111.! , .vl o. \ JP _ I. I‘m In. ’( _ J, .5 A . n I‘ll" lo III-1|. i 4 0|; "\ I It. . s: .((,.)\H. r [I I ‘- III| ll'lll‘.‘ dale], J t ! IF.- ’1 ' V; I o I "‘I ' ‘ u, - .. .‘, a ... . ‘ ‘I | ‘ E . . .5 v ' I ‘: ' ‘. ,u ”.v ‘ i ' 1 . t 0.. I1 A.. 5‘ 0. 5'. I .. . L. , . bl 5 . t v- n: .r . . -. 7} V41 f H i H? J! II : ,,‘. . i ..... I F?) "J { 2H9 if. § F: 1"” -f '7' | an? t1 IN" 4*, 1*ng f ' 73ml J .35).} I. J! ‘ 'i m H t’l ! \li II ‘I f u - | ‘ 'D l t H; ‘. ‘zhr ‘ 2 M 1 i ”L;0 "if! "1 bu“; ' ' 3' ' n ‘I , H .l '37 Hi, h‘ . '1!le L“‘l’ H 0..u.’6 0' v IIIIIIIIIIIIIIIIIIIIIIIIIIIIIII m“ I. 293 00563 0656 3|.IBRARY Mkihi'gcm State University This is to certify that the dissertation entitled ON ESTIMATION OF SOME DENSITY FUNCTIONALS UNDER REGRESSION AND ONE SAMPLE MODELS presented by Pathma Sarath Thewarapperuma has been accepted towards fulfillment of the requirements for Ph.D. degree in StatIStICS W Hira L. Kou] ‘*“"> Major professor Date W MSUI'J an Affirmative Action/Equal Opportunity Institution 0-12771 MSU ’ LIBRARIES .—:—. RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. ON ESTIMATION OF SOME DENSITY FUNCTIONALS UNDER REGRESSION AND ONE SAMPLE MODELS BY Pathma Sarath Thewarapperuma 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 1987 ABSTRACT In the analysis of the general linear model Y1 = xi'a + e1 , the scale parameter If2(x)dx appears as a factor in the asymptotic variance of the rank estimator g of B . A histogram type estimator was constructed in Koul, Siever and McKean (1987) for this parameter which does not depend on the symmetry of the error distribution. The window width h of the kernel function is random and is such that vfihn + constant in probability as n ->9. This is different from the usual assumption that lfihn-r(3 , hn 4- 0 as n 4~~ . Here it is shown that this estimator is asymptotically normal. An estimator is also constructed under one sample model for the functional ' [f2(x)dH(F(x)),which appears as a factor in the asymptotic variance of certain minimum distance estimators. The function R or its derivative h need not be bounded. This estimator is shown to be consist and asymptotically normal under mild assumptions on density function f and the function H . To my wife Premitha and my daughter Anushka iii ACKNOWLEDGEMENTS I wish to express my sincere thanks to Professor Hira L. Koul for his guidance, advice, encouragement, and specially for his patience during the preparation of this dissertation. I would also like to thank Professors Dennis Gilliland, Habib Salehi and Joseph Gardiner for serving on my committee. Finally, special thanks go to Ms. Carol J. Case for her excellent typing of the manuscript. iv INTRODUCTION AND SUMMARY Consider the linear model = t Y1 xi-B +81 1_<_i_<_n, where e1, 1 :_i.: n are independent, identically distributed with cumulative distribution function F and th density 1', .xi' is the pi row of aknown nxp design matrix X and 5 is a parameter vector. The scalar parameter Y = 7(f) = Ifd¢(F) appears as a factor in the asymptotic variance of the rank estimate of fi . Koul, Sievers and McKean (KSM, 1987) constructed a new estimator for y, which does not require the symmetry of the error distribution.