H 3% \NHI“WWWNWNWWWHWWilliMl \ 3 ° 06 LIBRARY Michigan State University This is to certify that the dissertation entitled SOME INFERENCE PROBLEMS FOR INTERVAL CENSORED DATA presented by TINGTING Yl has been accepted towards fulfillment of the requirements for the Doctoral degree in Statistics #47 Major Professor’s Signature '7 /2 2 / 2 00 S” Date MSU is an Affirmative Action/Equal Opportunity Institution ATERlAL IN BACK OF 500 SL‘lKPLEMENTARY PLACE IN RETURN BOX to remove this diedcout 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 SOME INFERENCE PROBLEMS FOR INTERVAL CENSORED DATA By Tingting Yi 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 2005 ABSTRACT SOME IN FEREN CE PROBLEMS FOR INTERVAL CENSORED DATA By Tingting Yi This thesis consists of two parts. The first part studies asymptotically efficient estimation of the baseline hazard parameters in a modified Cox model where covariate effects are nonparametric when data are interval censored. These estimators are obtained by maximizing the log-likelihood function with respect to both the finite dimensional and infinite dimen- sional nuisance parameters using method of sieves. The sequence of these estimators is shown to be consistent, asymptotically normal, with the asymptotic variance achieving the semiparametric lower bound. The second part of the thesis pertains to constructing tests for fitting a class of parametric regression models to the regression function of the log of the event occurrence time variable when the data are interval censored case I and when the error distribution is known. These tests are based on certain martingale innovations of a marked empirical process. They are asymptotically distribution free in the sense that their asymptotic null distributions depend on neither the null model nor the covariate and the inspection time distributions. ACKNOWLEDGEMENTS I would like to thank my advisor Professor Hira L. Koul for his guidance and many helpful discussions on the subjects of this thesis. He was always available when I had doubts and questions. His general thinking of statistical problems and ways to solve the problems will help my future research and working. I would also like to thank all the other committee members, Professor Habib Salehi, Lijian Yang and Yijun Zuo, for serving my guidance committee. Finally I would like to thank the department of Statistics and Probability for offering me graduate assistantship so that I could come to the states and complete my graduate studies at the Michigan State University. iii _"\ . ' , .- w l 1" ‘ i b , 1 i' ‘I 3‘ ‘l I ‘k' M 3" ii" i" , 5‘“ ‘\ ‘ u ' i } '.‘ : ., . ‘l i ' r u M 3 5 its. 877 s i i ‘W a f _‘ ‘V V" \ ' ‘31 ‘. . a 1'4"?» , '3‘”; ”w 5% .. if R. "‘ ‘3' ‘ l ' ‘ ' .. ' i~ a. U f Q E F 3" ”’3 ‘r- ”#6:; ‘ --'[~ I‘.‘. a "S a , "‘3“ .’ ""xf—Ih r-«h W PM i sf x.-. l Contents LIST OF TABLES 1 Introduction 2 Sieve Estimation 2.1 The model ................................. 2.2 Estimation ................................. 2.3 Consistency and Asymptotic Normality ................. 2.4 Information bound for 60 ......................... 2.5 An Extension ............................... 2.6 Simulation ................................. 2.7 Proofs ................................... 3 Model Check 3.1 Introduction and Main Results ...................... 3.2 Estimation of 6 .............................. 3.3 A Simulation ............................... Bibliography iv at: cm. vi 11 13 16 20 22 41 41 47 48 57 List of Tables 2.1 3.1 3.2 3.3 3.4 3.5 Simulation results for the Sieve MLE .................. 21 Empirical sizes and powers of Kn, test, 5 ~ logistic (0,;3) ....... 51 Empirical sizes and powers of Kn test, 5 ~ Normal (0, 02) ...... 52 Empirical sizes and powers of En test, 5 ~ Normal (0, 02) ...... 53 Empirical sizes and powers of Kn test, 5 ~ DE (0, fl) ......... 54 Empirical sizes and powers of Rn test, 8 ~ DE (0,3) ......... 55 Chapter 1 Introduction In an interval censoring set up one only knows that an event time X lies in a random interval. In the case I interval censoring model, the event time X is known to be either to the left of the observation time T or to its right. This type of data is also known as the current status data. In the case II interval censoring set up, there are two observation times T and U with O S T _<_ U, and one knows that either X S T, orTU. Interval censored data occur frequently in clinical trials and longitudinal studies. For example, in a long-term follow up study, the subjects are given yearly screening to detect cancer. Cancer onset can only be known to occur between screenings. Hoel and Walburg (1972), Finkelstein and Wolfe (1985), Finkelstein (1986), Diamond et a1. (1986), Diamond and McDonald (1991), Keiding (1991), among others, contain several examples of interval censoring data sets from clinical, tumorigenicity and demographic studies. The recent review article by Jewell and van der Laan (2004) contains some additional applications to health related studies. The first part. of the thesis studies asymptotically efficient estimation of the base- line hazard parameters in a modified Cox model where covariate effects are nonpara- metric when data are interval censored. The Cox’s regression model is widely used in survival analysis. In this model often the baseline hazard is assumed to be nonparametric while the covariate effects are modelled parametrically. In many applications the shape of the baseline hazard is thought to be well understood but the covariate effect is rarely specified precisely. For example, in insurance problems the Gompertz-Makeham hazard has a long tradition of successful application, [Jordan (1975), page 21]. Meshalkin and Kagan (1972) claimed that the logarithm of the baseline hazard is approximately linear for a number of chronic diseases. As an alternative to Cox’s regression model, Nielsen, Linton and Bickel (1998) studied a model where the baseline hazard rate belongs to a parametric class of hazard functions but the effect of covariates is nonparametric. They obtained an asymptotically efficient estimator of the underlying parameter by profile maximum likelihood method when the data are randomly right censored. The estimator of the baseline hazard parameter, called sieve maximum likelihood estimator, is obtained by maximizing the log-likelihood function with respect to both the finite dimensional and infinite dimensional nuisance parameters while the infinite dimensional nuisance parameter is constrained to a subset of the parameter space which increases with the increase in the sample size. The sequence of these estima- tors is shown to be consistent, asymptotically normal, with the asymptotic variance achieving the semiparametric lower bound. This work thus generalizes the work of Lu (2000) from the current status data case to the general interval censoring case. The nonparametric and semiparametric models for interval censored data have been studied in the literature. The monograph of Groeneboom and Wellner (1992) provides some basic results about the information bounds and nonparametric maxi- mum likelihood estimators of a distribution function with current status and case II interval censored data. Huang and Wellner (1995) and Huang (1996) study N PMLE of the linear functionals and underlying