ 8 Let . n Un(t) 2: 12—1/2 ZI i=1 (T <1) 11—1 Z é(T-+T-)/2 TjTie— z j (1 _ e—gTi)—1/2(l _ 8—19Tj)—1/21(Tj S tATfl] 5“- ,. __ —ér —éT _ "1211:17135 11(1—1 k) 11(Tk2Tj) Let Gn denote the empirical distribution of the design variable Ti, 1 S i g n and t0 = 99th percentile of On. The KY test statistic is Rn: sup M. OStStO V 017,00) As shown in KY, the limiting null distribution of Rn is the same as that of 3111303131 |B(t)|, where B is the standard Brownian motion. The 95th percentile of this distribution is approximately equal to 2.24241, which is obtained from the fact P( sup |B(t)| 2.24241} / 1000. Table 3.8 shows comparison of simulated significance levels for M91271), CV1 and Rn. For the simulated significance levels of CV1 we used the percentiles given in 71 Table 3.4. It shows that the empirical levels of statistics C V1 and K n tests are better than Mn(61n), when sample size is less than 200. But when the sample size is 200, the significance levels of all the three tests are comparable to each other. Table 3.1: Mean and MSE of bin, X, T ~ exp(1), 90 = 1 Sample Size 50 100 200 500 Mean 1.0442 1.0172 1.0025 1.0016 MSE 0.3324 0.1780 0.1216 0.0748 Table 3.2: Empirical sizes of Mn(én), X, T ~ exp(1) c1, 02\n 50 100 200 0.5, 0.2 0.024 0.044 0.049 0.8, 0.6 0.061 0.058 0.046 0.8, 0.7 0.088 0.062 0.056 0.9, 0.8 0.07 0.061 0.05 0.9, .9 0.07 0.059 0.049 0.9, 1 0.094 0.058 0.045 72 Table 3.3: Power of Mn(dn), T ~ exp(1), (cl,c2) = (.9, l) X\n 50 100 200 c(2,1) 0.975 1 l G(1,3) 0.927 0.999 1 W(1,5) 0.211 0.404 0.628 W(1,2) 0.452 0.693 0.929 Table 3.4: Simulated percentiles of CV1, X, T ~ exp(1) Percentile \ n 50 100 200 99.5 0.0412 0.0245 0.0152 99 0.0359 0.0216 0.0131 97.5 0.0298 0.0177 0.0112 95 0.0236 0.0156 0.0095 90 0.0183 0.0125 0.0077 Table 3.5: Empirical sizes of CV1, X, T ~ exp(1) True level\ n 50 100 200 0.005 0.004 0.003 0.004 0.01 0.011 0.009 0.008 0.025 0.025 0.029 0.026 0.05 0.048 0.056 0.049 0.1 0.101 0.106 0.09 73 Table 3.6: Mean and MSE of (An, X, T ~ exp(1), 90 = 1 Sample Size 50 100 200 Mean 1.6815 1.4594 1.3088 MSE 0.5558 0.3152 0.1878 Table 3.7: Power of CV1, T = exp(1). Dist. of X\ n 50 100 200 G(1,3) 0.95 0.999 1 G(2,1) 0.975 1 1 W(1,.5) 0.413 0.624 0.891 W(1,1.5) 0.394 0.45 0.575 W(1,2) 0.619 0.779 0.957 Table 3.8: Empirical sizes, X,T ~ exp(1), (c1,c2) = (.9, .8) Tests\ 11 50 100 200 Mn .074 0.07 0.055 1%,, .04 0.049 0.052 CV1 0.049 0.048 0.051 74 CHAPTER 4 Testing the equality of two distributions with Current Status Data 4.1 Introduction This chapter discusses the problem of testing the equality of two distribution func- tions based on current status data. Accordingly, let X, S (Y, T) denote the event occurrence and inspection times, respectively, from the first (second) population. Let F1 (F2) denote d.f. ofX (Y) and G1 (02) denote the d.f. of S (T). Let X1,.. .,Xn1 (Y1, . . . 1Yn2) be i.i.d. F1 (F2) and S1, . . .,Sn1 (T1, . . . ,Tn2) be i.i.d. G1 (02) ran- dom variables. Assume all random variables are mutually independent. In the two sample current status data set up, one observes (6,38,), 1 g i g 111, and (nj,Tj), 1Sjgng,where6=I[XSS],n=I[YST]. 