‘ The window width hn of the kernel that is used is 0(n-i) and therefore nibn does not tend to zero as n tends to w . This means that the methods that are available cannot be applied to obtain the asymptotic normality of this estimator. In Chapter I, it is shown that this estimator is asymptotically normal in the special case ¢(u) c u . During the course of the proof, we obtain a result similar to that obtained by Scheweder (1975) for a special kernel under a different set of assumptions using Hajek's projection technique. Next, consider the one sample model. Suppose that 1, ..., Xn are random variables with the common distribution function F and density f . X The density functional T s T(f,F) = ff2(x)dH(F(x)) appears as a factor in the asymptotic variance of the. minimum distance estimators of various models (Boos 1981, Koul and DeWet 1981). The function .H is defined on (0,1). The function H or its first derivative h is not assumed to be bounded. However, some assumptions are made on the behavior of H (and h) near 0 and 1. An example of the type of functions considered is given by H(x) = log ng , h(x) = §f%:§7 . In Chapter 2., we construct an estimator for T . Estimators have been constructed for similar functionals by various authors (Ahmad & Lin 1976, Scheweder 1975, Cheng & Serfling 1983). In all these it is assumed that either H or one of its first two derivatives is bounded and therefore these methods cannot be applied for constructing estimators for T . The estimator defined in Chapter 2 is shown to be consistent and asymptotically normal under mild conditions on density f and the function H . vi Chapter 0 1 TABLE OF CONTENTS INTRODUCTION AND SUMMARY. ASYMPTOTIC NORMALITY OF AN ESTIMATOR OF THE SCALE PARAMETER IN LINEAR REGRESSION. 1.1 Introduction and Preliminaries. 1.2 Asymptotic Normality, Theorems and Proofs. AN ESTIMATOR FOR THE SCALE PARAMETER APPEARING IN THE MINIMUM DISTANCE ESTIMATION OF THE LOCATION PARAMETER. 2.1 Introduction and Preliminaries. 2.2 Consistency of the estimator. 2.3 Asymptotic normality of the estimator. BIBLIOGRAPHY vii Page (v) 31 34 36 42 CHAPTER I 1.1. Introduction and preliminaries. Consider the linear model = ' ' = Yi xi 8 + e1 1 1, 2, ... n where el, e2, ... e are independent identically n distributed (i.i.d.) random variables with cumulative distribution function F and density f, xi' is as is th usually the case, the 1 row of a known nxp design matrix K and‘ B is a pxl parameter vector. The rank estimates of B are the values of 8 which minimizes Jaeckel's (1972) dispersion function given by 545 D (B) = ¢ a(R(Yi — Xi'B ))(Yi - Xi'B-)y i=1 where R(Zi) denotes the rank of Zi among Zl’ Z2, ... Zn’ and the rank scores are generated by a(i) =¢KF$I) for a non-decreasing function ¢ defined on (0,1). For a complete rank analysis of B one requires an estimate of the scale parameter v = Ifd¢ For example this appears in the asymptotic variance of the R-estimates of B . (See Koul (1984, monograph) or Hettmansperger (1984) for a complete discussion in the case of Wilcoxon scores.) In Koul, Sievers and McKean (KSM, 1987) an estimate of 'y was propsed based on the following arguments. If g and C are independent random variables with distribution functions F and ¢(F),respective1y,then the distribution function of k_;| is H(y)===]iT(y + X) - F(-y + x)}d¢(F(X)). y > 0 . AHere and throughout the following if limits of integration are not specified then they extend from -w to +00 When F has density f, density of H at 0, from the definition 0f 7,18 27. This suggests that an estimate of y may beaohtained by estimating the slepe of H(y) at y = 0. Note that, by a probability"transformation we can write H(y) =(!1{F(y + F_1(t)) - F(-y + F'1(t))}d¢(t) where F—1(t) = inf {xz F(x) 1 t} . Let E‘ be an estimate of 8 . Then the residuals Zi , are defined by = ... ' ° :: Zi Yi xi 8 , 1 1, 2, ... n Let Fn be the empirical distribution function of the Zi's. An estimate of H(y) can thus be defined as l _ _ I {Fn(y + Fn 1m) - Fn(-y + Fn 1(t))}d¢(t). Hn(y) o y > 0 —1 where Fn (t) = inf {xz Fn(x) 1 t} n . Then H ( ) = .1. X E { J. _ 1:}. , 1= 3= 1 (12(1) ‘ Z' i 3) If tn is chosen near zero, an estimate of y is Y = Hn O a.e. Lebesgue. -1 x - (A3) ILé (B - 8)” = Op(1) where A = (£15) % and ||-|| is the Euclidean norm. We will here obtain the? asymptotic normality of this estimator for the special case ¢(u) = u ._ In this case, H" (y) =91- . 22- 1 ' " n 2 - <-|,Z~--Z Loy) n ij 1 J ' 1 = — - 1 . , —1 ,< :3 Thus we want to show that . -é _ —§ ‘ 2 /K{Hn(n tn,a)/2n tn,a — If (x)dx} is asymptotically normal for O .< a '< 1 We modify Hn(:y)_ process in the following manner. Since [[A'1(B-B)H =z. 010(1) , (-xi-xjrAA’ltfi-s) (xi-xjmt where ”in ~ : a < .. II (I. and bij (.xi xj)A. Let Kn(t,y) - l DIH HM uM H (K (D I P I U‘ H‘ t... d A :3 to ‘< v un(t.Y) II E! IN :3 A ri- ‘< v m :3 0. let Xn(t,y) Kn(t,Y) - unct)y) We will therefore consider Kn(t,y) - 2niy ff2(x)dx and obtain its asympmotic Irormrtics as a process in t and y. In Theoren1.l weewill show that (1) sup sup [Xn(t,y) — Xn(0,y)[ -> o in probability as n+°°. t v G ('1 Here and throughout the following sup sup means that t y the suprema are taken over all t and y such that IItII'j: a and 0 < y.f b < m . This means that the asymptotic properties of Xn(t,y) can be obtained by considering Xn(0,y) . We obtain these properties in Theorems 2 and 3. Since un(t,y) = EKn(t.Y) ....1. _ l .. 2 — E H E §1(Iei - ej - bIj tI f_ n y) = Ii; . Z I [F(bi'j t'+ n'%y+X)-F(bi'j t-ne_%y+X)3dF(X) 1 J we have (t -1ZZ[Gb't+-% Gb't-é] un .y) - H i j ( ij n Y) ‘ ( ij ‘“ V) where G(y) = [F(y+x)dF(x) is the distribution function of e1 e2 In order to consider Xn(t,y) in place of Kn(t,y) — Z/Hy [f2(x)dx we will first show in Lemma 1 that sup lun(t,y) - 2yn% f f2(x)dxl + O as n + w tiy < Suppose that (Al) holds and that the second 0*”. meal: derivative" f" of ‘f is bounded. Then, sup lun(t,y) — 2yné I f2(x)dx| + O as t.y ; i 2 Proof. lun(t,y) — 2n y [f (x)dx| |% Z Z I{F(bij't+fiiy+x) - F(bij t-fiiy+X) - 2n'iyf(x)}dr(x)l i J Eli-i g Z I{F(x)+(bij t+n-éy)f(x)+%(bij“t+n-éin J '(X)-+ l(b_'t+n-§y)3f”(.x+el) — F(x) - (bfiFt—nnéy)f(x) - 6 u %(bij't-n‘iy)2f'(x) '.