parameters in Cox’s proportional hazards model with interval censored data. Klein and Spady (1993) use the profile maximum likelihood method to derive estimators of the regression parameters that achieve the semiparametric lower bound. Li and Zhang (1998) derive an asymptotical efficient M-estimator of the regression parameters in a linear regression model with interval censored data. Rossini and Tsiatis ( 1996) use the sieve method to obtain asymptoti- cally efficient estimators of the regression parameters in a semiparametric proportional odds regression model with current status data. The second part of the thesis pertains to constructing the lack-of-fit tests of a para- metric regression model when the response variables is subject to interval censoring case I. The proposed tests are based on certain martingale innovations of a marked empirical process developed by Stute (1997) and Stute, Thies and Zhu (1998). These two papers study the marked empirical process and its innovation martingale trans- formation for a general regression model. Our work extends the methodology to the interval censored case I data while the inspection time and covariate distributions are unknown but the error distribution is known. The tests are shown to be asymptoti- cally distribution free in the sense that their asymptotic null distributions depend on neither the null model nor the covariate and the inspection time distributions. Several other papers deal with model checking problems under censorship. N ikaba— dze and Stute (1997) use the Kaplan-Meier process to test the null hypothesis that the unknown distribution function of the true survival time is from a parametric family of distributions when the data are right censored. The Kapan-Meier process in their paper has also been transformed to be asymptotically distribution free. Stute (2000) constructs the tests based on the empirical process of the regressors marked by the residuals for model checking in right censored regression. The process in this paper is attached with the Kaplan—Meier weight and the weak limit of this process is a Gaussian process with a covariate function depending on the null model. Rabinowitz, Tsiatis and Aragon (1995) propose a class of score statistics that may be used for estimation and confidence procedures of an acceleration failure time model for interval censored data. Simulations are conducted for both parts of the thesis. To study the behavior of the sieve MLE, a finite sample simulation shows a very desirable behavior in terms of bias and standard error. For the second part of the thesis, the simulation studies assessing some finite sample level and power behavior of the proposed tests for small and moderate sample sizes are also given. Chapter 2 Sieve Estimation 2.1 The model Let (X, T, U, Z) be a random vector, where X represents the survival or event time, T and U are the monitoring variables and Z the covariate which could be a vector. Assume that, conditional on Z, Tand U are independent of X, with a joint continuous distribution function H such that T s U with probability one. In Cox’s regression model, the conditional cumulative hazard rate function of X, given Z, has the form A0(r)eB’Z, where A0, with unspecified form, is called the baseline cumulative hazard function, and [i is a vector of regression parameters. Nielsen, Linton, and Bickel (1998) proposed an alternative model with A0 depending only on some parameter 00 and the covariate effect is of an unspecified form. More specifically, the conditional cumulative hazard rate function of X, given Z, is of the form (2.1.1) Mac, 9o)9(Z), where A(.r, 60) is a known function with an unknown parameter 60 and g is an unknown function. Here do belong to O, a subset of Rd for some (1 2 1. They discussed the estimation of 60 and 9 under the right censoring. Lu (2000) discusses the estimation of 60 and 9 based on the interval censoring case I data. In this paper we discuss the estimation of 60 and 9 based on interval censoring case II data, where one observes independent random vectors (Ti, Ui, 62-, 7i, Z,), with 61-: [(XiSTi)’ 7,- : [(Ti i = 1 Z- . z- +rilog[e—A(Tiv9)ea( l) —e—A(Uz"6)ea( z)l (2.2.1) — (1 — 5,- — 7,)A(U,,9)ea(zz')}. Here a Z _ F(t,z,6,a)=1— e—A(t’6)e ( ’, F(t,z,6,a)=1— F(t,z,6,a). To maximize the log-likelihood over all possible 6 and or, we should set a(Zz-) to be positive infinite if 6.,- = 1, and negative infinite if 6,- = 0 and "y,- = 0. Hence a meaningful maximum likelihood estimator over all possible functions a does not exist. The log—likelihood function is maximized as a varies over a small set of functions which depends on the sample size. More specifically, we approximate a by a sequence of step function with known jump points and maximize the log-likelihood as a varies over these step functions. As the number of steps increases along with the sample size, the bias from the approximation disappears. Assume that the covariate lies in a bounded interval. Without loss of generality, it will be taken to be an interval [0,1]. To construct the step function, define a partition 0 = zo < zl < - - - < zk = 1, where It depends on n and increases with the increase of n. The step function is then defined as k (2.22) 071(3) 2 Z a'n,j1j(3)i i=1 7 where Ij(z) is the indicator function for the jth interval, defined by Ij(z) = 1 if zj __ 1 < z s zj and zero otherwise. For the fixed partition, the step function is completely specified by the parameters (anl, - ' - ,ank). Hence from here on, an will denote either the function on given by (2.2.2) or, equivalently, the vector or, depending on the context. The estimate (6, (in) is obtained by maximizing the approximate likelihood formed by substituting (2.2.2) for or in (2.2.1). Since k is an increasing integer-valued function of n, written as k(n), on will tend to a. The next section show that when k(n) = 0(nr) with 0 < r < 1/2, (6, (in) is consistent. Now define a step function, 0071, of the form (2.2.2) as an approximation to 010. Precisely, km) (2.2.3) cron(z) = Z 010(zj)1j(z). j = 1 Let fin : (6)0711, ' ' ° rank): [3071 = (003 00(21): ° ' ° )QO(Zk))i 180 = (60:00): :é'n = (éién)a and F30“, Z) = F“, Z, 60, (10), F371, (t, Z) = F(l, Z, 6, an), (2.24) FIBOTLU’Z) = F“, 2,60,00n), F371“, Z) = F(t, Z,é,(1n). We shall be assuming A(t, 6) to be twice differentiable with respect to 6, for all t. Let A(t, 6), A(t, 6) denote the first and second derivatives of A(t, 6) with respect to 6. The first and second partial derivatives of the log-likelihood are used to generate the estimates and their variance. In view of (2.2.1), the first derivative with respect 8 to6is Sn1 0(9, 071) _ 8Ln(9,a’n) _ as n 6_F(T’Z) . = 1 Z l . [3” ’ ’ A(T..0)e““(Zz) 7. - lF/jn(U.°.Zi) — FpaniaZill Fanazil - Fpn(T7;.Zi) (2.2.5) x [A(U,~, 6) — A(T,-, 9)}ean(Zi)} ()2 — Ffln(Ti’ Zi) + F312 (T2’ 21) and that with respect to anj is Sn, j (6: an) 8LT), (6, an) Han] - F T..z.