75 The problem of interest here is to test the null hypothesis that the two event occurrence distributions are the same, i.e. H02: F1(:r) = F2(:7:), for all a: E I, against the alternative H12: F1(:1:) 75 F2(:r), for some :1: E I, where I is a compact sub-interval of [0, 00). In this chapter we adapt the test proposed by Koul and Schick (2003)(K-Sh) to the two sample current status data. More precisely, let 0%(Si) = F1(S,-)(1— F1(S,')), 0%(Tj) = F2(Tj)(1 —- F2(Tj)) and consider the regression models 5,,- := 171(52') + 01(Silcli’ 1 S i _<_ n1, nj :: F2(Tj) + 02(lec2j’ 1 Sj _<_ n2. Here (11" (23- are i.i.d. r.v.’s such that E((1,-|Si) = 0 = E((2j|Tj) and E(C‘12i|Sz-) = 1 = E(C%j|TJ-), 1 g i 3 n1, 1 S j 3 n2. Assume also that 01, G2 have positive densities 91 and 92 on [0, 00), respectively, and that F1, F2 have bounded densities. Let U denote the set of all nonnegative functions that vanish off I and whose restrictions to I are continuous. Consider the integral r = / 11(11)[F1(11) — F2(x)]dx, 11 e u. A possible choice for u is the indicator 1: of the interval I. The integral 1‘ is 0 if the null hypothesis holds, and is non-zero under the alternative H12, for all u E U. 76 Let K be a symmetric density with compact support [—1,1] and a = an be a bandwidth sequence, and let 1 "1 "'2 WMMTflw T = _— n1712 i—1j_1 91(51)92(Tj )( 77j)-Ka(51— —T j) Observe that E(T)=//\/1KS—\/_lF1(t1)lKa(8-t)dsdt which is close to I‘ for small a. Thus I provides an estimate of I‘ if 91 and 92 are known, which is rarely the case. This suggests to replace the densities in T by their estimates. Accordingly, let n1 n2 1! )1/ (T9)( —17‘)K (S-—T-), (4.1) T=n11nzgjzllfi 2 J a Z J where 13k is an estimate of ”k = fi/gk, k = 1, 2, constructed from the pooled sample such that 1% (0:) = 0 for a: Q! I. Similar to K-Sh, the estimators of ”k can be obtained when u is known and when u = u7 as described in Remark 4.2.5 below. So that the asymptotic normality of I both under the null hypothesis and under local alternatives n1n2 l F=F+N_2, N:=———-———, 1 2 7 n1+n2 (4-2) can be obtained, where '7 is a non-negative continuous function such that 7(0) 2 0,7(00) = 0 and 0 < fu(x)7(z)dx < 00. The rest of the chapter is organized as follows. Section 2 discusses asymptotic normality of I under a general set of assumptions on the estimates 191 and 192. Section 3 reports the numerical results of the two simulation studies. The first one assesses 77 the finite sample level and power behavior of I test. The simulation results of the test statistic I are consistent with asymptotic theory. In the second study, the finite sample comparison of I and CV2 tests is made, where CV2 is defined in chapter 1. Since the asymptotic distribution of CV2 is not known, so in order to find Monte Carlo levels and powers of this test, we need to estimate its cut off points. Estimated cut off points are obtained by first getting 10,000 values of CV2 and then by finding percentiles from the distribution of these 10,000 values. Simulation results show that for all the chosen alternatives and bandwidths, significance level of CV2 is better than I, and power of I is better than CV2, when sample sizes are 50 and 100. But when sample size is 200, significance level and power. of I and CV2 tests are comparable. In our simulations, F1 and F2 are computed by the one step procedure for calculating the nonparametric maximum likelihood estimator, based on isotonic regression, cf. Groeneboom and Wellner (1992). 