%(bn"t'n’iy)3f"(x+92) ’ 2n-éyf (x)}dF(x)I where 61 and 92 are chosen appropriately. I§§§f21féybift f(,x) +l écbij: t+n§y)3f' (x+el) - écbij't—n'iy>3f"(x+ez)}dF(x)l const - % X Z (Ib1 ftI+ nély|)3 Z _ ( b..% 3 —3/2 1 J 13 Note that the second term tends to zero as n tends to m . Next it is shown that % Z I IbijtJB -+ o as n + m , i J This proves the lemma. Now 1 3 '1 a Z ; Iowan = h- z z |(x1 xj)' At]3 1 J 1 J _ 1' 2 ._ I - B’ 2 Z |(C1 - c )tl |(c — c )tl C.—xiA 1 J 1 3 :3- ;g HG, - an2 Hci - cjna, Htllja . 3 2a 2 2 g n §§ 3 8 2 :-—§-—max “Gill 2 H61” 1 i 2 1‘1 but 2 ”Ci” = Z xi'azixi = traceZAxi' xiA= p and 1 1 ' i is therefore bounded and max “(Gil + 0 as n + w, by assumption. 1 1 3 Thus BEE Ibij'tl +0 as n+oo. Let us also note that 2 . i i z IICiH _<_ due,” 21):. = mp“. i i 1 1.2 Asymptotic normality of Hn (Theorems and proofs) Theorem 1. Under (A1), (A2) and the assumptions in Lemma 1 P sup sup IXn(t,y) — Xn(0,y)| + O as n + w . t Y Proof: sup sup IXn(t,y) - Xn(0,y)| is first written as t Y a sup(max) over a discrete set in the following manner. See Mugantseva (1978) for details of the idea used in the following subdivision. Divide the interval [0.6] into Nn subintervals of equal length by the points y i = 1, 2, ... N i ' n Since t = A-1(§ - §§ 6 RP we define points sk in RP so that the ball of radius a in Rp is divided -k into smaller balls of radii 2 n. We will specify the values of kn and Nn later. Then sup sup lxn - xn<0.y)l t y :max SUP sup IX (.t.y) - X (s .y)| k llt-skll 16,, y n n k n + max sup an(sk,Y) - Xn(0.Y)I R Y Let us first consider the second term. Here we remark that the full force of the above subdivision is not needed to show that this term goes to zero as n + w This can be obtained using a finite partition. But this division is required to obtain bounds for the first term in the right: hand (side of theeahove inequality. Now max max IXn(sk,y) - Xn(0.Y)I k y '1 = max max I— X Z {1 _ , -% -1 _ . _ -% k y n i j (ei ejfbij sk+n y) (ei ejlpij Sk n y) -1 -§ + 1 —; (ei-ejjn y) (ei-ej i -n y) _G(bijisk+n—%y)+G(bijis —n-%Y)+G(n-%Y)sth—%Y)}I k (0 By Lemma .1, we have sup |G(bij t+n'%y) - Gcbijt-n_%y) — G(n-%y) + GC-n'iy)| tiy ' -% I -5. £3 2 5 sup |c;(bij t+n y) - G(bij t-n y) — 2n y [i (x)dx| t.y + sup |G(n'iy) - G(-n‘iy) — 2niy fi2(x)dx| t.y It is therefore sufficient to consider 1 . max sup I— Z Z {1 _ . , -§ - 1 _ , _ -§ k y n i j (ei ej:m.. sk+n y) (e. e.:b.. s. n y) + 1 _i (ei—ej:-n “y)}| Let y be such that yr i y j yr+1 for some r Then the (i,j)th summand of above -% < 1 —- (e.-e«b..'s +n y ) - 1 _ , —§ 1~ 13 k r+1 (ei-ejjmij sk-n yr+1) — —§ + 1 -% (ei—ejjn yr) (ei-ejjrn yr) (ei‘ejibij Ski“ yr+1) (ei'ejioij Sk ‘ “ yr+1) " 1(ei—e._<_’n—é + 1 -1 } J yr+1) (ei-eji—n 2yr+1) +'{1 g 1 (Gime‘j-f-n yr+1) _ -§ (61 eji'*‘ yr+1) 1 -. + - (ei-ejjn éyr) 1(ei-efign iyr)} 10 Using the same procedure a similar lower bound can be obtained for the above summand. Thus max mgx lKn(sk,y)-Kn(0.Y)I mix max IKntsk,yr,1) — Knc0.yr+1)| (2) IA + max max lKn(O’yr+1) — Kn(0’yr)| k r + Similar terms arising from the lower bound. We will now show that each term in the right hand side of (2) tends to,0 in probability as n tends to w . Given e > O , consider P(m:x max lKn(skyr+1) - Kn(0’yr+1)| r 1 . :_—2 g £‘Var(Kn(sk,yr+l) -'Kn(0.yr+1)), e Var(K n(sk’ yr+1) - Kn(0’yr+l)) till—i 2 E g Var(Knij Sk'yr+1) ’ Knij(0’yr+1)) /\ (A v + '1 - I n J #.