ON ASYMPTOTIC REPRESENTATIONS FOR REDUCED QUANTILES IN SAMPLING FROM A LENGTH-BIAS~D DISTRIBUTION. Pranab Kumar Sen

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1 ON SYMTOTIC RERESENTTIONS FOR REDUCED QUNTILES IN SMLING FROM LENGTH-BIS~D DISTRIBUTION hy raab Kumar Se Departmet of Biostatistics Uiversity of North Carolia at Chapel Hill Istitute of Statistics Mimeo Series No October 1982

2 ON SYMTOTIC RERESENTTIONS FOR REDUCED QUNTILES IN SMLING FROM LENGTH-BISED DISTRIBUTION * BY RNB KUMR SEN Uiversity of North CaroZia~ ChapeZHiZZ. Noparametric estimatio of the quatiles of a distributio based o a sample from the correspodig legth-biased distributio is cosidered. log with some represetatios of this estimator i terms of," e averages of idepedet radom variables, some limitig results are established. The case of the reduced quatile process is treated briefly. MS Subject CZassificatio : 60F05, 62E20, 62G05 KeyWords ad hrases: symptotic ormality, Bahadur-represetatio, empirical process, ivariace priciples, reduced quatiles ad quatile process, weak covergece. * Work partially supported by the Natioal Heart, Lug ad Blood Istitute, Cotract NIH-NHLBI L from the Natioal Istitutes of Health.

3 1 ~ Let YI""'Y be idepedet ad idetically distributed (igied.) oegative radom variables (r.v.) from a distributio G, defied o R+ == [0,(0). G = G p is called a lesth-biased distributio correspodig to a +. distributio F (alsa defied o R ), 1f (1.1) Gp(Y) = }J-l f~ xd(x), for every y E: R+, where (1.2) }J 00 = faxdf(x) is assumed to be fiite. The distributio fuctio (d. f.) G p arises aturally i may fields; see for example, Gox(1969), atil ad Rao (1977,1978) ad Colema (1979) where may life applicatios are cosidered. We assume that the uderlyig d.f. F admits a uique p-quatile ~ = ~ (F),i.e., p p (1. 3) F(x) = phas a uique solutio ~p' where a < p < 1. Our mai iterest is to provide a oparametric estimate of ~p based o Y l,, Y, ad to study its various properties. For a somewhat related problem (based o a mixture of a ordiary ad a legth-biased distributio), we may refer to Vardi (l982a,b) Let Y 1 < < Y be the order statistics correspodig to Yl,,Y : -. - : The, the sample p-quatile correspodig to ~p is defied by Y : k where k is a suitably chose (radom) iteger, depedig o all the order statistics. This estimator is formally defied i Sectio 2, where the basic regularity coditios are also itroduced. Sectio 3 is devoted to the study of the large sample properties of this estimator based o (i) the weak covergece of some related empirical processes, (li) Bahadur (1966) represetatio of sample quatiles ad (iii) some strog ivariace priciples for the empirical distributios. Some geeral remarks are appeded i the cocludig sectio. ~!~~vs~etl..~~!..~~. Note that by (1.1) ad (1. 2), -1-1 ~ -1 ~p. -1 ~p -1 (2.1) }J =}J fo df(x) =}J f O x xdf(x) = f~ x dg(x), so that (2.2) -1 = 00-1 }J fox dg(x)