4.2 Asymptotic behavior under the null hypothe- sis and local alternatives This section discusses the behavior of the test statistic I given in (4.1) under the null hypothesis and under the alternatives (4.2). Note that the choice 7 = 0 in (4.2) corresponds to the null hypothesis. To stress the dependence of local alternative on the parameter 7 we write Pry for the underlying probability measure and E7 for the corresponding expectation. 78 Arguing as in K—Sh, we shall describe the asymptotic behavior of I as the sample sizes n1 and n2 tend to 00. For this we need the following assumptions. (A.1) The function u E U, the set of all non-negative functions that vanish off I and whose restrictions to I are continuous. (A2) The densities 91 and 92 are bounded and their restrictions to I are positive and continuous. (A.3) For any pairs of d.f.’s (F1,Gl), and (F2,G2), P(0 < F1(S) < 1) = 1 and P(0 < F2(T) < 1) = 1. (A4) The weight function K is a symmetric Lipschitz-continuous density with com- pact support [-1, 1]. (A5) The bandwidth a is chosen such that a2N —+ 0 and aNc -+ 00, for some c < 1. Note that 01(5) 2 0 as, implies either F1(S) = 0 or F1 (S) = 1 as. In the former case E(6 S) = 0 implies 6 = 0 as. Hence 6 — F1(S) = 0 as. Similarly F1(S) = E(6lS) = 1 as. implies 6 = 1 as. and hence 6 — F1 (S) = 0 as. Thus 01 (S) = 0 as, implies 6-— F1(S) = 0 as, and, under (A3), P(01(S) > 0) = P(0 < F1(S) < 1) = 1. Similarly, P(O < F2(T) < 1) = P(02(T) > 0) = 1. The condition (A3) is a joint condition on the supports of (F1, G1) and (F2, G2). For example, if distributions of X and S are exponential with the scale parameter 61 > 0, then P(0 < F1(S) < 1) = P(0 < e_01S < 1) = 1. But if the distributions of X and S are U(0, 1) and exponential with the scale parameter 0 > 0, respectively, then P(0 < F1(S) < 1) = P(0 < S < 1) = 1 — e‘g, and hence in this case the 79 first part of (A.3) does not hold. A sufficient condition for (A.3) is that F1 (F2) be strictly increasing on the support of G1 (6'2). Also note that under (A2) and (A.3) the functions 91: 92 and 0%, 0% are bounded and bounded away from zero on interval I and so are the functions 01/ 91 and 02 2./92 To establish the asymptotic normality of I, rewrite this statistic as n2 _1 n1 1 .. N 2 igrlflg'SDC1i—EZT2(Tj)02(Tj)C2j+—£—1-—Zr1(Si)6(Sz-)+T4, i=1 i=1 where r19) = S1(z)— 12212091120. T—,-), 2j: —1 1 T2(2) = 172(x);1‘21:91(51)Ka(x—S,), xE[0,oo), and n1 n2 T4=,,,, ZZV1(S)02(T2).(F2(S)—F2T(2,—1)Ka(S T-) 1 2i=1j=1 As in K-Sh, the following additional definitions and assumptions are made .to analyze the asymptotic behavior of I. Let S = (311 . . .,Sn1), T = (T1, . . . .Tnz). 6 = (61, . . . ,6n1), ’7 = (171,... 177712). and 6,- (773') be the vector obtained from 6(1)) by removing 6,- (71]). Definition 4.2.1 We say the estimator rk is consistent and cross-validated (CCV) on I for the function rk if the following conditions hold: 111 N 22119) )E.1(r1(S.-)-r1(s .)) 131:0..(1) "Ii: 1 N ”2 ‘7. 2 12(1 1211.2(1 ) 211912111: 012(1) n2j= 1 80 N 1< atlaxnljugEmrnx)E.(1=1(x)ls,6.1>2151 = 0.2(1), N K131111212 :21} E,[(.2(.)_ E7IT2(T)IT. 12,1)2IT] = 0.,(11. We say i‘k is a modification of fk if P7(SuPa:€I Ii‘k(:z:) — fk(:c)| > O) —+ 0. We say fl: is essentially CCV on I for rk if there exists a modification of 72k which is CCV on I for rk. Assumption 4.2.2 The estimate 72k is essentially CCV on I for T1: = u/gk for k = l, 2. The following result gives a sufficient condition for Assumption 4.2.2. Its proof is similar to that of Lemma 2.4 in K-Sh and hence no details are given. Lemma 4.2.1 Suppose there are modifications 17k of 19k such that, for k=1,2 0 g 32(1) 3 K, :1: e I, (4.3) for some finite constant K, 51—1-2 2 E7101 — V1(S 5.))2IS] : 0.2(1), (44) 7,12}: E.