-> -ZZ]6{’ if; 2) A “Wt-(2.) =1 n 2’ --[F 3 (Uszi) "'Fp (TM->1- o —F, T-,Zo+ ’7 n F T-,Z- 2 ,3n( 2 2)+ WFflnfU it Zil" Ffin(TiiZi) 512(2 7.) (2.2.6) x [A(U,-, 9) — A(T,-,9)]e“n(zz')1j(z,-)} j = 1,2, . ~ ,k. The score vector is defined as ( Sn, 0(6,(.rn) \ ~ 5 1(9. an) 572(9. an) = n, ( Sn, k(6,(rn) ) The estimates (6, (in) are defined to be a solution to the score equation (2.2.7) sum, an) = 0. The derivative of Sn with respect to (6, an) is called the Hessian matrix and related to the observed information. This is defined as ~ _6S 6, which is the (k + 1) x (k + 1) matrix of partial derivatives with respect to 6 and an of the elements of Sn(6, on). Then the elements of Hn are defined by h00(69 an) = (2.2.8) h0j(03 an) : (2.2.9) BS”, 0(6, an) as ( 1 ” (Si-Fe Ti’Zi) ; Elf .. z: A(T,, 9)e“n(Zz') an(T iv Zi) 72—[Ffinwf’zil”Ffi)~.(:”z‘)Z.‘)l- . . F9. (U..Z.-)—Ffin(r,,z,) 9..(T..Z.) X[A(U.'. 6-) (AT 0)]ean(Z.-)} “ 3.22 {61%.an(7’. HZ-i)A2(T..9)9m"(Zil 6?. _ Ffln(Tii 22') + VZFIBn( TiiZ2)Ffln(Uia Zz) [1%,sz Zr) — Fpn(T.'. Zr)? 1M9. 9) — MT. 9))“e““9}, BS”, 0(6,an) Borj 1 n 5i—Ffln(Tini). a (2.) _ A T-,6 ’1 2 1- Z- ni§1{ Ffinm‘izz') ( z )8 ]( z) 72 — [93,,(11. Zr) - an(T.'. 2.)] F T,Z Ffin(Ui,ZZ-)-Ffin(T,-,Z,-) 3n( 9 2’ ><1MU.-.9)— MT.- 9))e“ ”(91.20) 1 (51F); (3,271). . z. —E Z { F2 211,2.) AlTiaglmTivé’wa 01.1%) i=1 fin Z z 79169an9 36%..“1926 [17an .3 Z.)- FanfT..Z.')l2 ><1MU.-. 9) — MT. 9)llA(U.. 9) — MT. 9))e““9I,-1 >1 ><[A(U-, 9) )—A(T-, 9) )je“n(Z.')IJ-(2,-)} —— 7’9: (62:61” TT”ZZ))A (T,,9)e““n(Zz')IJ-(2,-) fliFfinAT zi Zi)Ffin(UziZz) [1’9"sz Zr) - 1”’;3,,,(T.'.Zi)l2 (2.2.10) [A(U,-, 9) — A(T,-,9)]2e2“n(Zz‘)1-(2,-)} j = 1, . . . , k, Irv-(9,0...) = 0, iyéj=1,2,---,k. In the above expressions the expectation is taken with respect to the true parameters (60, 0’0). 2.3 Consistency and Asymptotic Normality In order to have the consistency and asymptotic normality of the estimator, we use some assumptions. We call the following assumptions Condition A. (1) The real parameter 60 is an interior point of O. (2) Let T, U and Z be the support of T, U and Z, respectively, where Z is a closed interval of R. A(:r,6) is bounded away from 0 and 00 over :1: E T, or :1: E U, and 6 E M, where N1 = {6: I6 — 60] g A} for some 0 < A < 00. The density of (T, Z) and (U, Z) are bounded on T x Z and U x Z, Lipschitz continuous in z, uniformly fortETanduEU. (3) The first and second derivative of A(:r,6) with respect to 6, A(r,6) and A(:r,6) 11 exist, are bounded for a: E T or a: E U and 6 6 N1, and continuous in 6, for any fixed :12. (4) a0(z) is Lipschitz continuous on Z. For any function 12(2) defined on Z, let ||b||oo = supzez|b(z)| and Nb” = W be sup-norm and Lz-norm respectively. Theorem (2.3.1) below states the existence of at least one consistent (in sup-norm) estimator, 6, which is a solution of the score equation. First, let’s define 160’". 2 flan [8071. ———A T, 6 Ffion(T’Z) ( 0) + Fflon(U,Z) ‘F,30n(T,Zl >< WU, 00) — [\(T, 90)]2] 6201mm), F T,Z _ F T,Z F U,Z )6071( )A(T, 60)A(T, 60) + )8071( ) 5077K ) D00(T, U, Z, 60, (iron) = [ D01(T1U)Z’907OOTI) = I: 950nm Z) Ffimw, Z) — Ffimm Z) X Wu, 90) — [\(T, 00)][A(U, 90) — A(T, 60H] amen”), F5 ”(T, Z) 2 Ffl (T, 2W), (U, Z) D11(T,U,Z,90,Oron) [WA (T90) 1230:7311, 2) _ F0929“, Z) (2.3.1) x [A(U, 90) — A(T, 90))2] em‘m(Z), where Ffl0n(T, Z) is defined in (2.2.4). Theorem 2.3.1 Assume that the Condition A holds, and the number of intervals is increasing at a rate k(n) = nr, with 0 < 7' < 1/2. Assume that (2.3.2) P(F(U, Z,6,a) — F(T, Z,6,oz) > c) = 1, V 6 6 9,0, for some 0 < c < oo. Assume also that for all k and don with llaon — Golloo < A0 for some positive and finite number A0, there exists a 0 < C < 00, not depending on n, such that (2.3.3) P(I.(Z)=1)=o(1), kP(I]-(Z)=1)>C, j=1,2,---,k 12 and E[Dm(T U Z 60,00n)1j (Z )12 [011(7" U 2 90,00n)1j(le > (2.34) E[D00(T, U, Z, 60, 0071)] “3.21 E Then, there is at least one consistent (in sup-norm) solution to (2.2.7), i.e. there exists one (6, din) such that I6 — 001+|lén — acne. = 0p(1)- Theorem 2.3.2 Assume that the conditions in Theorem 2.3.1 hold. Assume also k(n) = nr, with 1/4 < r <1/2, and (E[DOI(T, U, Z, 90, ao)])"’ 2.3.5 ED T,U,Z,6,a — ( l [0“ ° 0)] EIDu(T,U,z,9o,ao)J Then the estimator (6, (in) in Theorem (2.3.1) has the following convergence rate |9 - 90| = 01201—1“), lldn - aolloo = 0190771”)- Theorem 2.3.3 Assume that the conditions of Theorem 2.3.2 hold, and 02 defined below is finite. Assume also the third derivative of A(t,6) with respect to 6 exists for 6 in a neighborhood of 60, and is continuous at 60. Then fin; — 90) _. N(0, 92), where the asymptotic variance is given by (E(D01(T’ Ur Z: 90, 00)|Z)l2)] —l. 2.3. 2: ED TUZ9. —E ( 6) 0' ( 00( ’ ’ ’ 0‘00» ( E(D11(T,U,ZaHOaQO)|Z) 2.4 Information bound for 60 The true model has two parameters: 6, a finite dimensional, and a, an infinite dimen- sional functional parameter. The semi-parametric information bound for estimating 13 6 is based on the maximum of the asymptotic variance bounds of regular estimators for 6 obtained using parametric sub-models of a (Bickel et a1. 1993). It is shown in this section that the above asymptotic variance 02 achieves this bound. Projection method is used to find the efficient score for the semi-parametric model and hence the variance bound. The conditional log-likelihood of 6 and a. given Z based on (T, U, (5, '7, Z) is given by (1__ €_A(T’ 9)ea(Z)) [e—A(T’ 9)ea(Z) _ e—A(U, 6)ea(Z)] 6 log + '7 log (2.4.1) —(1 — 9 — 7)A(U, 6)ea(Z). Consider a general parametric sub-model with a = on specified by T (a real number) where %a7(z)|1=0 = a(z) for some function a(z) with Ea2(Z) < 00. Take derivatives of (2.4.1) with respect to 6 and r at (01 = (10, r = 0) to obtain the scores SO(T9 Us 29697160! (10) 6—F (T,Z), : B0 600(Z) Ffl0(T, Z) MT’QO) 7 — [Ffloaji Z) — F[10(T’Z)l Ffi0(U, Z) — Ffi0(T, Z) + 9 — 9130a, Z) + F300, Z) (2.4.2) x[/\(U,90) _ A(T,9O)]eao(Z), Sa(T) U1 Z) 6) ’71 60: (10) 6—FI (T,Z) ‘ _ 30 (l’OZ r _ F50(T,Z) A(T,60)e ( )a(Z) 7 _ [Ffioajizl — 380(71’ 2)] — + 5—H T,Z + , F. T it ) FpO(U,d)-F90(T,Z) 150‘ ) (2.4.3) xA(U, 90) — A(T,90)]e0‘0(zla(2). 14 To find the information bound, project So to the linear span formed from all square integrable Sa’s. This projection is denoted by S a‘ and is computed by solving the following equation, for all Sa’s, Note that E(6|T, U, Z) F50(T, Z), E(7|T,U,Z) = F50(U,Z)-F50(T,Z), Var(6|T, U, Z) = F30(T, 2)}?"30 (T, Z), Var(7|T,U,Z) = [FflO(U,Z)—F,,0(T,Z)] x [1—(F,,0(U,Z)—F,,O(T,Z))], (2.4.5E(76[T, U, Z) = 0. Substituting (2.4.2), (2.4.3) for So and 5,, in (2.4.4), taking conditional expectation given (T, U, Z) first, and then taking expectation with respect to (T, U, Z), we obtain E(D01 (T, U, Z, 60, (10)0.(Z)) = E(D11(T, U, Z, 60, ao)a'(Z)a(Z)), where D01 and D11 were defined in (2.3.1). Taking conditional expectation given Z first, and then expectation with respect to Z, we obtain (2.46) E[E(D01 (T, U, Z, 90, 00)|Z)G(Z)] = E[E(DII(T, U, Z, 60, 00)|Z)a*(Z)a(Z)]. It can be verified that E(D01(T, U, Z, 60,00)(Z) E(D11(T, U, Z, 60,00)|Z) (2.4.7) a*(Z) = solves (2.4.6) and hence also solves (2.4.4). Therefore, the efficient score is given by 15 SO(T3 U3 Z) 6,7160, 0’0) — 80‘ (T: U) Zr 63 7, 60) a0) 6 —- F (T, Z) . E _ 30 a (Z) _ — FIBO (T, Z) 8 0 (ACT, 00) A(T, 60) E (D01(T,U,Z,60,ao)|Z)) (D11(T,U,Z,00,ao)|Z) 7'- [F50(U, Z) *FpO(T.Z)l - a (2) F50(U,Z)—Ffi0(T,Z) F°(T’Z) e 0 + 6—F,BO(T’Z)+ X (Wu, 90) — MT, 90)] — (Aw, 90> — A(T. 90>JE(D°1(T’ U’ 2’ 6‘” “0””) . E(D11(T.U,Z,9o,ao)lZ) The semiparametric information bound is equal to the variance of the efficient score: E[SO(T1 U: Z, 6a 7) 60) a0) _ Sa‘ (T: U: Z) 617a 90) 00)]2 and the asymptotic variance bound is the inverse of the information bound. Take the conditional expectation of the square of the efficient score, given (T, U, Z) first, and then expectation with respect to (T, U, Z) to obtain E[SO(T3U12761776050'0) — Sa" (T, U, Z, 6, ’7, 90, 00)]2 = E[Sg + 8621.. — 2SOSat] (E(D01(T, U, Z1602 ao)|Z))2 E(D11(T, U1 Z) 60: 00)|Z) = E DOO(T1(]:Z)607QO) - In view of (2.3.6), it follows that 02, the asymptotic variance of 6 achieves the as- ymptotic variance bound. 