THE APPROPRIATE REDUCTION OF THE WEIGHTED EMPIRICAL PROCESS

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1 THE APPROPRIATE REDUCTION OF THE WEIGHTED EMPIRICAL PROCESS by Galen R. Shorack and Jan Beirlant TECHNICAL REPORT No. 59 February 1985 Department of Statistics University of 1fashington Seattle. Washington 98195

2 Abstract: Us a new transformation that the weighted This is shown to arise inequality of Marcus and to reduce dow~ to [0,1], it is shown processes [0,1] are always t a discussion of a new

3 1. ras PROCESS Let x n 1,..., X n n be independent with completely arbitrary df's Fn 1,...,Fnn' Let Cn 1>.,C nn denote known constants satisfying :Er cn~ = 1. Then for - 00 < x < 00 ( 1.1) is called the weighted empirical process. Let Xn1,...,X nn be the associated continuous rv's of van Zuijlen (1978), and let Hn t,...,hnn denote their df's. Then (1.2) where = n c; == :E cn~ Gni = I t (1.3) for 0 ~ t ~ 1 a.s., (1.4) [Xni~X]=[Pni~H:(X)]=[~ni~Gni(Hn(x))] for -oo<x<oo a.s., (1.5) = [f,,;~ [Xni ~ x] = [Pni ~Fn(x)] =;1Fni(x)] for -oo<x<oo a.s. (1.6) If we defined the reduced weighted empirical process 7Ln. by for 0 ~ t ~ 1, (1.7) then = on a.s, 1. Note that has covariance function =

4 THE THEOREM Let Q denote the class of all continuous functions g on [O,lJ such that g is 7' and g (t )/vtis ~ on [0, ~J and g is symmetric about t = ~. We assume the Cni'S are u.a.n: in the sense that max (c n ; ) = max (c n 1)/ 2: f c~ -l' 0 as n -l'00. Let D denote the collection of right continuous functions on [0,1] that have left hand limits at all points. Let denote the a-field generated by the finite dimensional sets. Let C denote the subcollection of continuous functions on [0,1]. Let Ii denote the sup norm. Now the 7ln's above are processes on (D, n ). Moreover, if 7l is a process with continuous paths a.s., then 7l is a process on (D, J). The statement 7Ln. converges weakly in II/gil, denoted 7ln => 7l on (D, J),I /q[i), will be taken to mean that Skorokhod constructions of these processes (also labeled tln and 7l) satisfy 1(71n - 7l)/qII -l'p 0 as n -l' 00. The statement that tln is weakly compact on (D, /q II) is taken to mean that every subsequence ri' has a further subsequence n" for which 7ln" => (some 7l) on (D,,1), /q). THEOREM 1, Suppose q E Q and the Cni'S are u.a.n. Then (i) 7Ln. is necessarily weakly compact on (D,, I /q ). (Ii) Any possible limit process 7l must satisfy P(71/q E:C) = 1. condition that,t as n -l' 00 for 0 ~ s t ~ is necessary wid sufficient for to ",,,,t",,,,!v =

5 ~ 3 ~ Moreover, 7l is necessarily a normal process with 0 means and covariance function K; we denote this by 7l :: N (OX). (iv) Suppose tl n => 7l holds and a Skorokhod construction has been performed so that tl n - 7li ~p 0 as n ~ 00. Let h. E 1 fa h 2(t )dt < 00. Then 2 in that 1 1 f lui 'lln ~p f hd 'll o 0 as n ~ 00 (2.3) 1 for some tv fa lui 7l on the Skorokhod probability space. Also note that a.s, (2.4) REllfARK 1. Shorack (1979) proved that a different reduction than this satisfied (i). (ii) and (iii) provided max (nc n 1) was uniformly bounded in n. In that reduction, tln was the weighted empirical process of the rv's ani == Hn<,Xnd where Hn == L:r Hni/n. 3. ras INEQUALITIES We have the LVLJ.VVYLUC< minor variation on inequality of and

6 - 4- For all '\,0 > 0 for which the rhs of (3.2) is a number in (0. 00 ). PO/v; (3.2) where E:b... E: n are iid Rademacher rv's (i.e., P(E:i =1) =P(E:i =-1) =Ji) independent of X n 1...,X nn and (3.3) Let V; == l[o,al(f:)/q(f n ) for 0< e ~ ~~ and q 7' on [O.~]. Then ET:; = i: l[o.ajcf::j[q (F:)]-2dF n ~ faa q(tt 2 dt. Let'\ = 0 = E:/2 in (3.2). Chebyshev gives P(It; I > '\/4) ~ (64/E: 2) faa q (t )-2dt. Then for 7 such that F: (7) ~ e we have from (3.2) that e P(IIIEn/q(Fn)II!", > E:) ~ (512E: 2 )f q{tt 2dt/[1-(256/g2 ) f q(t)- 2dt] o 0 a provided e is so small that fa" q (t )-2dt < E: 3/1024. Specializing to tl n gives (3.4) for all n, c n l ' '., 'c nn ' F n 1,.,.,F nn (3.5) Elementary computations done by many (see Shorack and Wellner (1984), for example) give the following result ( 2,.t we have E o r ~s t and