4 2 Let G (y) = -le~ 1 1= I (2.1) ad (2.2), G (2.3) ad for + I(Y i ~ y), Y E R be the empirical d.f. based o Yl,,Y we replace the true d.f. G by its oparametric estimator _I ~ give by -1 ~, we obtai the estimator _I 1 1 = - E~ Y-. ~ 1=1 :l Notig that the d.f. G is a step fuctio, we ow defie a (radom) iteger k (;:; K ) (2.4) K by = max{ k : k -1 E. 1 Y. 1= :l Note that depedig o the value of p (O<p<l) ad the Y.,i;:;l,,, the iequality 1 i (2.4) may ot hold eve for k=l This is particularly true whe may be small ad p is close to zero. We by lettig K ;:; may,however, remove this techical difficulty -1 -l 1 wheever Y : l > p( Ei=lY : i ). Thus, K is a positive iteger valued radol variable, ad, for every, p{ 1 < K < } = 1. The sample - - estimator of (2.5) We ~p is the take as ~,p = Y, where K is defied by (2.4). :K e- may remark that i the classical case, the sample p-quatile is defied by the kth largest order statistics, where k is a positive iteger such that ki is closest to p. However, i (2.5), the estimator comes out as a order statistic with a radom idex K whose properties are ot all that kow.i fact, K i (2.4) is defied i teris of partial sums of reciprocals of order statistics which are either idepedet or idetically distributed. Hece, the classical reewal theory results may ot "'_I -1-1 Note that by (2.3), ~ = ri=ly i has expectatio ) -2 (2.6) Y = f O y dg(y) - ~ = f O x df(x ~ apply directly to K -1 d ~ a var1ace y, where where we assume that E y- 2 = E X-I = v 2 < 00 The, by the classical cetral G F limit theorem, as (2.7) so that +00, 1 "'_I ~( ~ ~-l )1 y ~ x (0, 1), (2.8) ~-1) = 0 (1)

5 -1-2 Further, ote that a(x) = x has a cotiuous derivative al(x) = -x for all X E (0,00), so that by (2.7) ad (2.8), (2.9) ~ 2 ~ _I -1 ( ~- ~ ) = ~ (~ - ~ ) + Opel) 2 2 ~ " (0, ~ y ). These results will be useful i our subsequet aalysis. We 3 also assume that the d.f. F has a cotiuous probability desity fuctio (p.d.f.) f i some eighbourhood of t,; p G, is give by g(x) ad f(~') is strictly positive. Note that g, the p.d. f. of = xf(x), so that otig that ~ E (0, 00), we have p (2.10) 00 We may eed some other regularity coditios which will be itroduced as the occasios arise.,,~.v!y~"a!,j::.~p..:r:.e~~!.1t~t:~?~, '?f ~~p Let us deote by (3.1) H(x) = ~ y-ldg(y) ad H(x) = f~ y-ldg(y), for x (0,00 ). The, ote that both Had Hare odecreasig, ad, by (2.1)-(2.4), (3.2) H(~ ) = p/~ = p 1I(00) ad H(Y : K ) ~ ph(oo) < H(Y : K +1), where, covetioally, we let Y : = 00 Further, for every x (0,00 ), we + l let Y. (x) = Y. I(Y. < x), l=l,.,, so that ~ ~ -~ -1 (3.3) [H(x) - H(x)] = l:i=l[ Y i (x) where ~ 2.)('(0,. Yx ), (3.4) Yx 2 -_ fx o y-2dg (y) - (fx y- l dg(y))2. f ( ) o exists or every x 0,00. The~ we have the followig Theorem 1. If v < 00 (3.5) roof. Note that for every x > 0, ad (2.10) holds, the, as ~ 00 } -~ {-l -l} -~ {-1 -I}. - ~ ='p L 1 Y. ~ - l:. 1 Y.. ~ )-p~. +0 (1). 1= 1 1= 1 (3.6) ~H(x) - ~H(x) = ~{H(x)-H(;X)} +(~-~){HCx)-H(;X)} +(}1 -~)H(x). _1: lso, if we let x = ~p ±.C 2, for some fiite CC> 0), the, it ca be show that (3.3) holds with the further simplificatio that Y~ may be replaced by Y~ s such, by (2.9), (3.3) ad (3.6), we coclude that for every >0, there exist