2.5 An Extension In this section, we will extend the above results to multiple interval censoring case. Let (X, T, Z) be a random vector, where X still represents the survival time, Z the covariate, and T 2 (T1, T2, - -- ,Tp), are the vector of monitoring times with P(0 < T1 < T2 < < T p < oo) = 1. As before, T and X are conditionally independent, given Z. Let A = (61,62,--- ,6p) where 6]- : 1(Tj_1< X g Tj) j =1,--- ,p. Set T020, Tp+1=oo. 16 Suppose we observe n i.i.d. copies (Ti, Ai,Z,-) i = 1, - n ,n of (T,A,Z), where Ti = (Ti,1,---,T,"p), Al = (6i,l""’6i,p) and 62'“, = 1(Ti,j—1 < X, S T,,j),j= 1,--- ,p, and T,,0=0, T,,p+1 =oofor anyi= 1,--- ,n Then the conditional log-likelihood of (6,a) based on (Ti, Ai), given Z,,i = n p+l Ln(6,a) = —Z Z 6,-jlog(F( Tijv Z-,6,—a) (T j_1, Z-,6,a)) i=1j=1 . 0(Z') : —_:ZI{62 110g(1—e —A(T,,1,6)e z A(T,-j_1,6)ea(zz 2') -A(T,-,j,6)ea( —8 l “ 6i,p + 1A(Ti,p’ g)ea(Zz‘) 2)], + Z (5,-jlog[e_ j=2 Here 6i,p+1=1_23'=16‘i,j’ i=1,--- ,n. The first and second derivatives of this log—likelihood with respect to (6,071) are Sn, 0(02071) 3142(6) (In) 39 _ 1 " an(z.) , Ffln(Ti,leil- . _ _ . . _ nge z {6”1F9n(Tz',1,Zi)A(Tz’1’9) 6,,p+1A(T,,p,6) F" 17/3,,(T i jiz')A(Ti,ga'6)_Ffin(Ti,j—1,ZilA(Ti,j—ligl} + Z 6,- j=2 j Ffin(T7' J’ZZ) Ffln(Ti,j—1’Zi) Sn,s(9:01n) _ 8Ln(9,0n) _ aOlns F (T Z-) ._.. ea ~ ,dn 2,1’ ”t — —Ze TL( Wi){07’1F,3n (T T,:,1,Z)A(Ti,’-1‘9l 6i,p+1A(Ti,pa9) i=1) F3n(T ij’ Z,)A(T,- j,6)-F,3n(T,',]-_1,Z,-)A(T,',j_1,6)} F3n(T i ,J’Zi—l Ffln(Tiaj—1’Zi) , 8:1,"° ,k. 17 (100(9, an) 65,, 0(0, an) 66 1 n a (2.) Ffin(Ti,1:Zz')~ .. = _ en z 5- AT- ,0 —5- AT. ,0 n: {z’lFfin(Ti1wZi)(z’1 ) 1,P+1(Z,P ) + :p 6 F5710" i j,Z¢)A( Ti,ja9)“Ffln(Ti,j—1,Zi)A(Ti,j—1am} ’l . . . j—2 j Ffincr z J’Zi) Ffln(Tl,J—1’Zz) --Ze2“"(ZZ{21Fg:(%bz2:2)A(Tz-,1,0> j=p F31“ Tijazi)Ffln(Tij—1:Zi) 6 +3232 2j[F5n ( Tij’Zi‘) Ffin(Ti,j-1’Zi)]2 [A(T i j» 6)_A(Ti,j—li0)]2}, hosWfln) _ BSn,0(9,OIn) _ "Bans F (T ,Z,-,) . 6 i,1 = -Zea"(zi (321W 1Fa:(T T2- 1,-Z)A(Ti,1:9>‘5z',p+1A(Ti,pv9> + if)! 5. Ffin ( T239320“ Tm”) Ffinm j-l’ Z')A(Ti 21-1’0)} j=2l,j Ffin(Tiij i") Ffln(Tz,j—1’ZZ) 1 n 2071(2') . . Ffin(Ti,1’Zi) . - 72:8 z 13(Zz){6%1FE (Ti 1,Zi)A(Tz,1’6)A(Tz,1’9) ._ n 3 j:p62' Ffincrzjaz z)Ffln(sz—1,Zz) 2’j[F/3n( TWZ) PM Tin—1,3012 ><[A(Tz',j,9)-A(Tz',j—1a9)HA(Tz' j29)—A(Ti,j—1a9)]}, 18 hssw, an) 357% 0(6. (11;) 66 F' (T 1,Z) = — earl” i)! (Z-) *3" 2’ ‘AT. ,9 —5. AT- ,9 ”2:: 3( {6i’1Ffin(Ti,liZi) (2,1 ) 2,p+1 (2,1) l + 32:5 6- 'Fb’nm' j’Zi)A(Tz°, j60)z—Ffin(Ti,j_1,Zi)A(Ti,j_1,6)} 1 n 2an(Z-) Ffln(Ti,1’Zi) 2 “2 :e 2 13(2) 5- A (T- ,9) z { z’11’§n(Tz',1,Zz‘) 2’1 n i=1 F? F ,Z-F i'—— :Zz' + Z 62]. fin (T 2,] ZZ),8n(T J 1 )]2 [A(T ij, 6)—A(Ti,j_176)]2}a j—Z [Ffin(Tz,j’Z z)‘ Ffinm j—I’Zi2) where s = 1, - -- ,k, and F572 (T, Z) is defined in (2.2.4). Under the assumptions similar to Condition A, we can prove analog of Theorem (2.3.1) to (2.3.3) for this multiple interval censoring case. Note that the bold T is the random vector while T z’ is its ith coordinate. First we need to define Z>- ” F50n(Tj’Z)F60n(Tj—1’Z) 2 Ffim(Tj,Z) - Fflonaj — 1,2) X [A(Tj, 00) — A(Tj _ 1, 00)]2JB 200n(Z) || ":3: 3:9 3 5 D00(Ta Z, 001 0071) F3011 (T3 ' Z)F30n(T_7 -1» Z) ”F50n(’ Zl" Fflon (Tj_1,Z) X[A(Tja 60) _ A(Tj _ 1, 90)][A( T3300) - A(Tj _ 1,00)]] e2aOn(Z) F (71,2) P F! (T,Z)F (T-_ ,Z) fion 2 r3071. .7 6071. .7 1 = —-————A T,0 , [Ffi0n(T1,Z) ( 1 0)+jZ::2Ffl0n(T—J,Z)—Ffi0n(Tj_1,Z) x [1“qu 90) _ A(Tj _ 1,60)]2] e2aon(Z), FBOn(T11Z-) Ffi0n(T1vZ) P D01(T,Z,00,aon) = [ A(T 00mm 90) )+ j: Dll(Ta 270010071.) where Ffi0n(T, Z) is defined in (2.2.4). The consistency of the sieve MLE can be proved similarly as Theorem (2.3.1) 19 when we change the assumption (2.3.2) to P(F(Tj,Z,9,a)—F(Tj_1,Z,g,a)>C)=1, j=l,---,p+1, V669,a, and the assumption (2.3.4) to ED01(,,TZ€0,007113 Z _Z l )()]2 > C. E[D11( (T Z 90,00n)13(Z)] E[D00(T, Z, 90, 0011)] The consistency rate of the sieve MLE is still 0p(n”1/4). The proof is analogous to that in Theorem (2.3.2) except that we need to change the assumption (2.3.5) in Theorem (2.3.2) to be (E[D01(T,Z,60a010)])2 E[D11(T, 2,60100” E[Doo(T, 190,00” - The proof of the asymptotic normality is similar to that of Theorem (2.3.3). The asymptotic variance now is 02 = E(DOO(T, 2, 90a,» — E((E(D°1(T’Z’OO’QO)|Z))2)]—1. E(D11(T1 Z190,00)]Z) and this asymptotic variance also achieves the information bound. 2.6 Simulation A simulation is presented before we go to the proofs of the stated asymptotic prop- erties of the estimator. Assume the conditional distribution function of X given Z is the Weibull distrib— ution 1_ e_$60600(2) where (10(3) 2 log(z). Also assume that U and T are uniformly distributed on the upper triangle of [1,2] X [1,2] and Z is uniformly distributed on [0.2, 1.2], and true parameter 60 = 2. 20 For each fixed sample size (n=30, 60, 100, 200 respectively) and appropriate k’s, k is the number of jumps in the step function and increases with the increase of the sample size n. 100 replications of the estimate of 60 based on the sieve maxi- mum likelihood estimator are obtained. The means and standard deviations of these estimators thus computed are reported in the following table. Table 2.1: Simulation results for the Steve MLE n=30 n=60 n=100 n=200 mean std mean std mean std mean std k=1 1.9855 0.2673 1.8380 0.3417 2.0189 0.3150 1.9880 0.2565 k=2 2.0357 0.3969 1.9430 0.2608 1.8871 0.3106 1.8022 0.1886 k=3 2.0975 0.4351 2.0002 0.1892 1.8456 0.2399 1.7890 0.1522 k=4 2.0529 0.2704 1.8907 0.2638 1.8676 0.2185 1.7865 0.1633 k=5 2.0067 0.3790 1.8841 0.1979 1.7821 0.1715 k=6 1.9103 0.3303 1.9250 0.2121 1.9425 0.2278 k=7 1.9241 0.1781 1.9200 0.1571 From the above table we can see that the mean is around the true value for all the sample sizes and the standard deviations decreases with the increase of the sample size n and k. 21 2.7 Proofs Before we go through the proofs, first we need some notation. If a is a vector with elements aj,1 g j _<_ m, then Han... = max [0]"- ISjSm If A is an m x m matrix whose (1', j)th element is denoted by aijv then A 00 = .. II n 1 £23? m 231w Proof of Theorem (2.3.1) Recall the definition of 01071 in (2.2.3). The Lipschitz continuity of (10 implies that (2.7.1) “0071 — OOHoo = 00600—1)- Note that $710371) = 0 is equivalent to ( Sn, 00371) \ (2.7.2) Sn(fin):= k3,, .163") :0. ( k8”, 1.9312) ) The derivative of 571(371) with respect to fin is ( (1000371) (10103") hokfgn) \ (2'7'3) Hn(/3n) := 010371) “(’3") O 0 hkhokmnl 0 0 kthfinU where hij is defined in (2.2.8), (2.2.9) and (2.2.10). The lower-right k x k sub-matrix is a diagonal matrix. 