((z>2(T,- > - u.(T,-)>21T1 = 0.20), (4.5) N 1 3.1%.] :gflylm (T) 571171 (T)IS.5.'])2IS] = 0127(1). (46) N 12.3.22 :21; E.((u2(a:)— E.{T2(x>IT, 2,112.11 = 0.20). (4.7) Then, Assumption 4.2.2 holds. The next result gives the asymptotic distribution of I under the alternative (4.2) for any 7 including the case 7 = 0. 81 Proposition 4.2.1 Suppose the conditions (A.1)-(A5) and Assumption 4.2.2 hold. Then, under Pry, N1/2(T — F)/T converges in distribution to a N(0, 1) r.v., where 2 _ 2 r — f u lq1¢1(x>+q2¢21dx, 1/21 = 1710- F1)/91, $2 = F2(1— F2)/92, C11: N/n1 and (12 = N/n2- Details of the proof of this result are similar to those appearing in K-Sh and left out for the sake of brevity. Remark 4.2.3 The above result suggests a test which rejects H02 for large values of ITI. To implement such a test we need a consistent estimate +2 of 72. Given such an estimator f2, we have under the above assumptions that N1/2(’j' — I‘)/7“' is asymptotically standard normal under Pry, where I‘ = 0 under H02. Let now (I) denote the standard normal distribution function and z a /2 be its (1 — a/2)-quantile. Then a test that rejects H02 if |(N1/2’j')/%| 2 Zak/2’ has the asymptotic level a. Moreover, from the above result, the asymptotic power of this test, under P7, is 1— (za/2 — H) + (—za/2 — n), where _ f u(:r)'y(:r)d:1:. K— T Note that the value of I: does not change if we replace u by cu, with c a positive constant. Remark 4.2.4 Optimal u. Similar to K-Sh, the optimal u can be achieved such that it maximizes the asymptotic power, or equivalently the function It, under (4.2) for a specific function 7. An application of the Cauchy-Schwarz inequality shows that 82 n is maximized by the choice it. = VII 7 q1w1(x)+ q2¢2(z)’ (4.8) u: and the maximal value of r; is 2(x)I (2:) 1/2 W = (/ (111/1135) +€2¢2($)dx) ' The optimal ury depends on the sample sizes, the density functions 91 and 92, and the distribution functions F1 and F2. Next, we shall present the estimates of gk, 0%, 7'2 and ”IV Estimates of 9k, 0120’ T2 and Vk. Similar to K-Sh, estimates of Vk, k = 1,2, can be found for fixed given u and for the (unknown) optimal u = u7. For this we need estimates of the inspection time densities and variance function. The inspection time densities 91 and 92 can be estimated by the kernel density estimates "1 . 1 91m = a Z Kh1(t — 5,), i=1 . 1 92(t) = -- Z Kh2(t — Tj), t 6 RN, With bandwidth hk, k = 1, 2. Its expected value is 5k“) = /9k(t+hky)K(y)d(3/). t6 R+. Lemma 4.2.2 Suppose (A2), (A4) hold and the bandwidth hk is such that hk —) 0 and hkng ——> 00 for some c < 1. Then the following hold: sumezlékm - §k(t)| = 0p7(1)7 (4-9) /(‘g‘k(t) — gk(t))2dt _. 0. (4.10) 83 Details of the proof of this result are similar to those appearing in K-Sh and left out for the sake of brevity. Next, consider the following estimator of 0,26, k = 1, 2: Tl . 62(t) = 2,2110% 7'; H1(Si))2K61(t — Si) Ziil K610E " 51;) n2 ._ . . 