6 a c «00) ad a iteger, such that for every (fixed) C ad >, 0-0 (3.7) O the (3.8) while, k I k k { 2 ~ H (s ±C- 2 ) - ~H(~ ±C- 2 ) I> c } < /2. p p k other had, if we let -!:2 J = {x: ~ - C- 2 < x < ~ +C p - - k 1 ~ H (~ -C- 2) < ~ H. (x) < ~ H (~ +C -~ ), V x p - - p by (2.10), as + 00 (3.8) ad (3.9) that,by virtue s J, k -!:2 (3.9) 2 {~H(~ ± C )_ ~H(~')} + ± C~f(~ ). p Hece, if we choose C so large that ~Cf(~) > c we coclude from (3.7), p. }, the of (3.2), amely, llh(y : K ):: p< ~H(Y:K +1),. (3.10) p{ Y : K S I } ~ 1 -,for every ~ o Havig obtaied this weaker boud fory : K,we like to exted a result of Ghosh(1971) o (a weaker) Bahadur represetatio of sample quatiles to our H ad employ the same for the proof of (3.5). Note that by (3.6), for every x,y E: I, (3.11) where, by '" ~ H (x)- ~H(x) - ~ H (y) + ~H(y) = ]J { H (x)- H(x) - H (y) + H(y)} + (J1 -]J ){. H(x)- H(y)}, (2.9), (2.10) ad the defiitio of I ' 4 e -, (3.12) sup{ I( ~ -]J ){H(x) - H(y)}1 : x,y S I } = Ope-I), 0~1 =Op(l),.while, usig the idetity that for x >- y, (3.13) H (x) - H (y) - H(x) + H(y) = fxu-ld[g (u) - G(u)] y = x-l{g (x)-g(x)} _y-l{g (y) _ G(y)} + fx u- 2 {G (u)-g(u)} du. y alog with (2.10) ad the Bahadur (1966) represetatio: (3.14) sup{ IG (x)-g (y) - G(x) + G(y)1 : x,y s J } = (- 3 / 4 log ), p we coclude that (3.15) sup { ~ IH(X) - H(x) -. H(Y) + H(y) I} = 0p(- 1 / 4 10g ) x,y sj By virtue of (3.10).ad (3.15J,,we.c(1)c1ude that k (3.16) 2{H (Y K) - H(Y K)} = 2 { H (~ ) - H(~ )} + 0 (1). : : p p p' so that, by (3.6), (3.15) ad (3.16), we have k

7 5 (3.17) where H(~ ) k (3.18).2{ Fially, H I (x) = Note k.2 h.l H (Y:K ) - j..ih (Y : K ) } k k =.2 {H (~ ) - H(~ )} + H(~ ).2 (j..i p p p -1 = j..i p ad j..ih(y:k) = p + -1 } j..i p - H(Y : K ) - j..i) + Opel) ~{ k = ~ H (~ ) - H(~ )} + p.2( j..i -j..i) + 0 (1). p p p -1 ope ). Hece, as + 00, (3.5) follows from (3.18), (2.7), (2.9) ad the fact that for x J, -1. x g(x) = f(x) + f(~ ) as Q.E.D. that {py~l.:.p j..i-l _ y~l(~ ) + pj..l-l = py~l_y~l(~ ), i > l} form a ]. 1 IIp sequece of i.i.d.r.v.'s with mea zero ad variace 2 2 ~p (3.19) a = (l-p) f O Y dg(y) + p J~ Y dg(y) ad hece, by (3.5) ad the cetral limit theorem, we arrive at the followig. Theorem 2. Uder the hypothesis of Theorem 1, (3.20) This result as well as (3.5) ca be exteded to several quatiles uder o extra regularity coditios. 4. Some geeral remarks. I (3.5), we have cosidered a weak represetatio for.,._ ,~. _~.T... _ - _...ot-... '... ' ,~ '..._J... ~ where the precise order of the remaider term has ot bee studied. If, i,p additio to (2.10), we assume that i some eighbourhood of ~, the p.d.f. f has p a bouded first derivative f', the, otig that g'(x) = (d/dx){xf(x)} = f(x) + xf'(x), we coclude that g' is also bouded i the same eighbourhood. s such, * k ~ we may let J = {x : I x -.~ I <C-.2 (log) } ad parallel to (3.10), we may p - * claim that Y : K I almost surely (a.s.), as + 00, while i (3.14), we may ot oly use the order as we obtai that as --7 oo, (4.1) I I - 3 / 4 log a.s., but also, by Taylor's expasio, *} sup{ -3/4 g(~ ){x - ~}- G(x) + G(~p) : x I = O( log ) a.s. p p s such, we obtai that uder this additioal regularity coditio, i (3.5), we have a almost sure represetatio with a remaider term O(- 3 / 4 log ).