22 Let #(fin) = ESan). and F130(T,Z) - F611 (T’ Z) 071(2) W1(T, U,ZaIBn) = Ffln(T Z) 8 a W2(Ta U1 Z1671) = [F180(T’ Z) - Ffin(T’ Z) Fa (U, Z) — F50(T, Z) —- (Ffin(U, Z) _ Ffin(T, 2)) _ R Z + 0 1773,,(03 Z) _ anC’X Z) Ffln(T, Z)]ea ( ), Ffio (T, Z)Ffin (T, Z) 2an(Z) 17511012) 6 ’ W4(T, U, 2,571) = H) [Ffinag’m _Ffi:(T,Z)]2 e2 ( )’ W3(T,U,Z,fin) = (2.7.4) where Ffl0(T, Z) and Ffln(T’ Z) is defined in (2.2.4). Then by (2.7.2), (2.2.5), (2.2.6) and by (2.4.5) we obtain (fl ) kE(W1(T» U, Z, [inlMT’ 9)11(Z) + W2(T,U, Z,/1n)[A(U,9) ‘ MT, ”111(le # n = ( kE(W1(T,U, Z, fin)A(T,0)Ik(Z) + W2(T, U, Z, Bn)[A(U, 9) — A(T,0)]Ik(Z)) ) (2.7.5) By the Condition A, F(T,Z,6,a) is Lipschitz in 0, a, uniformly for (t,z) E T x Z. It is easy to check that p.030) 2 0 and Hymn)“00 = 0(1) if “fin — 50n|lm = 0(k‘1) and P(Ij(Z) = 1) = 0(1) for j = 1, - u ,k. Let 2(3)?) = EHn()’3n). By (2.7.3) and the definition of hijmll)’ 0 g i,j S k, 23 ( boo<6n> balmy.) b0k(.3n)) kbmwn) kbuwn) 0 0 O 0 \kbokmn) 0 0 bkkWn) ) SW”) = where on .- 600%) = E(W1(T, U, Z, fin)A(T, 6) + W2(T, U, 2. 6n)[2i(U, 6) — MT, 6)] — W3(T, U, Z, Un)A2(T, 6) — W4(T, U, z, 6n)[A(U, 6) —- A(T, 6)]2), b0j()3n) = E( [WI (T, U, Z, 6n)/\(T, 6) + W2(T, U, z, 6n)[A(U, 6) — [\(T, 6)] — W3(T, U, Z, fin) x A(T, 6)A(T, 6) — W4(T, U, Z, 6n)[A(U, 6) — [\(T, 6)] x [we — AU", 6)1]Ij<2)), hymn) = E( [W1(T, U, 2. 6n)A(T, 6) + W2(T, U, z, 6n)[A(U, 6) — A(T, 6)) — W3(T, U, Z, 6n)A2(T, 6) — W4(T, U, Z, 6n)[A(U, 6) — A(T, 6)]2] 13(2)), Qflmazo i¢j=Luwh with W1 to W4 as in (2.7.4). The inverse of BUM) is (100 (3.4901 (2.7.6) 23—106”) = , (161 ((16111 where k b2. ’1 0.7 = b _ _ (100 00 .2 b . . , qm is a row vector with its jth element b . 0 . _qOOb—iy.’ ]:1,2)..'7k3 J] 24 and q“ is a k x k matrix with its (2', j)th element b -b - . . —l 02 0] . 1: ... Since #030) = 0 by (2.7.5), and by Condition A , g(fin) is continuous in 131), by (2.7.1), (277) lll‘(»3on)iloo = 0(1)- Since E(,3n) is continuous in fly; by Condition A, 2’1 ([3071) exists and ”2’1 ([3071) ”00 < c for large n by (2.3.2), (2.3.3) and (2.3.4), it follows from (2.7.7) and the inverse func- tion theorem (IFT) with sup-norm (Lemma 1 of Rossini and Tsiatis (1996), which is stated in the following lemma) that there exists fin = (5,577)), with an of the form (2.2.2), such that (2.7.8) (1%) = 0, and (2.7.9) “.371 — fiflniioo = 0(1)- Next, suppose that there exists some finite constant c such that (2.7.10) IlSn(3n)lloo = 019(1) (2711) Palm-.1665“... < c) —> 1, then by IF T again, with probability tending to 1, there exists solution fin = (9,677)) of the equation 371.0311) 2 0 such that “.372 — :371iioo = 012(1)- 25 This, (2.7.9), (2.7.1), and the triangle inequality imply that ”372 “ fiolloo = 012(1)- This completes the proof of the theorem as soon as we verify (2.7.10) and (2.7.11). Next, we shall prove (2.7.10) and (2.7.11). To prove (2.7.10), fix a 5 > 0, from the definition of 372(572) in (2.7.2), we observe that (2.7.12) Panama“... > a) s P 05., 0(3n)| > g) + P( sup Iksn, Mn» > g). ISJSk Rewrite S,(nO(BTI)=n 12A2,n ”2:1 where {AZ} n}15¢'sn is the 2th summand in the r.h.s. of (2.2.5) with [3n replaced by fin. By the Condition A and (2.3.2), there exists a constant C such that supl < 2' < 72 [A4, 12' S C for all 72. And by (2.7.8) we observe that Then Chebyschev’s inequality implies that ~ 5 402 (2.7.13) P 0872, 0(,3n)l > 5) < ”—53 The second term in the upper bound of (2.7.12) converges to zero by Bernstein’s inequality: (27-14) P( SUP lkSn,j(/3n)| > 5) Sk SUP PSnOk J‘Wnll > 52-) 1 S j S k Here rewrite where {82, n}ISiSn is the 2th summand in the r.h.s. of (2.2.6). Similarly, by the Condition A and (2.3.2), there exists a constant C such that supl < ,- < n le', nl g C for all n, we also have E33,” = 02 < C2, EIBMIP g Cp’zp!EBf,n v 2: 1, - -- ,n,v p 2 2. By (2.7.8), EB), ”((321) = ES”, 3(6),) = 0 v 2'=1,--- ,n. Apply Bernstein’s inequality to obtain: P(|kSn,j(/§n)[ > g) = P0: BMW” > 3) 2:1 I/\ n52 (2.7.15) 2exp (— 1616202 + 40165) . This together with (2.7.14), we obtain ~ 5 n52 2.7.16 P 163 ' 3' > - < 2k - . ( l (1 £33.93 kl 71,20 ”ll 2) - eXp ( 16k202+40k5) Combine (2.7.12), (2.7.13) and (2.7.16) to obtain: Pnw > a) = o (i + [mm—g) This proves (2.7.10) upon taking k = 0(n7) for some 0 < '7 < 1/2. Now we shall prove (2.7.11). Since ( 600(6)» hm") hog/37.)) ~ khmOrjn) kh11(/3n) 0 0 41720371) 1: , I 0 ‘.° 0 ( kh0k(_3n) 0 0 (chm/3n) ) 27 ~ the inverse of Hn(/3n) is - ‘ k-l 7' (2.7.17) H5103”): (200(572) 0010312) , 061(372) (94011672) where 1 ~ ~ I: h2- ~ '— 000(13n): ((1000372) — 2 72049370) , a01(fin) is a row vector with its jth element §~ ~ —a00(/8n) 0] (1377,), j =1a21' ° ' 7k) .7 h] and 0110672) is a k x k matrix with its (2',j)th element 1. -h‘l(3)+a <6>h°ih°j<6> -—12--- 6 By the definitions of hijwn) and bij([3n),0 g 2', j S k, and the law of large numbers for triangle arrays, we obtain Ihoown) — b06671» L o lh0j(3n) - boj<6n)| —”—> o Ihjjwn) — bfi-(fan)! 1» 0 0 S j S 12. Together with the definitions of 000,001,011 and qoo,q01,q11, we get (2'7'18) i000(3n) — qoo(,3n)| —p—) 0 ”0010272) _ C101mm)” L 0 Had/3n) — q11(3n)II—:> 0. By the facts that Z‘1(,13n) is continuous, llz-l(50'n.)lloo < c, and H377, — 6072““, = 0(1), we obtain HE‘ICS’nHIQO < c, this together with (2.7.6) , (2.7.17) and (2.7.18), we proved ||H731(/3n)||00 < c with probability approaching 1. 28 Lemma 2.7.1 (Inverse Function Theorem with Sup-norm). Let A(x) be a contin- uous differentiable mapping from Rm to 72'" in a neighborhood of 1:0. Define the Jacobian as the m x m matrix H (x) = 6A(x) { derivatives of the elements of H with respect to the elements of x). If there exists constant C and 6“ such that ”114(30le < C and sup HHCC) - H($o)||oo S (20)”1, x : ||x — x0||¢o < 6* then for d < 6*/(4C) and all y such that My - A($0)Hoo < d, there exists a unique inverse value x in the 6* neighborhood of x0 such that A(x) = y and Hx — xoll < 4Cd.(Rossini and Tsiatis 1996) Proof of Theorem (2.3.2) We are going to use some general results on the convergence rate of sieve esti- mations. The following lemma is a part of Theorem 1 of Shen and Wong (1994). To state the lemma, we introduce some general notation. Let Y1, - - - ,Yn be a se- quence of independent random variables (or possible vectors) distributed according to a density p0(y) with respect to a o-finite measure it on a measurable space (37, B) and let 9 be a parameter space of the parameter 6. Let f : 9 x y —+ R be a suitably chosen function. We are interested in the properties of an estimator fin over a sub- set 8n of 9 by maximizing the empirical criterion C7103) = £22; ((6, Y2), that is, ann) = man/3697, 071(8). Here 9n is an approximation to 6 in the sense that for any [3 E 8, there exists 7rn/3 E 9n such that for an appropriate pseudo—distance p, p(7rn)6, )3) —+ 0 as n —> 00. The following additional assumptions are needed for the lemma. 