Zuijlen, M.C.A. van

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stichting mathematisch centrum AFDELING MATHEMATISCHE STATISTIEK SW 34/75 JANUARI M.C.A. VAN ZUYLEN SOME PROPERTIES OF THE EMPIRICAL DISTRIBUTION FUNCTION IN THE NON-I. I .D. CASE ~ MC 2e boerhaavestraat 49 amsterdam fH[r/:::_ l i'.:)c:11 l....Li''-4 l l{Uit1 Af¥iS fSHU;~J-'4 P,un..te.d a..t .the. Ma..thema.:tlc.a.i. Ce.n,t,r,,e., 49, 2e. BoeJr.haa.vv.,.tluutt, Am1.:deJr.dam. The. Ma..thema:ti.c.al. Ce.n,t,r,,e., 6ou.nde.d .the. 11-.th 06 Fe.bll.u.all.y 1946, -l6 a. n.on.-pll.o6il ..i..m.:U:tc.it.fon. cunu.n.g a..t .the. pll.omo:Uon. 06 pllll.e. ma..thema.:tleo a.n.d ..i..:t6 a.pp.U.c.a.UonJ.:,.. I.t M .6pomoll.e.d by .the. Ne..thw.a.n.d6 GoveJr.n.me.n..t .thll.ou.gh .the Ne..thvr.1.a.n.d6 01tga.nlza.Uon. 6oll. .the. Adva.n.c.eme.n..t 06 PUll.e. Rue.Mc.h (Z.W.0), by .the. Mu.nlcipai.,i;ty 06 Am.6.teJr.dam, by .the. UnlveJl..6ily 06 Am.6.teJr.dam, by .the. Fll.e.e. UnlveJl..6ily a..t Am.6.teJr.dam, a.n.d by ..i..n.du..6.:tJuv.,. AMS(MOS) subject classification scheme (1970): 60Gl7, 62G30 SOME PROPERTIES OF THE EMPIRICAL DISTRIBUTION FUNCTION IN THE NON-I.I.D. CASE by M.C.A. van Zuylen ABSTRACT For N = 1,2, ..• let x 1N,XZN'"'"'~N be independent rv's having contin-uous df's FIN'FZN'''"'FNN' For the set XIN,XZN''""'XNN' let us denote by ~I :N:::;x2 :N:::; .. ,:::;~:N t~e or~er stati:1i~s, by ]FN the empirical df and by FN the averaged df, i.e. FN(x) = N ln=I FnN(x) for x E (-00 , 00 ). It is shown that for each E > 0 there exists a O < B(=B) < I, independent of N, E such that for N = 1,2, •.• , (a) P(JFN(x) :,; -1-/3 FN(x), for X E (-00,00)) ?:: 1-c, (b) P(JFN(x) ?:: SFN(x), for x E [Xl:N'oo)) ?:: 1-E, Moreover, these assertions hold uniformly in all continuous df's F 1N,F 2N, ..• ... ,FNN' The theorem can be used to prove asymptotic normality of rank statis-tics in the case where the sample elements are allowed to have different df's. KEY WORDS & PHRASES: Errrpirical distribution function~ order statistics. 1. NOTATION AND RESULTS For N = 1,2, ... let x 1N,x2N, ... ,~ be independent random variables (rv's) having continuous distribution functions (df's) F1N,F 2N, ... ,FNN, where all these rv's are supposed to be defined on a single probability space (rl,A,P). Let us define for N = 1,2, ... the empirical df JFN of x 1N,x2N' ... '~N by taking NJFN(x) to be the number of elements in the set {XnN: XnN $ x, n = 1,2, ... ,N}. The order statistics of x 1N,x2N''"''~N ~re denot=~ ty Xl:N$X2 :N$ .• ,$~:N and let the averaged df FN be defined as FN(x) = N In=l FnN(x) for x E (-00,00). --1 --1 - -Define FN by FN (s) = inf{x: FN(x) ~ s}. Because FN is supposed to - --1 be continuous we have FN(FN (s)) = s for s E (0,1). THEOREM. For each E > 0 there exists a O < S(=S) < 1, independent of N E and of the continuous df's F1N,F 2N, ... ,FNN' such that for N = 1,2, .•. , (a) -1-$ s FN(x), for XE (-00 ,00)) ~ 1-E, (b) Replacing XnN by -XnN for n = 1 ,2, ... ,N, N = 1,2, ... the following corollary is irrnnediate. COROLLARY. For each E > 0 there exists a O < B(=B) < 1, independent of N E and of the continuous df's F1N,F2N, ... ,FNN, such that for N = 1,2, ... , (a) -1 -- s (1-FN(x)), for XE (-00 , 00)) ~ 1-E, (b) The theorem and the corollary are useful for proving asymptotic nor-mality of rank statistics in the case where the sample elements are allowed to have different df's. For the i.i.d. case these results are well-known. They are given by SHORACK in [3] and proved in [2]. The present theorem will be proved in an entirely different manner. The basic