#g COV(Knij(-Sit'yr+1) ‘ Knij(0’yr+l)’ 2 i Knmj(sk,yr+1) " Knmj(0’yr+1)) ’1 . ‘ . . = ...x where Yn13(t'y)%Ie -.3- a J tl 5 n 4y). Consider the two terms in (3) separately. n _ r . Var Knij(°kpyr+1) Lnlj(0iyr+1)) IA . - - ‘ 2 (lei'ej‘bij skii“ yr+1) (lei ejli“ yr+1' A. -1 - I{F(bm +n ayr+1+x) — F(bij'sk—n 2yr+1+x) _: -1 + F(n y 1+x) — F(-n 2y + "% I r+ _‘ r+1 x) — 2[F(n y +b s r+1 ij kAoix) _ v(—n-éy + +bij'skY0+X)]}f(X)dX 1U 11 But F(b 's +n-éy +x) — 2F(b 's AO+n-%y +x) ij k r+1 ' ij k ° r+1 -i,“ , = ‘ I _ " I ‘ I _% {bij sk (bij skAO)}f(bij skA0+n yr+1+x) I , i "'2 I (bij skAo)f(biJ. skA0+n yr+1+x) + 0(1) Ibij'sk|f(bij's Ao+n'i +x) + 0(1) k yr+1 A similar expression may be obtained for the remaining three terms. But we observed that 1221' 1 - ' ' b- S | < - X I H0 -C H IIS H n i j 13' k —-n i j 1 J k i né (constant) Thus. l X Z Var(K (s ) - K (O p < ( t) "i (4) n2 . j nij- k’yr+1 nij ’yr+1 ’—~ cons n - Covariance term. 1 3‘2 2 X Z COV(‘Knij(Sk’yr+l) " Knij(0’yr+l)’ iriim - ' I! _ unmj(sk.yr+1) Knjm(0.yr+1)) 1 ‘2 i i i Cov(K ..(s.,0,y ) - K .(s ,O,y .) (say) h i#j#m.- n13 k r+1 nmJ k} r+1 12 égfgfg {E Kn i w(sk,0 yr+1)Km j(sk'0' yr+1) — EKnij(sk,O,yr+1)EKnmj(sk,O,yr+1)} We will obtain a bound for the first term. From what follows it is easy to see that a similar bound may be obtained for the second term. Consider, 12 2 Z X E {KnijCSk’yr+1)‘-’ j.,(0 yr+1)}' ifjfm n ‘ {Knmj (.Sk! yr+1) "' Knmj(0 , yr+1)} _ 1 _ . ' _% _% _ g2 i$§f§ I {F(bij Sk+n yr+l+x) —IF(bijskan yr+1+x) . _1 _ (5) - F(n §yr+1+x) + F(-n J‘iyrflthxH' tot- . -% _ o ' " _ _ {F(bmj sk+n yr+1+x) F(bmj'sk n y +x) r+1 -F(n-iyr¥1+x)£F(-n-&yr+1+x)}f(x)dx Note that F(b 's +n-%y' -4- -F(b 's —n-%y +x) » ij k r+1 X) - ij k; r+1 p “g? "ii - *(9 ,yr+1+x) + F("n yr+1+x) _ -iy -i -1 2 -5 -g g -1 2 , ' 2n yr+1f(-n yr+1+x) -—4n yr+1f'(x+t2) _ a *1 ' __ 2n yr+1hij t Skf' (”1,1,3 ) + O,(n )., 51110 9 III and yr+1 are bounded. 0 does not depend on k and r 13 Then (5) can beawritten as 7 '3 ‘ ' -1 2 . '7 izai E Z I4n yr+lh1j‘skpmjtskf'tx+w3)f'dx+ocn > 1mm .3 = M1 t'ls Z 2 Z bij'skbmjtskv+ 0(n 2) where; M1 is a constant. 17‘3'7‘111 1 '1 '. But I;3 Z Z X b13 skbmJ'skI - ;3IZ(§(xi'xj)'Ask)(g(xm-VJ)UhfiM i#j#m J 1 2 'M2 2 p (6). ‘ :3- ;(x‘j'Ask) 3: K 3;]ch = H where M2 is a constant. From (3), (4), (E), and (6) it follows that kn . ~ f . . 5 ’2 Z {Var(,Kn(_sk,yr+1) - Kn(,0,yr+l)_) 5113 2 Nn n , k r where M3 is a constant. k . Hence if kn and Nn are such that 2 n'Nn“ ni'd‘ for some fixed 6 > 0 then k . 2 n-Nn-n-i + 0 as n + w~. This completes the proof that the first term of (2) tends to zero in Irobability as n tends to w . Next we want to show that mix [hn(0,yr+1) - Kn(0,yr)|+ O in Irobability as n + m. It suffices to show that % é ) _ .