29 CO. 8 is bounded. C1. For a given 1’30, 3 constants A1 > O and a > 0, such that for all small 6 > 0, 'f E8 ,Y —6,6,y 22A 2a. p<6,60)‘>“.,fi€en ((60 ) ( )) 1. C2. For a given [30, 3 constants A2 > 0 and b > 0, such that for all small 6 > 0, . b inf Var(€()30,Y) — [(13, Y)) S 2A262 . p(/j2180) S 635 E an C3. Let Qn = {£03, .) — “rm/30,) : B E @n}. For some constant r0 < 1/2 and A3 > 0, H(e, Qn) S Agnmllogé) for all small 6 > 0. where H (e, Qn) is the Loo-metric entropy of the space Qn, i.e. exp(H(e, Qn)) is the smallest number of c-balls in the Loo-metric needed to cover the space Qn. Lemma 2.7.2 Suppose Assumptions CO to C3 hold. Then (Kiln, (30) = 0p(ma-X("_T: “Wm/30, (30), K216 (Wm/30, (loll) where K(7rn[)’0, [30) = E(€(;30, Y) -— “fin/30, Y)) and 1426231 - 165%.? .f b 2 a, fig, if b < a. From the proof of Theorem 1 of Shen and Wong (1994), it is noted that the global T: maximizer could be replaced by a local maximizer around the real parameter and the convergence rate is still true for the local maximizer. In this situation, the sieve 9n is a sequence of shrinking neighborhoods of the real parameter fig. To apply the above Lemma (2.7.2) to our case, let Y = (T, U, Z, 6,7), {3 = (9,0), 7777,73 = (than) where an is of the form (2.2.2) with O'nj = a(z]-). Also let an ={(9,Otn)1|9 — 60| S an, “an — aOHoo < bn}, 30 where an and fin are chosen such that, with probability approaching 1, the MLE (6,031) is in 97). Define the metric as follows (27-19) P03, fio) = l6 "- 9o| + H0 - aolloo, and let up, Y) = {610g (1 - e—A(T’ ”60(2) ) + 'ylog [e—A(T’ ”60(2) — e_A(U’ g)ea(Z)] —(1 — 5 — 7)A(U, 0)ea(Z)}. We shall now verify the conditions C0 - C3 of Lemma (2.7.2) for this 3. Note that in the proof below we always denote C as some finite and positive number. Under our assumptions, CO holds. Note that Elm, Y) _ — E,“_,_...,..,.a.6,,og,,_e_,,,.,,,.(z>, +[e-A(T. 60)e0‘0(Z) _ e-A(U. 60)e00(Z)] 10g [e—A(T, 6)ea(Z) _ e—A(U, 0)eo‘(Z)] ._e—AfU, 90)BOO(Z)A(U, (”80(2) } The Taylor expansion of [(6, Y) with respect to 6 and 0 around (90,020), and the fact that the expectation of the first derivative of [(6, Y) w.r.t fl vanishes at 30 and the matrix of the second derivatives is negative definite by (2.3.5), we obtain (37-20) ENG/30, Y) — ((5, Y» Z Cp2(fi. 50). for some finite and positive number c. Hence the condition C1 is satisfied with a = 1. Note that the Condition A implies that for all y (2.7.21) 12630.6) — 6.6, 361 s C(I6 — 601+ Ha — Gellool- Hence Va"‘(f(/30,Y) - ((5, Y)) S E(€(/3o, Y) - €03, Y»? S 06203.30)- 31 Thus the condition C2 holds with b = 1. By (2.7.21), we also have where H (7), 9n) is the metric entropy of the space 972 with respect to the norm |0 — 90' + Ila — OtoHoo- Since 972 is a sequence of shrinking neighborhoods of 60 = (60, do), there exists a positive and finite number C0 such that [6| S Co and Han” _<_ C0, (6,an) 6 8n, and an is of the form (2.2.2). For any n > 0, divide the interval [0, Co] into small intervals, with length at most 27 / 2, such that the number of intervals is less than or equal to 2C0/n + 1. Then it is easy to see that (.7... 1......) g (13:2 .1) (2% .1. 9“”) . 61.... G) as n is small enough. Hence, by (2.7.22) and (2.7.23), for all small 6 > 0, H(€1Qn) _<_ 0km) log (1) = On.) log (1) . Z 2 Therefore C3 is satisfied with 70 = 521. Thus lemma (2.7.2) is applicable to this f, which in turn yields that (2.7.24) p(1§n,fio) = 0p(maX(TI—T1P(7Tnl30s 130), K1/2(7Tn,80, :30)», where 1—7 loglogn T: 2 Zlogn ' Note that, for large n, 1/4 < 7 < 1/2 implies that 1/4 < r < 3/8. Since 60 = (60,020), nnfio = (60,a0n), where am is of the form (2.2.2), by (2.7.19) and (4) of the Condition A, we obtain that p2(7'l'n,3(),,80) = ”(10-71 — Gollz _<_ Ck(n)72 = 071—27. 32 Thus (2.7.25) p(7rn,,z30,fio) S C72”. The same argument as that leading to (2.7.20) gives that, (27-26) K(7Tnl30150) = E(€(fio1 Y) _ [(Wnflo, Y» S Cilaon — 00”2 = 071-271 which is of order between 0(n’1/2) and 0(n’1) for 1/4 < 7 < 1/2. It follows then from (2.7.24), (2.7.25) and (2.7.26) that for 1/4 < '7 < 1/2, 10(3n150) = 0p(n_1/4)1 thereby complete proving Theorem (2.3.2). Proof of Theorem (2.3.3) Recall the definition of Sn, 0(6, 0) from (2.2.5). Furthermore let a be a measurable function on Z with Ea2(Z) < oo, " 5'-F (T2321) . Sn(0,a)[a] = 71—1,;1{ 21835.2.) A(Ti,6)ea(Zzla(Zz-) 72' — [FfifUszz'l ‘ F.6(Ti’zm Ffi(U,—,Z,-) - PAP-Fuzz“) + 6,- —F.(T,-.Z.~)+ F6062.) >< [A(U1-10)- A(T,,e)).a(Z.->.(z,)} where Ffi(t, z) is defined in (2.2.4). Denote the expectation of Sn,0(6,a) and Sn(6.a)[a] by u0(6,a) and u(¢9,a)[a] respectively. By (2.4.5), we obtain F547", Z) — F50", Z) ' a Z F(T,Z) A(T,6)e ( ) 611W) = E{ + [1760(T1Z) — 12x12) FIL30(U’Z) - F80(TvZ) — lF(3(U»Z) — FS(TvZ)l _ + Paw. 2) -— F)3(T, 2) 123(2, 2)] (2.7.27) x [A(U, 6) — A(T, 6)).G(Z)} 33 and 6130(7", Z) —— F50, Z) a Z F),(T,Z) A(T,6)e ( )a(Z) M91000] = E( + [Ffi0(T, Z) — Ffi(T, Z) F50(U, Z) — Ffi0(T, Z) — [F3(U, Z) — Ffi(T, Z)] F5(U, Z) — Ffl(T, Z) F(:r, 2)] (2.7.28) x [A(U, 6) — A(T, 6)]ea(Z)a(Z)}, where F60“, z) and Ffi(t, z) are defined in (2.2.4). The method used here is similar to that described in Huang (1996). From Lemma (2.7.3) below, we obtain the following stochastic equi-continuity results, for every 0-FBR(T621) Ffin(T,-,Z,) x x [AM-16> — A(T.,é>ieén(zz‘>[a*(Z.-> — aw.» def = I+II, where ) n 6-—F.~ (T- Z _1 7’ Illn 2, Z a Z a: t I 1.12, F. (71.2.) Mame "( 61. (Zn—awn]. i = , n n 7. — [F ((11122) - anm" Z.,-)] 1 fin II = — 6-—F~ T-,Z.- + , Ff T- Z- 712. = 1 2 ,.3n( 2 z) FBnaji, 57;) — F3n(Tz, ZZ) )3'n( 7’ l) MAR/1.6) — A(T.,é>1e“n(zi)[a*(Z.-> — a;.J. Rewrite I as 1 n 6. _ F (T.,Z.) . A, . .., . I = 1;. 1F. f%.,2.)‘ A(T119)e"”(21)[a —an1 ’I. :1 167?, Z ’1. n F/ (T',Z‘)—F" (T,Z) . 1 230 2 2 fl 2 1 - z. . ‘ 412.21 Ft? (T232?) A(Ti19)ea"( z)[a (Zr) -an(Zz')l- = n By (2.7.36), Theorem (2.3.2) and the Lipschitz continuity of F with respect to 0 and a by Condition A, the second term is op(n’1/2) . For the first term, by Theorem (2.3.2) and Condition A, we can write it as 1 Z": 6i—FBO(Ti’Zi) 350(Ti,Z,-) n2.___1 F50(Ti1Zi) F3n(Ti1Zi) 1 n 6._F (TH,Z) . ,. = _ Z z [’0 ‘ ' (1+0p(|9-90l+||51n‘00H))(A(Ti’90)+019(l6’60l)) 712,:1 Ff30(Ti’Zi) (.aaz.) + 01.616. - (mill) 1.12,.) — awn}. A(Ti, .).an— —— C __ {f(6,7,t,u,z,6,a) — Ffi(t,2) A(t16)ea Z Ffio(t,3) 8/ ‘ [Fg(u,2) - Ffl(t,z)] _ ' — . 0(z) F13(u12) — FB(t,z) Ffi(t,2)] X 1401.0) A(t,0)]e - F u, — F t,z) _ . . 7 #50752)? Ffiolalfiz) lFfio(t’Z)l X [A0590 ’ A(t,60)]600(2) 1 (2.7.38) |6 — (Jul 3 Cn—1/4,||a — aoll s Girl/4}, +[6 — Ffi(t,z) + —[5 — Ffl0(t,z) + that is, by the functional notation used in van der Vaart and Wellner ( 1996) for the empirical processes, 777(5)?” 0(910) — #0(910)) — 75091;, 0(90: 00) — #0(90100)) (2.7.39) = flu)”, — P)f((5,'), t,u, z, 9,02), where Pn is the empirical measure based on (62-, '72-, Ti, Ui, Zz),i = 1, - -- , n and P is the probability measure of ((5, 7, T, U, Z) with respect to the real parameters (60, 00). Note that under Condition A, functions in C are uniformly bounded for large n, and (2H740) if(6271 ta us 2, 61 C!) _ f(6aA/a tau: 2160100” S. 00(i6 _ 00' + Ha — OOHOO), for some finite and positive number C0. Therefore, C is a set of functions which are Lipschitz in parameter (19,0) 6 D, where 'D = {(0 — ()0, oz — (10) : a is of form (2.2.2), If) — ”0| S Cit-1M, Hoe — (70” S 011—1/4} and the norm in LOAD) is |l(01,al)—(02, 02)le = (01—02|+||al—02H00. By Theorem 2.7.11 of van der Vaart and Wellner (1996), the metric entropy of C with bracketing 38 with respect to L2(P) norm H[ 1(e,C,L2(P)) S H(e/c,’D,Lo.), for some finite and positive number c. When we prove Theorem (2.3.2), we already obtained H(e, D, Loo) S C1k(n)log(1), c for some finite and positive number C1. Hence H[ ](€,C,L2(P)) S C2k(n)log(1), c for some finite and positive number C2. It follows that for any 6 > 0, there exists 0 < C3 < 00, not depending on n, such that J[ )(6,C,L2(P) (1g foe “17+ H[ ](t,C, L2(P))dt S C3k(n)1/2el_’7, for any n > 0. This and the fact that k(n) = n”, with 0 < 7 < 1/2, in turns imply that (2.7.41) J[ 1(Cn'1/4,C,L2(P))= 0(1). Note that f(6,7, t, u, z, 60, 020) = 0 by (2.7.38). This fact and (2.7.40) imply that, for any f E C, (27.42) P(f2(6, 7, t, 1..., 6a)) 3 Curl/2, for some finite and positive number C4. Apply Lemma 3.4.2 of van der Vaart and Wellner ( 1996), which is stated in the following Lemma (2.7.4). Let Y, = (6,-,7,,T,-,U,~,Z,-),i = 1,--- ,n,.7: = C and e = Cn‘l/d. By (2.7.42) and f is bounded, f E C, the conditions of the lemma hold. It follows from the lemma and (2.7.41) that 772.13% sup |(Pn—P)fl)=01‘1(1)- f E C In view of (2.7.39), we obtain (2.7.37). The lemma is proved. 