2 __ . Zj=1(’7] “2(TJ» K62“ T3) t6 IR+ .2 02“) z n Zjil K02“ " Tj) where uk, 1: = 1,2 is the kernel regression estimate n ,(t) 2.116.Kb, 0, Ck ——> 0,11%!”c + Ck) —) 00, for some c < 1%. The following lemma gives the needed properties of this estimator It follows from Lemma 3.3 of K-Sh. Lemma 4.2.3 Suppose (A.2)—(A.5) hold, and the conditional fourth moment ”k is 2 is essentially CCV on I for 0% and bounded on an open interval I. Then 6k (4.11) superIo?C -— oil 2 019.7(1), k = 1,2. Now, consider the following estimator of variance function T2: 111 .2 n2 ~2 . . 1 u (S) . 1 u (T ). T2 = (11;.— Z .2 z ”i629 + ”n— .2 J “22(le (4'12) 1i=1 91(Si) 2 j=1 {12(le The following Lemma proves the consistency of this estimator. 84 Lemma 4.2.4 Suppose the assumptions of Proposition 4.2.1 and Lemma 4.2.3 hold 2 = 7'2 +0p7(1), where and a be a uniformly consistent estimator of u on I. Then, i 2 __ 2 + 2 7' — Tl 72 u2(x)F1($)(1 — F1($))d01($) + (12 / u2(x)F2(a:2)((1)- F2($))d02(x). 92 33 = (11/ 9%(93) Proof. Note that (4.12) can be written as 7"2 = i1 + 1‘2, where Let f2 = $12 + i22, where 1 n1 112(3)) ~2 T1 = ‘11— . "1 2:21 9%(52') 6%(‘S'Z)a In order to prove $2 = 72 + op7(1), it suffices to prove that (4.13) 2 + emu), %2 = 72 + 0,9,0). .2 T=T For the first claim in (4.13), it suffices to show that (4.14) 2 1 = + rf+op,(1), ‘22 =e22+op,(1). By the choice of n1 and n2, 0 < ql < 1. Thus, for the first claim in (4.14), it suffices to show that n1 .2 2 «2 1 [u (32;) - u (3010160 51— : [ .209) = 0mm). (4.15) i=1 91 2 Now, the left hand side of (4.15) is bounded above by «2 . 0 (a?) sup [who — uze): .; l =-— 0127(1), :36 91(55) 85 by uniform consistency of a, 61, 91, (A2) and (A.3). This completes the proof of the first claim in (4.14). The proof of the second claim in (4.14) is similar, thereby completing the proof of the first part of (4.13). To prove the second claim in (4.13), it suffices to show that 12 = T12 + op,(1), “22 = 13+ 01.70). (4.16) By the choice of n1 and n2, 0 < ‘11 < 1. Thus, for the first claim in (4.16), it suffices to Show that n . i i aha-was.) _ / man) dc (x) "1 - 612(5 ') x 1 i=1 1 Z By the Law of Large Numbers, " MS )4 he) 42(x>a%(x) —— n11: / —T——dGl (at), in probability. 924(3 ) 91W) Thus, to prove (4.17), it remains to prove that n . i 21: 1,2,5.) [4%) _ sass] n1 ,2, ‘ 4345.) gas.) By the triangle inequality, the left hand side of (4.18) is bounded above by the sum of the following two terms: 1 "“1 "119 1 2 1 A := —— s o , —— . 2 ”1 22111 ( )1(S)l91(154) £090] [ 2(8.)— ms») , n Note that, for uknown andu = u7, ”—1122- =11u2(SZ-) = 0197(1), and by (A2), infxel'gl (3:) > 0. Hence .2111 o :1: 2(2: AISsup|1(?2—(I1 ”172112“, xEI 91(3) i=1 = 0P7(1)0P7(1)— _ 0137(1)) 86 by uniform consistency of 61 and (4.11). Similarly, because 0%(S) < 1, VS, l§¥(x)—g¥(x)l 1 ”1 2 --—- u Si er £04440) ”1.; ( ) = op,(1)op7(1>=om<1>, 242$ by uniform consistency of 61 and (A2). This completes the proof of the first claim in (4.16). The proof of the second claim in (4.16) is similar, thereby completing the proof of Lemma 4.2.4. Remark 4.2.5 Estimation of ”k' Estimation of ”k when u is known. As- sume (A.1), (A2) and (A4) hold. Then Vk can be estimated by fi/gk with his as mentioned in Lemma 4.2.2. We shall now show that these estimates satisfy the assumptions of Lemma 4.2.1 and hence Assumption 4.2.2. It follows from (A2) that gk(t) > 4,6 for all t E I and for some ,6 > 0. Thus, by (A4), §k(t) > 26 for all t E I. In view of (4.9), 17k = fi/(fik V6) is a modification of 19k. We then obtain (4.3) from the boundedness of u, while (4.4) and (4.5) follows from (4.9) and (4.10). Of course, (4.6) and (4.7) holds as 17k does not depend on 6 and 1). Estimation of Vk when u = u/y. Here we shall discuss the estimation of ”k = WIT/9k) where '7 is a known non negative continuous function. In view of (4.8), an obvious estimate of 11.7 is . , where ibk=:—, k=1,2. . _ 711 ’7 — A (11¢1 + 021/12 9k 21. Similar to K-Sh, we can easily verify the assumptions of Lemma 4.2.1 for Vk = M/gk by using Lemma 4.2.2 and 4.2.3. 