A Short and Elementary proof of the main Bahadur-Kiefer Theorem

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1 EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science Memorandum COSOR A Short and Elementary proof of the main Bahadur-Kiefer Theorem John H.J. Einmahl Eindhoven, August 994 The Netherlands /

2 A SHORT AND ELEMENTARY PROOF OF THE MAIN BAHADUR-KIEFER THEOREM John H.J. Einmahl Eindhoven University of Technology A short proof of the lower bound in the strong version of the famous Theorem A in Kiefer (970) 0 the Bahadur-Kiefer process is presented. The proof is elementary and does in particular not use strong approximations. AMS 99 subject classifications. 62G30,60F5. Key words and phrases. Bahadur-Kiefer process, empirical and quantile process, strong law. Running head: Proof of Bahadur-Kiefer theorem.

3 Let [, U 2, be a sequence of independent uniform-(o, ) random variables and for each n E N, let n Fn(t) = ~ L: l[o,tl(ui), 0 ::; t ::;, i=l be the empirical distribution function at stage n. The uniform empirical process will be written as On(t) = n 2(Fn(t) - t), 0 S; t ::; ; On(t) = 0 for t < 0 or t >. Also for each n E N, Qn(t) = inf{s : Fn(s) ~ t}, 0 < t ::; ; Qn(O) = 0, denotes the empirical quantile function and we write J!3n (t) = n'i ( Q n (t) - t), 0 ::; t ::;, for the corresponding uniform quantile process. The so-called Bahadur-Kiefer process is defined by Rn(t) = on(t) +!3n(t), 0 ::; t ::;. This process is introduced in Bahadm (966); in Kiefer (970, Theorem A) the "inprobability-analogue" of the following statement is proved () n"4 II Rn II -_ n--+oo (log.)2 lion 'i JIll a.s., where II = SUPO<t< (t) for any real-valued function on [0,]. In the latter paper a proof of () itselfls-claimed but not presented. However, it is proved in Shorack (982) that, indeed, the expresssion on the left in () (with 'lim' replaced by 'limsup') is not larger than, almost surely, (note that lion =!3n II) whereas in a. recent paper by Deheuvels and Mason (990) it is established that the same expression is not smaller than, almost surely. The short and elegant proof in Shorack (982) is based on the Kiefer process strong approximation of an, but in Shorack and Wellner (986, pp ) a similar, direct proof of the "upper-bound-part" is given. The ingenious proof of the "lower-bound-part" (which finally led to a complete proof of ()) in Deheuvels and Mason (990) is extremely technical, moreover it is again based on the Kiefer process strong approximation of an. It is the purpose of this note to give a new, short proof of the "lower-bound-part" of (), i.e. we will prove t.hat

4 (2) I liminf -4--;:- II Rn II > - n... oo (log II an 2 a.s. Our proof is rather easy and not based on strong approximations. It uses as tools the following well-known facts on empirical and quantile processes, although most of them are not required at their full strength. FACT (Mogul'skii (979»). (3) liminf (loglog n)~ II an = r/8~ a.s. n--+oo FACT 2 (easy). (4) a.s. FACT 3 (Kiefer (970». (5) Define the oscillation modulus of an by let {an} ~=l be a sequence of positive numbers with an 0 and nan i. FACT 4 (Mason, Shorack and Wellner (983». If log(l/a n )floglog n ~ c E [0,00), then (6) limsnp n-+oo (an log log n) (2( + C»2 a.s. FACT 5 (Stute (982». Iflog(l/a n )/ log log n ~ 00 and rw n / log n ~ 00, then (7) I. wn(an) n I = n-+oo (an 0g(/ an»2 a.s. FACT 6 (Mallows (968». If (N I,..., Nk), ken, has a multinomial distribution with k parameters m and ]),..., ])k, where men and ]),...,])k are non-negative with L Pi ::::, then for all )\},..., Ak k P(N ~ AI,..., Nk ~ Ak) ~ II P(Ni ~ Ai). i=l 2 i=l