39 Lemma 2.7.4 Let Y1, Y2, - -- , Yn be i.i.d. random variables (or possible vectors) with distribution P and let Pn be the empirical measure of these random variables. De- note Gn = fi(Pn — P) and “Cu“; = supf E j: ICan for any measurable class of functions F. Denote J[ ](€,.77,L2(P)) = [0‘ \/1+ H[ ](t,.7:,L2(P))dt. Let f be a uniformly bounded class of measurable functions. Then 1: J[ )(62f2L2(P)) Ellonllfsm, ,(.,r,r.(p))(1+ 6% M , if every f in .7: satisfies Pf2,e2 and ||f||oo S M. Here E" means outer expectation with respect to P. 40 Chapter 3 Model Check 3.1 Introduction and Main Results The purpose of this chapter is to develop tests of lack-of-fit of a regression model when the response variable is subject to interval censoring case 1. Now let Y0 denote the times of the onset of an event and T0 the time of inspection. Suppose, additionally, one is interested in assessing the effect of a covariate Z on the time of the onset of the event, for example one may wish to asses the effect of the age of a patient on the time of onset of a disease in the patient. One way to proceed is to use the classical regression analysis where one regresses Y := log Y0 on Z but one observes only (6, T) with T = log T“. But then the question of which regression model to chose from a possible class of models becomes relevant. More precisely, assume Y has finite expectation and let 12(2) := E(Y|Z = 2) de- note the regression function. Let M = {m9(z) : z 6 IR, 9 E 9} be a given parametric family of functions, where 9 is a subset of the q—dimensional Euclidean space Rq. This class of functions represents a possible class of regression models and the problem of 41 interest is to test the hypothesis H0: u(z) = m90(z), for some 60 E 9, V2 6 IR, based on n i.i.d. observations X,- = (6,-,T,,Z,-),1 S i S n on ((5,T, Z), where 6 = I (Y S T). The alternative of interest is that H0 is not true. In the case Y,’s are fully observable tests for the lack-of-fit hypothesis H0 have been based on the marked residual empirical process (3.1.1) "171: Y 86:22? )I(Z, g z), z 6 IR, where 6n is fl-consistent estimator of 60 under the null hypothesis, and 33(z) is the conditional variance of Y —— m9(Z), given Z = 2, under Ho, cf., An and Cheng (1991), Stute (1997), and Stute, Thies and Zhu (1998), among others. The last paper showed that the tests based on its innovation martingale transforms are asymptotically dis- tribution free. Our focus here is to develop an analog of this transformation for the current status response data when the error distribution is known. Since the Y,-’s are not observable, we need to replace them in (3.1.1) by 17,-, a cepy of (3.1.2) 1? = E(Y|6, T, Z) = 5130/15, T, Z) + (1 — 6)E(Y|6, T, Z). To proceed further, let F denote the d.f. of the error 5 := Y — ,u(Z). Assume that (3.1.3) F is continuous, 0 < F(y) < 1, for all y E R, E5 = 0, E52 < oo. 5 is conditionally independent of T, given Z, and T is independent of Z. 42 Then, with F := 1 — F, we obtain, t [_.. y dF(y - #(le F(t - (1(2)) t- f .. “My my) (3.1.4) ems =1,T = t, Z = z) = +/.l,Z, F (t - 11(2)) ft°° y dF(y - 14(2)) 1 - F (t - U(Z)) If: ”my dF(y) F(It - 74(2)) E(Y|6=O,T=t,Z= z) + 12(2), t, z E R. Let R(6, t, z) :2 E(Y|6,T = t,Z = z) — ;i(z), V(Z) :=/_ y dF(y) 02(2) 2: Var(R(6, T, Z)|Z = z), t, z E R. From (3.1.2), (3.1.4) and the fact V(OO) = 0, we obtain t— ”(2) oo dF R(5,t,z) = (sf-0° dely) +(,_,)ft—_,1(z)y (y) F(t - 11(2)) 66 — 14.2)) 6 1 — 6 ___ ”(t ‘ “(We — 11(3)) ’ F(t - #(lel V(t - 6(2))[6 -F(t - U(z))l F(t - 14(2))“t - 11(2)) ' By the conditional independence of e and T, given Z, E{R(6, T, Z)|T, Z} = 0 and V2(T - #(le (3.1.5 0 B o G, D[—oo, oo], in uniform metric. Thus, for example the test that would reject f1 whenever Kn := sup, 6 IR |V72(z)| > b0, where ba is 100(1—a)th percentile of the distribution of supO S t S 1 |B(t)| would have the asymptotic size a. To discuss the more interesting problem of testing H0, we proceed as follows. For convenience, let P9 denote the joint distribution of (6, T, Z) when p = mg, and E9 and Vera denote the corresponding mean and variance operations. Let R9, 0,9 stand for R, a when [.2 = mg and 6n denote a fi-consistent estimator of 60, under H0, based on (6,, T,, Z,);1 S i S n. Tests of H0 will be based on the process 971(2) 2: Vn(z, 67.), where 1 " R9(5-,T-,Z-) V(z,6) = — z 1 21(Z-Sz), zElR,6€€). n «77,; 06(Z2') 2 To analyze asymptotic behavior of Va, we need to make the following assumptions. (3.1.6) «6116. — 601 = ope), (P9,) (3.1.7) The d.f. F has a continuous density f and m6 is differentiable in a neighborhood of 60 with its q x 1 vector of derivative moo, so that there exists a family of q x 1 vectors of functions 960(t, z, 6), z 6 IR, t 6 IR, 6 E 8", which is R t,z.(5 the derivative of 9( ( ) ) with respect to 6 at 6 = 60, such that \7’ 0 < b < oo 06 Z R (T,Z-,6) R9 (Til-.64) , sup U17 9 1,2? I — 0 ‘ ’ — (6 — 60)960(T,-,Z,,6,) = 0,,(1), fillejgolléh 06 l) 060(Z,) 1 S 2 Sn 44 (3-1-8) E00”9()0(Tia 26156)||2 < 00- Let V(x) denote the derivative of V(x) with respect to x E IR and R9 denote the vector of the first derivatives of Hg with respect to 6. Direct calculations show that with x = t — m9(z), - —1'/x (5—Fx ux x x1—2Fx , R,(,,,z,=[ ()1 “358% ()f()+V(,)[5_F(I)]f((1)($)p($)()2)) ",6,, Let h.(z) ;= 36133E980(6,T,z), zElR,6E6 lI(T - m6(2)) f(T - m6(2))) 897’”) ‘= E( (F-F)(T—m.(z)) Use a conditioning argument and the independence of e and T, given Z, to obtain . . Th90(z) EQORQOM, T, z) = [90(z, H)m90(z), h90(z) = 690(2, H) 00 (z) . 0 Next, let 129(1), T,Z) D := —— < 2. IR . g(z) E),{ 06(2) I(Z _ )}, z e ,6 e 9 Note that the independence of T and Z enables one to write (3.1.9) D90(z) = /-z h90(u) dC(u), z E IR. Arguing as in Stute (1997), which uses a standard Taylor expansion, a Glivenko- Cantelli type of an argument and a weak convergence argument, we obtain the fol- lowing Theorem 3.1.1. Under the assumptions (3.1.3), (3.1.6) - (3.1.8), we obtain that uniformly in z E IR, under P90, Vn(Z) = W1(Z,60)+n1/2(6n — 90)’D60(Z) + 019(1). Moreover, Vn(-, 60) =:> B(C(-)), in D[—o<:, 00], with respect to the uniform metric. 45 Next, we develop an analog of the linear transformation of Stute, T hies and Zhu (1998). Let 6.,(z) = [”h.0(.)h.0(.yda(.), / 00,, (u,H)m9°(1:,)m6°(uy dG(u), - zEIR. z 0 090W) Note that this is a nonnegative definite q X q-matrix. But we shall assume that 1190(2) is nonsingular for all z < 00 and define the linear functional transform (Qchz) = 6(2)— / h..