- 1 - - 1 - } + O as n + w . 14. Now ‘Var (Z z'{1 1. j (leiCejl—fn yr+l) = g g var(1(n-§yr1 2 '1 < const — Z X Z 2(y -y ) = N - — — n3 r r+1 r n N 2 + O as n + m if N + w This is true if Nn : n 1 for some small 51 > 0 Since the second term may be treated in a similar manner, these two results now imply that max max IK (sk,Y) - K (0,y)| + O probability as n + w . k n n We will now consider max sup sup IX (t,y) - X (S..Y)| k Ilt—skllsén y n “ “ This is where the division of '{t :IItH i a} into small balls is used. "Recall that sk are the points such that A IIt-skll - 6n an(t.y) - chsk.y)| '{1 — 1 3H4 =l g L-IoM - -1 (ei-ej_ In Theorem_2,.we obtain the asymptotic normality of Xn(0,y) . We will prove this for a general function ¢ since this result is of independent interest. It is assumed that the function ¢ is twice differentiable everywhere and that both ¢' and ¢" are (uniformly) bounded. W‘firu ... x 20 Scheweder (1975, 1981) has obtained this result for a general kernel under different assumptions using afiVoneMises type.expansionr In his {roof he assumes that /th + 0 but in ours /th + constant. Of course, our result is valid only for the special kernel considered. 'Theorem'Z: Let ¢z [0,1] + [0,1] bela nondecreasing twice differentiable function such that is first two derivatives ¢' .and ¢" are uniformly bounded and continuous. Then ' J. _ '.J_-_1_ '_ t . £32 {¢(n) Mn “lfllegiregfliy/VH) for vy 1s asymptotically enauivalent in probability to 1 , ‘ nilcb (F(.€j))1(.|ei-ej|_<_y//n) + 2n'5y I, ¢"(.u)f<.F-‘1(-u)>i{1 -u}du [0,1] (F (- 93 )1“) and hence we) - ¢ci=l)}1 ' a - ¢'(F(,x)>f<.x)dF(x) H n “ ”eurecmlf-n Y) I 42> N(,0,02) where 2 _ . X - a = 4y20f1¢"(x)f(r 1(.x)>{2(.1—x)of u¢"(u)f(F 1(a))du + 4Ofx'¢'(u)f(F—1(_u)_)du - 4x0f1 ¢"(,u)f(F-l(__u))du}dx 1 + 16y20I1 4>' (mm-1(1)}? ax - 16y2
    )f(,x)dF(x). Let wn and 22 -1 1(.It-SI :in “y) I wn(t,s)an(s) . n&(.2§l)—1 (t.s): fn(t)% Following Scheweder (1975) let us consider the difference 1 n where' and i g ¢'(anu‘fl ' ¢'(F'(F}1(|t_sl_f n—éy)an(t)an(s) zn'éy I l{¢'(Fn(t)) — ¢'(F(t))}wn(t,S)an(t)an(S) I ¢'(Fn(t))-¢'(F(t)) % v ‘ Fn(t) _ F(t) fn(t) n‘{Fn(t)-.(t)}drn(t) 2y 2y I ¢h"(F(t))-¢"(F(t))fn(t)Wn(t)an(t) + 2y I¢"(F(t)){fn)fdFl 24 + O as n + w under the assumption that ¢' is bounded. = ' . . .. ,1 —-f ¢'(F(x)){Fcn’%y+x) -F(-n-éy+x)}f(x)dx, _* .1 T1 ’ H X 2 Tij ' *3 II ‘% _ n -1 _ 2 2n ygé, ¢ (,u)f(F (“)){1(F(;ej)-_'(F(.u))+<1>'(F(6’k))}1‘(l(gym—(rial),)f(u)du - expected value} + 2n-iy if ..¢"('u)f(F-1(u)){1(F((1)511) - uldu . ELIJ. I. II Now‘l§[¢'(F(u)) 1(Iek’ulj n-%y)f(u)du = E I _i _, ¢'(F(u))f 1 H But Z(¢F)"(ek+...) - expected value + O a.s. k by the SLLN. Similarly Z{f¢"(F(ek)) 1 4g - expected value} k ( ek-ulirl y) is asymptotically = 2n'5y £{¢'f N(O,02) where 02 is given in theorem 2. A This shows the asymltotic normality of T . T will have the same asymptotic distribution if ‘Var(T—f) + 0 as n + w . This is shown next. 26 lei’e; I: {fl-$3?) Var(T - 'i‘)= Var(—- Z 2 ¢' (F(ej )) 1( .J J - XI{¢‘CF(ej)).+ ¢'(F(u))}1(|e f(u)du) J l ..— . — "*~ U. n- 2y ) Let. L1(ei.ej) = {¢'(F(ei))+¢'(F(ej))}1(lei_ejl;3n-%y) L2(ei) = f{¢'(F(eil)+¢'(F(u))} 1(lei_u[:,,- ey)f(u)du L3.)