tool is a result of HOEFFDING [1] which is given in the lennna below. Suppose that z 1 ,z2 , ••• ,ZN are independent rv's, each assuming the values O and 1 only, with Pr(Z. J = I ) = p. for J = 1,2, ... ,N J and N -I 0 < N I p. = p < 1. j=I J LEMMA (HOEFFDING). If f is a strictly convex function defined on (-00 , 00 ) then N E(f ( I j =I z.)) :,; J where equality holds if and only if p 1=p 2= ... =pN = P• 2 In particular this lemma together with Chebyshev's inequality implies that for n > Np, ( 1 • I ) N N Pr( I j=I z. ~ n) :,; Pr(J I z.-NpJ J j=I J N :,; (n-Np) - 4 E( L j=l - 4 Z. - Np) J 2 . PROOF OF THE THEOREM ~ n - Np) (a): For all N ~ I and all O < S < I we have ( 2. l) -1-P(JFN(x) ~ S FN(x), for x E (-00,00)) -1-~ S FN(Xn:N), for n = I,2, ... ,N) ~ l -Here we have used the Bonferroni inequality and the fact that JFN(Xn:N) = = nN-l with probability one for n = 1,2, ... ,N. Now for n = 1,2, ... ,N, (2.2) P(F (X ) < 0 nN- 1) = P(X < FN- 1( 0 nN- 1)) N n:N µ n:N µ N = Pr( l j=l Z. ~ n), J where z1,z2 , ... ,ZN are independent rv's each assuming the values O and 1 only, with for J = 1,2, ... ,N. -1,N --1 -1 From (2.2) and (I.I) with p = N lj=l FjN(FN (SnN )) = SnN-l it 1.s now immediate that for n I,2, ... ,N, so that (2.3) 3 where M1 is a finite constant, independent of N and of the df's F1N,F 2N, ... ... ,FNN. Assertion (a) in the theorem now follows from (2.3) and (2.1). (b): For all N ~ l and all O < S < l we have (2.4) P(JFN(x) ~ SFN(x), for x E [X 1 :N'oo)) N ~ P( n [JFN(Xn:N-) ~ SFN(Xn:N) ]) n=2 N ~ I - I P(FN(Xn:N) > s-l(n-l)N-1) n=2 [SN]+l , - -1 -1 - L. P(FN(Xn:N) > S (n-1 )N ) , n=2 4 where [BN] is the greatest integer in BN. The terms with n > [BN] + I -1 -I may be omitted because B (n-l)N > 1 for these terms. Since n ~ 2 we are -1 guaranteed that O < B(n-l)N ~ 1 for every term, for each N ~ 1 and 0 < B < 1. Now for n = 2,3, .•• ,[BN]+l, (2.5) N = P( l j=l Z. ~ N - n + 1), ] where z1,z2 , ••. ,ZN are independent rv's each assuming the values O and 1 only, where in this case --1 -I -1 Pr(Zj = 1) = pj = I - FjN(FN (B (n-l)N )) for J = 1 , 2, .•• , N. From (2.5) and (1. 1) with now p = 1 - B- 1(n-l)N-I it is immediate again that for n = 2,3, ... ,[BN]+l, P(F- (X ) > B- 1( l)N-l) ~ 4B-2(1-B-l)-4(n-l)-2 , N n:N n- -so that (2.6) [BN]+l _ 1 _1 L P(FN(X ·N) > B (n-l)N ) ~ n=2 n. as B ➔ 0, where M2 is a finite constant, independent of N and of the df's F 1N,F2N, •.• ••• ,FNN' Assertion (b) in the theorem now follows from (2.6) and (2.4). ACKNOWLEDGEMENT. I wish to thank Dr. F.H. RUYMGAART for suggesting the problem and Prof. Dr. W.R. VAN ZWET for his careful reading of the manu-script and his valuable connnents. REFERENCES [1] HOEFFDING, W., On the distribution of the nwnber of successes in inde-pendent trials. Ann. Math. Statist. 27 (1956) pp. 713-721. [2] SHORACK, G.R., A uniformly convergent empirical process. Tech. Report No. 20, Math. Dept., Univ. of Washington, 1970a. [3] SHORACK, G.R., Functions of order statistics. Ann. Math. Statist. 43 (1972) pp. 412-427. LIST OF SYMBOLS LATIN GREEK MATHEMATICS Normal Italics Script A a A a A - : "bar" B b B b 8 0 : zero C C C C ] : one D d D d 00 : infinity E e E e E I I : summation F f F f E . element ·of . G g G g n : intersection H h H h Q lFN : empirical distribu-I i I. i tion function J j J j K k K k L 1 L l M m M m N n N n 0 0 0 0 p p p p Q q Q q R r R 1' s s s s T t T t u u u u V V V V w w w w X X X X y y y y z z z z 

Some properties of the empirical distribution function in the non-i.i.d. case

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