8 6 Secodly, istead of the behaviour of a sigle quatile, if we like to study!,; the same for the etire process { 2 ( ~ - ~ ) ; O<p<l}, we may eed a more,p p striget momet coditio. Note that v < implies that as x f 0, x G(x) ~ 0, though the rate of this covergece is ot precisely kow. If we assume that oo -2-0 (4.2) = f Y dg(y) a the, we obtai that < 00 for some 0 > 0, (4.3) x- 2 - O G(x) ~ a as x f O. lso, by (3.1), for every x > 0, (4.4) (4.6)!,; 2{H (x) - Hex)} = -1 ~ = x IG (x) Further, it follows from O'Reilly (1974) that for every > 0,!,;!,; + (4.5) sup{ 2IG (x)-g(x)]/{g(x) [l-g(x)]} 2- : x R } = 0 (1), p. so that by (4.3) ad (4.5), for every x (0, ), as ~ 0, x-l~[g(x) - G(x)] = (~[G(X)-G(X)J/{G(X)[l-G(X)]}~-f)G(X)[l-G(X)J}~-E/x e. coverges to 0, i probability, where we choose so small that (2+0)(~ - ) > 1. s such, the represetatio i (4.4) is valid for every x > 0, ad, usig (3.6) ad (4.4), we have x -2!'; (4.7) ~[J..l H (x) - J..lH(x)J = J..lx- ~[G (x)-g(x)j +J..lf O y 2 IG (y)-g(y)]dy. If we let WO(G(x» (4.8) 1 + H(x)~( J..l - J..l) + opel) a.e. -I!,; x -2!'; = J..lX 2 IG (x)-g(x)] + }lf O y 2 [G (y)-g(y)]dy + p (x,y) = q !,; - H(x)J..l fa y d{ 2 IG (y) - G(y)]} + opel) a.e.!,; + = 2 [G (x)-g(x)], x R ad defie the p metric by q sup{ Ix(t)-y(t)l/q(t); O<t<l} ;"q(t) ={t(1-t)}~-, the, the weak covergece of WO = {Wo(t),O<t<l} to a tied-dow Wieer process W O, i the p metric i (4.8), follows from O'Reilly(1974), ad hece, usig q.!,; + (4.3) ad (4.7), the weak covergece of 2[ll H (x) -. ph (x) Lx R to a Gaussia. process follows readily. Thus, if we defie W * = {w* (t);o<t<l} by lettig!,; W*(G(x» = 2 + [ II H (x) - llh(x)], x Rad W * = {W* (t);o<t<l} by lettig

9 7 o. bet) W*(t) = }Jb(t)W (t) +}Jf o b (s)w (s)db(s) - H(b(t))}J fa W (s)b (s)db(s), for O<t<l where bet) = G-l(t), t (0,1), the, we have * ~. * (4.9) W ~ W,i the Skorokhod Jl-topology o D[O,l]. * Note that the covariace fuctio of W is give by (4.10) * * EW (s)w (t) = (1-5) (l-t) vst + st (V l - vsf\t). where (4.11) -(Sl\t}(V svt - VSf\t)' t.:t 2 v = fa y- dg(y),for t (0,1). t l.j 2 (s,t) (0,1), lso, i (2.4), we deote the solutio K by K (p) ad assume that f(t.: ) is. p strictly positive (ad fiite) for all p :O<p<l. The, istead of the Bahadurrepresetatio i (3.14), if we use its geeralizatio by Kiefer(l967) ad 0_ use (4.9), the, we are able to show that for every c > 0, as + 00, (412) sup 2!,,; I I!,,; y r: j(log)2 < c a.s., p (O, 1). :K (p) - '""p ad further, (3.18) holds simultaeously for all p E (0,1). s such, the weak 1 ivariace priciple for { ~[ t.: - ~ ]; O<p<l} follows from this exteded,p p versio of (3.18), (4.9) ad the radom chage of time results i Billigsley (1968,pp.144-lS0). Further, usig the results of Csaki (1977), (4.7) may be stregtheed to a a.s. represetatio,ad hece, strog ivariace priciples (as well as the law of iterated logarithm) for the reduced quatile process also follow uder the additioal regularity coditios (4.2) ad that sup{ Iff (x) I :x E R+} < 00

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