} 1(lel_u|1n—%y)f(u)du - I{¢'(_F(62)) + ¢'(F(u>)}1(|92_u|:n-%y)f(u)du Again, the boundedness of the function I¢'I = ¢', (since ¢' is non-negative)imp1ies that 92 = Var W(e1,e2) + O as n + w . This proves that 'Var (T - T) +-O as n + w . Combining these results, we now have that (-1) " Quill-i“ Y) 1 v 'l _ '3-1 329th) ¢(.-—fi-)}1(lre - 2y/H f ¢'(F(x))f(x)dF(x) 28 L" 2 2 . . . ':::aN(0,a ) where o is given 1n theorem 2. In order to grove the.weak convergence of the process to Gaussian process of the form (yZ), Q i y i b, where Z is a Gaussian random variable, we need to prove ‘ the following:“We will now consider the case ¢(u) = u Lemmajci. lim sup Var (Xn(0,Y) — Xn(0,x)) i constant~(X-y)2,05x,y:b. n+oo Proof of Lemma 2: Recall that Xn(0,y) = Kn(0,y) - ZKn(O,y) 1 h K 0 = — " w ere n( .Y) n E g 1(|ei - ejl j n éY) Hence it suffices to prove that Var (Kn(0,Y) - Kn(0,x)) i constant (x-y)2 n-l 1 Since Kn(0,y) - Kn(0.x) - —§-(g) g g £1¢;i_ej|: n‘éy) 4 1 moaHA in =( Z'il < -a -% j (Iei - ejl _ n y) (Iei — ejl i n x)} ) Z i < and Wu = constant. We have ' _ 2 Var (Kn(0,y) - Kn(0,x)) - (n—l) Var‘Vn '1 _ 2 . — . - (“-1) (g) 2(n—2)C3 + C4 9 c3 = Var E(1 and c4 = Var 1 For large n Var (Kn(0,y) - Kn(0,x)) = 4(n—2)53 + C4 As in theorem 2, it is easy to prove that C4+O asn-rao. -5 -.1 and C Var F(e2+n y) - F(eZ—n 2Y) - F(e +n.%V) + F(e -n_éx) 2 18 > 2 Var [?n’é(y-x)f(e2) + 2n—1y2f!(e2+e'n—éy) + 2n—1y2f'(e2+e"n—éy)] where 91 and 62 are chosen according to Taylor's theorem. Thus, Var (Kn(0,y) — Kn(,o,x)_) = lGVarBy-x)f(e2)] for sufficiently large n which implies that lim sup Var (Xn(0,y) - Xn(0,x)) : constant (y-x)2 n + on This proves the Lemma 2. 30 From Corollary 1, Lemma 2, and Corollary 15-5 from Billingsley we deduce Proposition.: Under the assumption stated in Lemma 1 and Theorem 1, with ¢(u) = u xn(;0,~) =—_—_-_,._x¢o), on D[0,b]_ 2 where X(y) = yZ, Z is a N(O’01) random variable. From Theorem 1, Corollary 1 to Theorem 2, Lemma 3 and Lemma 6 (KSM, 1987) it now follows that /H{HnCn-étn,a/2n-étn,a - ff2(x)dx} :22: N(O,of) where of = 16{ff3(x)dx - (ffz(x)dxf% CHAPTER II 2.1 Introduction and preliminaries, Let X be a random variable with distribution function F and density function f . The density functional f2(X) I dF F(x)(1-F(X)) appears as a factor in the asymptotic variance of minimum distance type estimators in various models (See Anderson - Darling (1952), Boos (1981), Koul and DeWet (1981)). (No limits of integration will be shown whenever the integration extends from -m to +w.) In this chapter, we proposean estimator for this type of functionals and obtain its consistency and asymptotic normality. Note that the density functional of interest is of the form T(f) f f2(x)dH(F(x)) I f2(X)h(F(X))dF(X) where H is a nondecreasing measurable function definedcxlbgfl h is the a.e. derivative of H and need not be bounded. We will assume that the function H satisfies the following properties. (1) I dH(x) ~ C log n for large n , [n-a, 1-n-a a > 0 and C is a constant which does not depend on n. (2) O < h(x) < nza for