(z1)'A.;(zl)[/°°690(z2)1o(dz2)]da(zl), wk. 1 -00 When we apply Q to Brownian motion BOG, the inner integral needs to be interpreted as a stochastic integral. Observe that (3.1.9) readily implies Q(D’00U) = 0, for any random vector U. Arguing as in STZ, one can also verify that Q maps B o C to B o C. Consequently, we have Q(B o G + DSU) = Q(B o C) = B o C, for any random vector U. These observations together with Theorem 3.1.1. suggest that under H0, Q17” would also converge weakly to B o G. But the transformation Q depends on the unknown parameters 60, H and G. Let hn, An and on denote the hgo, A60 and 090 after 60, H and G are replaced by 671, and the empirical d.f.’s Hn and Gn, respectively, in there. Define an estimate of Q to be 2 oo (an)(z) = 99(3) — /_oo h12(ZI)IAnl(Zl) [f2] h72(22)99(d32)] dGn(ZI)- 46 To verify the weak convergence of Qn 1777, we need the following additional smooth- ness condition on h6(z). For some q X q square matrix h60(z) and a non-negative function K0(z), both measurable, the following holds: Ellh6.(Z)lleo(z) < .0, Ellh6.(Z)llllh-6.(Z)||j < oo, 1': 0,1, and Vc > 0, there exists a (5 > 0 such that ”6 — 60“ < 6 implies “(16(3) — 1290(2) - (16062)“? - 6’0)“ S 5K0(z)ll6 - 90H, for almost all z(C). Using the methods of proof of STZ or Koul and Stute (1999), one can verify that under the above assumed conditions and under H0, QnVn => B o G. Hence, under H0, sup, E IR lQnVn(z)| => supO _<_ u S 1 |B(u)|, f[QnVn(z)]2dGn(z) => fol 82(2i)du, and the corresponding tests are asymptotically distribution free. 3.2 Estimation of 6 In order to apply the above results, it is important to have an estimator 6 of 60 under H0 satisfying all the assumptions. Li and Zhang ( 1998) constructed an asymptotical efficient M-estimator of the regression coefficients in a linear regression model with interval censored data and when the error d.f. F is unknown. Since here F is assumed to be known and since their estimator is computational much more involved, we shall instead use the conditional least square estimator defined by n 2 = argming E 9 Z [6, - F(T,‘ — 772.6(Z,))] . 2 = 1 Assume that F has a continuously differentiable density f and 260 :2 E(f2(T — 111.0(2)) m.90(Z)m60(Z)') 47 is positive definite. In addition assume that me is continuously differentiable with the matrix of derivatives 7726),)(2) satisfying “771.0(2)” S Mgo(z), with f 11/190(z)dC(z) < 00. Then using the classical Cramér type of argument one can verify that n1/2(6 — 6.) : 25371—1” EM, - F(T,’ — Tn.60(Z,'))]f(T, - m’60(Z'i))m60(Zt) + 0p(l), (P60)' Consequently, under H0, "WWI - 90) => N'.(0, 90), 0° ‘2 Ball/10266 M. ;= E{(FFf2)(r — 112.0(2)) m90(Z)r'n90(Z)’}. See, e. g., Liese and Vajda (2004) for a general method of proving asymptotic normality in nonlinear regression models. 3.3 A Simulation Here we shall exhibit results of a finite sample simulation. For simplicity we took M to be simple linear regression model. Thus q = 1 and Let M = {m(-,6) :m(z,6) = 62}. In this case then several entities simplify as follows. Let Z“) S Z(2) S S Z(,,) denote the ordered Z,’s and T (.)’s, 6(0’8 denote the corresponding T,’s and 6,’s. Also, let [fly-(z) E €67,620), Hn), an 2= R6n(6(j),T(j), 2(3)), 0'an I: 0n(Z(J-)), and Anj I: An(Z(,-)), where now 2 n . . 1 72(Z(i)) 2 An(.~) .= 7 Z , (Z(i))Z(,)I(Z,- 2 2.). i =1 f§ " - an 48 n 7?. - 1 Z(.-)€ - 1 Z(j)gannj = Vn(Z)—; —A Uni—”U2 E _02. 1(Z(j)/\ZZZ(«)) i=1 m 7” j=1 n] n n 1 1 Z(i)Z(j)€nien ' Rn ' = _n1/2 E : {1(Z(j)S Z) - g E A 31(Z(j)/\Z Z Z(.))}—1 1 2' = 1 n2 071?; aflj Uth To test the hypotheses H and H0, we consider the following two tests based on process ”9(2) and an/n(2) where Kn 1= SUP “49(le Rn 1= SUP lan/nlel- 2 E IR 2 E IR We reject H (H0) whenever Kn > ba (Kn > b0), where be, is the 100(1—Oz) percentile of the distribution of supO S t S 1 |B(t)|. Note that j R 6 ,T ,Z Kn = 1 max 2 6( (k) (k) (a) , x/filSkSniz1 09(Z(k)l kA ' - 1 k 1 ‘7 Z(i)Z(J‘)gnienj an Kn = — max E [1— — ] Next, we examine the finite sample performance of the test statistic Kn and Kn through some simulations. We generate the covariate Z,’s from the uniform distribution on the interval [0, 1], and Y,’s according to _ . 2 . ' Y,—BZ,+aZ,-+e, 1S2Sn, 49 while e,’s are simulated independently from the following distributions. I : logistic(0,fi): logistic distribution with location parameter 0 and scale parameter )6. II: Normal(0,02). III : DE(0,B): double exponential distribution with location parameter 0 and scale parameter )3. We also generate the censoring time variable T,’s from the uniform distribution on the interval [0, 3]. Hence H : u = 3Z, H0 : m E M hold with 60 = 3 if and only if a = 0. . We compute the empirical sizes and powers for different values of a and different error distributions. The results represent the Monte Carlo levels when a = 0 and the Monte Carlo powers when a 75 0. The sample sizes used in the simulation are n=100, 200 and 400, each replicated 1000 times. 50 Table 3.1: Empirical sizes and powers of Kn test, 5 ~ logistic (0, fl) a=0.1 a = 0.05 a=0m n=100 n=200 n=100 n=200 n=100 n=200 0.088 0.092 0.101 0.262 0.134 0.131 0.641 0.327 0.209 0.095 0.098 0.079 0.488 0.237 0.149 0.922 0.568 0.348 0.043 0.046 0.047 0.162 0.090 0.077 0.527 0.221 0.130 0.043 0.048 0.040 0.348 0.150 0.085 0.869 0.455 0.242 0.022 0.021 0.021 0.108 0.054 0.045 0.406 0.146 0.083 0.021 0.022 0.021 0.252 0.104 0.052 0.784 0.341 0.166 51 Table 3.2: Empirical sizes and powers of Kn test, 5 ~ Normal (0, 02) 020.1 a = 0.05 a = 0.01 n=100 n=200 n=100 n=200 n=100 n=200 0.105 0.089 0.095 0.416 0.208 0.155 0.890 0.581 0.363 0.100 0.091 0.086 0.742 0.385 0.260 0.987 0.870 0.639 0.057 0.043 0.045 0.313 0.135 0.085 0.808 0.472 0.265 0.045 0.040 0.044 0.630 0.270 0.173 0.976 0.782 0.528 0.034 0.017 0.026 0.233 0.089 0.045 0.722 0.363 0.186 0.018 0.019 0.019 0.053 0.187 0.107 0.964 0.686 0.423 52 Table 3.3: Empirical sizes and powers of Kn test, 5 ~ Normal (0, 02) a = 0.1 a = 0.05 a = 0.01 n=200 n=400 n=200 n=40 n=200 n=400 0.066 0.092 0.031 0.031 0.012 0.014 0.058 0.093 0.022 0.035 0.005 0.016 0.083 0.156 0.035 0.075 0.019 0.037 0.058 0.107 0.022 0.048 0.008 0.017 0.127 0.171 0.065 0.093 0.028 0.057 0.119 0.229 0.052 0.123 0.026 0.077 0.225 0.509 0.124 0.344 0.056 0.204 0.211 0.472 0.122 0.290 0.064 0.195 0.195 0.344 0.113 0.219 0.056 0.124 0.305 0.619 0.169 0.448 0.087 0.302 0.288 0.541 0.019 0.414 0.110 0.291 0.243 0.396 0.157 0.279 0.091 0.192 53 Table 3.4: Empirical sizes and powers of Kn test, 5 ~ DE (0123) a = 0.1 a = 0.05 a = 0.01 a B n=100 n=200 n=100 n=200 n=100 n=200 0.088 0.078 0.102 0.350 0.205 0.146 0.747 0.508 0.345 0.089 0.099 0.080 0.574 0.342 0.242 0.968 0.792 0.040 0.041 0.045 0.231 0.135 0.082 0.644 0.388 0.241 0.045 0.045 0.049 0.433 0.246 0.170 0.939 0.691 0.458 0.025 0.022 0.023 0.147 0.075 0.046 0.537 0.264 0.159 0.020 0.020 0.027 0.340 0.172 0.102 0.904 0.591 0.349 54 Table 3.5: Empirical sizes and powers of Kn test, 5 ~ DE (0,13) a = 0.1 a = 0.05 a = 0.01 n=200 n=400 n=200 n=40 n=200 n=400 0.102 0.083 0.058 0.038 0.030 0.019 0.083 0.089 0.047 0.041 0.025 0.025 0.086 0.093 0.042 0.042 0.014 0.019 0.106 0.169 0.051 0.089 0.017 0.050 0.080 0.135 0.040 0.067 0.019 0.040 0.081 0.122 0.042 0.058 0.021 0.025 0.177 0.477 0.091 0.301 0.042 0.171 0.191 0.355 0.098 0.233 0.050 0.144 0.161 0.286 0.082 0.175 0.042 0.109 0.217 0.534 0.090 0.343 0.042 0.217 0.274 0.548 0.165 0.401 0.082 0.263 0.231 0.470 0.135 0.323 0.080 0.203 55 From the above tables, we can see the empirical sizes are close to the nominal level when sample size is large. Under the alternatives, the power decreases as B or 0 increases, while it increases as a increases and sample size increases. 56 (2......1 '11.-..- ( 3111. *5"“‘°j"l"‘f u()1))))())))jj)((1)11