5 FACT 7 (Kolmogorov (929)). Let men and t E (0, ~). exist B:j,](2 E (0,00) such that for ](Id ::; ). ::; ](2mb Then for every 8 > 0 there P(am(t) > ).) ~ exp( -( + 8)).2/(2t( - t))). FACT 8 (Dvoretzky, Kiefer and Wolfowitz (956), Massart (990)). Let n E N. Then for all ). ~ (8) P(II an II~ ).) ::; 2 exp( - 2).2). PROOF OF (2). Let I denote the identity function on [0,]. From (4), (3) and Qn = + (3n/nt we see that it suffices to show that (9) I n4 II an(i + (3n/n"2) - an II n-= (log nf II an "2 liminf ----::- I ~ a.. s. Using (5), (7) and (3), (9) can in turn be repla.ced by (0) I /."4 II an (I - an/n"2) - an II liminf ,-- ~ (log nf II an Wi a.s. Set, for 0 ::; t ::;, a (t)' if lan(t) > / log n, on(t) = { / og " n, if an ()I t ::; / log. O,I,...,[log nj, where [xj denotes Define the following grid on [0,]: ti,n = i/[log n],i the integer pa.rt of :: E R. Now using (6) and (7), and again (3), it follows that instead of proving (0), it suffices to prove that () liminf a.s Ush]g t.he Borel-Cantelli lemma a proof,of () is established if we show that for ill = E (0,), L PAn < 00, where.=3 3

6 ::; (( - E) Vhite Cn = Cn( Cl,n, C2,n,..., C[log n]-l,n) = {O'n(ti,n) = Ci,n, ::; i ::; [log n] - }, Ci,n E [- log., log.] and Ci,n such that. ti,n +. 2" Ci,n E {O,,..., n} and such that. ti,n + n2" Ci,n is non-decreasing in i. Observe that PCn > o. Set cn = (. max ICi,nl) V (/log n) l~t~[log nl-l and, on Cn, let tn be the smallest ti,n,o ::; i ::; [log n], such that lan(ti,n) = cn, write dn = O'n(tn) and dn = an(tn)j set t~ = tn + /[log.] and d~ = O'n(t~). Now we have (2) Write nn = n/[log.] + n2"(d~ - dn). Note that, on Cn, n n is the number of observations falling in the interval (tn, t:j Now it is not hard to see that on Cn, the process am" defined by is a uniform empirical process based on nn observations. Hence the right hand side of (2) can be written as (3) P(n"4 sup ",,,[lot n] ::; S ::; _ ",,,[lot n] n2 n2 I(!l!:.u.)~ {a (s - dn[log n]) - a (s)}, tnn n~ mn Now observe that ::; 2(og n)2.-"4 -+ 0, as n Therefore, for la.rge n, the expression in (3) is bounded from above by (4) P(/."4(7)2" sup ",,,[log nj < < _ ",,,[log nj. _8_ I 7.2 n2 I - (- dn [log n] ) _ - () I O'm" S nt O'mn S which by Fact 6 is less tha.n or equal to 4

7 It is easy to check that, for large n, Fact 7 applies to the probability in (5). This yields, with (j = [/4, the following upper bound for the expression in (5) Now we are ready to complete the proof. Combining (2)-(6) we have P(AnICn) ~ l/n 2 (. large). Set Dn = {II an II> log n} and note that (8) implies that PDn ~ l/n 2 (n ~ 4). Hence for large n where sup* denotes the supremum over all Cn as defined before. Now, of course, L:~=3 PAn < 00 because of (7). This proves () and hence (2). 0 References Bahadur, R.R. (966). A note Oil quantiles in large samples. Ann. Math. Statist Deheuvels, P. and Mason, D.M. (990). Bahadur-Kiefer-type processes. Ann. Probab Dvoretzky, A., Kiefer, J.C. and Wolfowitz, J. (956). Asymptotic minimal charader of the sample distribution function and of the classical multinomial estimator. Ann. Aalh. Statist Kiefer, J.C. (970). Devia.tions between the sample quantile process and the sample df. In Non-]Ja7'((.mct.l'ic Techniques in Statistical Inference (M. Puri, ed.) Cambridge Univ. Press, London. Kolmogorov, A.N. (929). ttber das Gesetz des iterierten Logarithmus. Math. Ann Mallows, C.L. (968). An inequality involving multinomial probabilities. Biometrika Mason, D.M., Shorack, G.R. and Wellner, J.A. (983). Strong limit theorems for oscillation moduli of the uniform empirical process. Z. Wahrsch. Verw. Gebiete

8 Massart, P. (990). The tight constant in de Dvoretzky-Kiefer-Wolfowitz inequality. Ann. P'Obab Mogul'skii, A.A. (979). On the law of the iterated logarithm in Chung's form for functional spaces. Theory P'Obab. Appl Shorack, G.R. (982). Kiefer's theorem via the Hungarian construction. Z. Wahrsch. Ve7'w. Gebiete Shorack, G.R. and Wellner, J.A. (986). Empirical Processes with Applications to Statistics. Wiley, New York. Stute, W. (982). The oscillation behaviour of empirical processes. Ann. Probab Department of Mathematics and Compo Science Eindhoven University of Technology P.O. Box MB Eindhoven The Netherlands 6