7 THEPROOF As in Shorack and Wellner (1984), Koul (1970), or van der Zanden (1980), parts (i). (ii) and (iii) of the theorem hold, with q == I, provided IIn (s,t ] == L:r Cn1Gni (S,t ] satisfies lim max IIn ((k - l )/ m, k/m] -l' 0 as m -l'oo. n... l:liik:liim. In the present case this is trivial since lin (s,t] = t -s for this reduction. Thus Zln is weakly compact on (D,:J, 1/) and Zln => (somezl) on (D,:J), II) if and only if (2.1) holds. (4.1) Moreover. Zl must be N(O,K) with P(7l E C) =1. Now let Zn == 7ln/q. From the cr-inequality and (3.7) ( ] 4 31 E Zln (s.t ]4 ( )4[ EZn s.t ~ 2 q4(t) +E7ln s q(s) - q(t) (4.2) Thus if Zln' -l's.d. 7l on some subsequence n' then EZ(s,t] 3 = limeizn.(s,t] 3 ~ lim ~EIZn.(s.t] 4~3/4 1 r 4 2 [ )4)1 3 / q4(t) q(s) q ~ 8t 3 (t - S ) +3s2 _1 If for O~s ~ t ~ ~ (4.3) t since (t-s)/q2(t) ~.is q )-2du for q -r and for g(t.'" also we have s 1 q S t

8 - 6- for O~S ~t ~~. Now (4.3) implies that if 'lln' => (some'll) on (D, 3J, ), then P('ll/q E.C) = 1. (4.5) Thus'lln' => (some'll) on (D, Ii) implies for () small enough that P(!i('lln' -'ll)/q ;;;; g) + symmetric terms on [~, 1] < 6g for n;;;; some n e (4.6) using (3.5), (4.5) and 'lln' => 'll on the three terms in (4.6). Thus '!In' => (some'll) on (D, J), Ii!i) implies '!In' => 'll on (D,:JJ.11 /qli>. (4.7) Combining (4.1), (4.5) and (4.7) gives parts (i), (ii), (iii) of the theorem. We now turn to (iv). Suppose'lln => 'll. We follow Shorack (1985). Thus if h(t) = I:~ a;1(t i_t,t j ] where 0 == to < tl <. " < t k == 1, then 111 J nau; = - J'lln dh -"'p J hd'll (4.8) 1 since J (Zln -'ll)dh I ~ Zln - 'lll! fa d h "» O. Now let r, denote a step function of the above type for which (4.9) for n ;;;; some n t

9 (two analogous terms)m 1 = 2e + 2 f (h -ht)2dt/e 2 o < 4e. Thus Y n is Cauchy in probability. and hence Y n -lop some Y. This completes the proof of (iv). As a point of interest we note that = f h;(t)dt - lim f; en; f htdgni 1 [ o 1 0 (4.10) Even (4.10) is not particularly nice.

10 - 8 - References Koul, H. (1970). Some convergence theorems for ranks and weighted empirical cumulatives. Ann. Math. Statist Marcus, M. and Zinn, J. (1984). The bounded law of the iterated logarithm for the weighted emprirical distribution in the non-lid case. Ann. Probability Shorack, G. (1979). The weighted empirical process of row independent rv's with arbitrary df's. Statist. Neerlandica Shorack, G. (1985). Empirical and rank processes of observations and residuals. Canadian J. Statist. 13. Shorack, G. and Wellner, J. (1984). Empirical Processes. To be published by John Wiley and Sons. van der Zanden, A. (1980). Some results for the weighted empirical process concerning the law of the iterated logarithm and weak convergence. Thesis. Michigan State University. van Zuijlen, M. (1978). Properties of the empirical distribution function for independent nonidentically distributed random variables. Ann. Probability

d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N

d(x n, x) d(x n, x nk ) + d(x nk, x) where we chose any fixed k > N Problem 1. Let f : A R R have the property that for every x A, there exists ɛ > 0 such that f(t) > ɛ if t (x ɛ, x + ɛ) A. If the set A is compact, prove there exists c > 0 such that f(x) > c for all x