x glé’fl, 1-n-:]. For example, H(x) log 1-x , h(x) x(1-x) satisfy these properties. Estimators for this type of density functionals have not appeared (to the authors' knowledge) anywhere in the statistical literature. The density functionals of the form T1(f) = f¢i>i2dx were considered by many authors. For example, Ahmad and Lin (1976) proposed a consistent estimator for T1(i) under the assumption that the function w has a bounded first derivative. Cheng and Serfling (1983) also considered estimation of T1(f); They have shown the consistency as well as the asymptotic normality of their estimator with the assumption that the function w has a bounded second derivative. Scheweder (1975, 1981) has a similar result but under more restrictive assumptions. None of these methods can.be applied to T(f) since the derivatives of H(-) are not bounded. Note that when ¢ 1 the functional T1(f) reduces to T1* 2 LWJ is the proposed estimator for T(f), (3 < a <:% and fixed. 34 In the remainder of this note the integral f’°' [p-a,1—n‘§] will be denoted by f... , Thus, Tn = ‘f'fnz (Fn'lcx) 1 dH(x) 2.2 Consistency of the estimator. Let the kernel K be such that (3) K, is a symmetric probability density on R and is of bounded variation u, flu|K(u)du < w . (4) K(1) , the derivative cf K is a continuous function of bounded variation pl, (5) Window width an is such that an + 0 and n&(log n)"é an + w as + w . Theorem 1: Suppose that f has finite Fisher Information and that the derivative f' of f is bounded. Suppose that K satisfies (3) and (4) above. If né(log n)-1an-+ an) : Clexp‘{—C2nen2an2r+2} X 35 Proof of Theorem: Since T(f) is finite, given 5 > 0 select n sufficiently large so that T = I r2(r’1(;t))dH(.t) O, in Lemma(2.4, Schuster) we see that Hfh - fH(log n) + O 2 +°° as 11"”. in probability provided h(log nf2 n This completes the proof of Theorem 1. 2.3 Asymptotic.normality.0fizTn'. Theorem 2: In addition to above assumptions on K and F suppose that (6) ofllpi(t-s) - o'ln(t>dH(t) + o as s + o (7) 0f1f4(F'—1(t)){h(jt)}2dt. <03 < ... and (8) f(x)h(F(_x)). )}f(.F‘1(t))dH)dH-t) - p'(t+62(FFn-1(t)-t))}d1{(t)l 1 Znélfi‘Fn-l- 1“ "(an o')o|| C (log n) = 2.1% c “nu-1 - In. Ilrn' - runes n: . 39 Setting r 1 and En = (log n)e , where e > O , in Schuster's lemma we see that and- ffllclog n) + O . . .. k —l in prob. as n + w prov1ded n (log n) ‘an + w as n + w . Also [2“; f f(F-4(t)(FFn'1(t)-t) '{p' (-t+echFn‘1(t)- t) — q'(t)}dH(.t)| -1 _< 2n§ IIIFFn 1 _ - III! pcth' (“92(an 1m - t) 0 , - o'<.t)ldn(.t)‘ Assumption (6) implies that this term tends to zero as n + w . These reductions now imply that we need to obtain the normality of ., -N1 '10:) - 1:}f (F (U) dH(t) 2n; I.f(_F-1(t)) {FFn _1 ' f(F (13)) + 2ng.f.f(F-1(t))u{fn(.F_1(t)) - f(F"1(t))}dH(-t) . s2n§ fawn-la) - t}f' (F—1(t))dll(t) +2né of.f(_F-1(t)){fn(F-1(,t)) - f(F-1(_t))}dH(t) . Note that “F "1(t) — t = 5-{F F-J'FF “1(t) - FF ’1(t)'} + F F ‘1(t) — t ‘ n ' n n. ' ' n ' n n ' ' . -1 ~i Since Illi‘nFn - III = Op(a) and ni‘mnF’lFFn'ltt) - FFn'lun = n*(FnF'1(t> - t) 40 it suffices to consider 2n§;I f(F-ICI)){fn(F_1(I)) -f(F'1