87 4.3 Simulations This section examines the Monte Carlo comparison of the test statistics I and C V2 based on 10,000 replications. For simplicity we took I = (0,5) and u(:r) = 11-. The simulations are done using Matlab. The kernel function used for w, 91 and 92 in the simulations is g(l — 2:2)1 (le S 1). Let c1 be the bandwidth used for w. Similar to K-Sh, the values chosen for CI are 0.2 and 0.25. Also the bandwidths used for densities 91: 92 in the simulations are hl = h2 = c2(log(n)/n)1/5. In the tables below, exp(A) denote the exponential distribution with parameter /\ and wei(a,b) represents the weibull distribution with density w(t) :2 ba‘btb_1ezp(—t/a)b. The asymptotic level is taken to be 0.05 in all the cases. We used 7:2 of (4.12) to compute I. The entries in the table for I test statistic are obtained by computing the number of (If) 2 1.96)/10,000. Since the asymptotic distribution of CV2 is not known, so in order to find the Monte Carlo levels and the Monte Carlo powers of this test, we need to estimate its cut off points. Estimated cut off points are obtained by first getting 10,000 values of CV2 and then by finding percentiles from the distribution of these 10,000 values. After that, for CV2, the significance levels and powers are obtained by computing the number of (CV2 2 estimated cut off point) / 10, 000. Table 4.1 summarizes the empirical levels for test statistic I when sample sizes for both the populations are the same with chosen values of Cl and c2. The sample sizes chosen here are 50, 100 and 200. It shows that as the sample size increases the simulated levels are getting closer to the asymptotic level 0.05. 88 Table 4.2 represents the empirical levels for test statistic I when sample sizes for the two populations are not the same for all the chosen values of Cl and c2 and chosen inspection time densities. It shows that the simulated levels are consistent with the asymptotic theory when sample sizes are not the same for the two populations. Table 4.3 shows the simulated power of I for six different alternatives and chosen values of Cl and c2 when sample size for both the populations is 50. It shows that the power is getting better as the parameter of exponential distribution increases. Table 4.4 represents the simulated 95th, 97.5th, 99th, 99.5th and 90th percentiles of CV2 for sample sizes 40, 80, 100, 200 when distribution of X, Y is exp(1) and distribution of S, T is exp( 1.5). Table 4.5 represents the simulated significance level by using the corresponding simulated percentiles given in table 4.4 for sample sizes 40, 80, 100, 200 when distribution of X, Y is exp(1) and distribution of S, T is exp(1.5). It shows that the simulated significance levels of CV2 for different chosen sample sizes are very close to the true nominal size. Table 4.6 represents the simulated 95th percentile of C V2 for sample sizes 50, 100, 200 and for all the chosen inspection time densities. Table 4.7 shows comparison of simulated significance levels for I and CV2 for different inspection time densities and different sample sizes. For the simulated sig- nificance levels of CV2 we used the percentiles given in Table 4.6. It shows that the empirical levels of statistics CV2 is better than I, when sample size is small. But when sample size is large, then the results of I and CV2 are close to each other. Table 4.8 represents the comparison of power between I and CV2 for different chosen alternatives and sample sizes. For the power of CV2 we used the percentiles 89 given in Table 4.6. It shows that the power of statistics I is better than CV2, when sample size is small. But when sample size is large, then power of statistics I and CV2 is comparable to each other. Table 4.1: Empirical sizes of I, X, Y ~ exp(1), S,T ~ exp(1) cl,c2 n1=n2=50 n1=n2=100 n1=n2=200 0.2, 0.6 0.1151 0.0852 0.062 0.2, 0.9 0.1112 0.0812 0.0564 0.25, 0.8 0.0763 0.0698 0.0585 0.25, 0.9 0.0843 0.0655 0.0595 0.25, 1 0.1016 0.086 0.052 Table 4.2: Empirical sizes of I, X, Y N exp(1), (n1, n2) = (180, 200) c1, c2 S,T := exp(1.5) S,T :2 exp(1) S := exp(1),T := exp(1.5) 0.2, 0.6 0.061 0.0589 0.062 0.2, 0.9 0.056 0.0551 0.0573 0.25, 0.8 0.0573 0.058 0.0549 0.25, 0.9 0.0598 0.0586 0.059 0.25, 1 , 0.0535 0.0522 0.0560 90 Table 4.3: Power of I, S,T ~ exp(1), X ~ exp(1), n1 2 n2 = 50 c1, c2\Y exp(.5) exp(1.5) exp(2) exp(3) exp(4) exp(5) 0.2, 0.6 0.5381 0.2779 0.5762 0.9095 0.9634 0.9916 0.2, 0.9 0.4875 0.2767 0.5454 0.8891 0.9688 0.9940 0.25, 0.6 0.5253 0.2684 0.5960 0.9197 0.9772 0.9965 0.25, 0.8 0.5213 0.2777 0.5687 0.8980 0.9800 0.9913 0.25, 0.9 0.5227 0.2739 0.5900 0.8732 0.9761 0.9947 0.25, 1 0.5040 0.2645 0.5459 0.8943 0.9676 0.9848 Table 4.4: Simulated percentiles of CV2, X, Y ~ exp(1), S, T ~ exp(1.5) Percentile\n1 = n2 40 80 100 200 99.5 0.1896 0.108 0.0866 0.0466 99 0.1433 0.0934 0.0755 0.041 97.5 0.1413 0.0787 0.0649 0.0318 95 0.1189 0.0671 0.0563 0.0321 90 0.0974 0.0571 0.0471 0.0275 Table 4.5: Empirical sizes of CV2, X, Y ~ exp(1), S, T ~ exp(1.5) True level\n1 = n2 40 80 100 200 0.005 0.00498 0.0053 0.0051 0.0049 0.01 0.0099 0.0105 0.011 0.0121 0.025 0.02456 0.0254 0.0255 0.0249 0.05 0.05 0.0502 0.0501 0.0510 0.1 0.1022 0.1015 0.1014 0.0998 91 Table 4.6: Simulated 95th percentile of CV2, X, Y ~ exp(1). Dist. of S, T n1=n2=50 n1=n2=100 n1=n2=200 exp(1) 0.0999 0.0551 0.031 exp(1.5) 0.1008 0.0563 0.0321 exp(1), exp(1.5) 0.1011 0.0556 0.0311 Table 4.7: Empirical sizes, X, Y ~ exp(1), (c1, c2) 2 (25,1). S, T exp(1.5) exp(1) exp(1), exp(1.5) n1 = n2 T CV T CV T CV 50 0.1013 0.0510 0.0965 0.0486 0.1115 0.0480 100 0.853 0.0497 0.0729 0.0494 0.0876 0.0504 200 0.0521 0.0482 0.0559 0.0466 0.058 0.0465 Table 4.8: Power, S,T ~ exp(1), X N exp(1), (c1, c2) = (.2, .9). n1=n2 50 100 200 Dist. 61v T CV T CV T CV exp(0.5) 0.5016 0.2845 0.7067 0.5328 0.8941 0.8465 exp(1.5) 0.2677 0.1478 0.4136 0.2365 0.6511 0.4085 exp(2) 0.5624 0.3606 0.8223 0.6200 0.9756 0.8699 exp(3) 0.7976 0.7209 0.9812 0.9429 1 1 w(.2,1) 0.9468 0.9488 0.9997 0.9993 1 1 w(.5,1) 0.5487 0.3876 0.8281 0.6586 0.9804 0.8999 w(1.5,1) 0.2138 0.1298 0.3438 0.2129 0.5120 0.3922 w(2, 1) 0.4445 0.3086 0.6568 0.5418 0.8673 0.8488 92 BIBLIOGRAPHY [1] Ayer, M.; Brunk, H.D.; Ewing, G.M.; Reid, W.T.; Silverman, E. (1955). 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