A shortt of a one dimensional probability distribution is defined to be an interval which has at least probability t and minimal length. The length of a shortt, U(t), and its obvious estimator, U_n(t), are significant measures of scale of a probability distribution and the corresponding random sample, respectively. The shortt process is defined to be $ \sqrt{n}(U_n(t)-U(t)) / U'(t) $, similarly to the definition of the quantile process. It is known that this process converges weakly, under natural regularity conditions, to a Brownian bridge. In this note a strong approximation of the shortt process by a Kiefer process is established, which yields the weak convergence as a corollary. Applications of the result to the global and local strong limiting behaviour of the shortt process are also presented

Einmahl, J.H.J.

Geilen, M.

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Einmahl, JHJ John

Geilen, M Mario

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A strong approximation of the shortt process

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 A strong approximation of the shortt processCitation for published version (APA):Einmahl, J. H. J., & Geilen, M. (1998). A strong approximation of the shortt process. (Memorandum COSOR;Vol. 9826). Technische Universiteit Eindhoven.Document status and date:Published: 01/01/1998Document Version:Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)Please check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. 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Oct. 2023taB Eindhoven University of Technology Department of Mathematics and Computing Sciences Memorandum COSOR 98-26 A strong approximation of the shortt process J .HJ. Einmahl M. Geilen Eindhoven, November 1998 The Netherlands A Strong Approximation of the Shortt Process John H.J. Einmahl* Eindhoven University of Technology November 24, 1998 Abstract Mario Geilen Brabantia Nederland A shortt of a one dimensional probability distribution is defined to be an interval which has at least probability t and minimal length. The length of a shortt, U(t), and its obvious estimator, Un(t), are significant measures of scale of a probability distribution and the corresponding random sample, respectively. The shortt process is defined to be J1i(Un(t) - U(t))/U'(t), similarly to the definition of the quantile process. It is known that this process converges weakly, under natural regularity conditions, to a Brownian bridge. In this note a strong approximation of the shortt process by a Kiefer process is established, which yields the weak convergence as a corollary. Applications of the result to the global and local strong limiting behaviour of the shortt process are also presented. AMS 1991 subject classification Primary 62G20, 62G30, 62G35; secondary 60F15, 60F17. Key 'Words and phrases Kiefer process, length of shortt, oscillation modulus, robust scale estimation, strong approximation, upper and lower class. 1 Introd uction and main result Let Xl, X2,'" be a sequence of i.i.d. random variables with common distribution function F. A famous robust scale estimator based on the first n of these random variables is the length of a shortest closed interval containing at least fraction t (shortt) of these n observations. So the length of a shortt is defined to be Un(t) = inf{b - a: Fn(b) - Fn(a-) 2: t}, 0 < t < 1, where Fn denotes the right-continuous empirical distribtution function (df). Similarly we define the the theoretical counterpart of the length of a shortt (Le. our parameter of interest) by U(t) = inf{b - a : F(b) - F(a-) 2: t}, 0 < t < 1. The estimator Un (!) can be shown to be a least median of squares estimator (cf. Croux and Rousseeuw (1992» with asymptotic breakdown point 50% (see Rousseeuw and Leroy (1988)) *Research partially supported by European Union grant ERB CHRX-CT 940693. 1 and it is approximately min-max bias robust (see Martin and Zamar (1993». A functional central limit theorem (CLT) for the shortt process has been established in Grubel (1988). In Einmahl and Mason (1992) a functional CLT for a generalized quantile process is derived, which contains Griibel's result as well as the functional CLT for the classic quantile process as special cases. It is the purpose of this note to refine the functional CLT for On by establishing a strong approximation with a Kiefer process of the sequence of shortt processes {on}. We also briefly present applications to the strong limiting behaviour of the shortt process. In order to present our results we introduce more notation and some assumptions. Throughout let J = F' be continuous, strictly increasing on (a,xo], strictly decreasing on [xo,") and 0 elsewhere, for certain -00 :5 a < Xo < f3 :5 00. In addition let J' be continous on R with IJ'(x)1 > con x E [Xl, X2] U [X3,X4], with a :5 Xl < X2 < Xo < Xa < X4 :5 f3 for some c > O. Also let o < ", < 1/2 be such that the endpoints of the interval determining U(",) and U(1 - ",) are contained in [Xl, X2] U [X3, X4]. Set, as above, On{t) = vIn(Un(t) - U(t»IU'(t) = v'nh(U(t»)(Un(t) - U(t», 0 < t < 1, where h = HI with H = U-l; cf. the definition of the classical quantile process or see Einmahl and Mason (1992). Let K be a Kiefer process, i.e. K(n, t) = W(n, t) - tW(n, 1), n E N, 0:5 t :5 1, where W is a two-parameter Wiener process, see, e.g., Csorgo and Revesz (1981), section 1.15. Observe that n-1/2 K(n, t) is a Brownian bridge for any n E N. Theorem 1.1 Under the above assumptions there exists a Kiefer process K such that -1/2 ((1ogn)2/3) sup IOn(t) - n K(n, t)1 = 0 1/6 a.s. 1/:9$1-1/ n REMARK 1. We only present the approximation with a Kiefer process since our method gives the same results for the approximation with a sequence of Brownian bridges. Note that the functional CLT for On in Grubel (1988) immediately follows from Theorem 1.1. REMARK 2. It is not clear if the rate in Theorem 1.1 is optimal. However, in the recent paper Deheuvels (1997) it is shown that in the Kiefer process approximation of the classical quantile process the power of lin in the approximation cannot be higher than \. Therefore we expect that the power of lin in Theorem 1.1 cannot be improved beyond 4' since the shortt process is a generalized quantile process. In section 2 we present two applications of Theorem 1.1. The proof of Theorem 1.1 is given in section 3. 2 2 Applications We first present the analogue of Chung's (1949) upper and lower class result for Dn = sUP1199-1116n(t)l· Theorem 2.1 Let An too. Then P(Dn > An i.D.) = { ~ according as ~ A! _2).2 { < } L...,,;-e n 00. i=l n = Proof We present a brief sketch of the proof. It is well known that Theorem 2.1 holds with Dn replaced by sUPO<t<1 In-1/ 2 K(n, t)l. Now by the law of the iterated logarithm (LIL) for SUPtE(O,71)U(l-71,l) In-1/ 2K(n,t)1 it can be easily shown, using An > Vllogiog n eventually, that Theorem 2.1 holds with Dn replaced by sUP71:5t9-'1 In-1/2 K(n, t)l. Finally, it rather easily follows from Theorem 1.1 that we can replace the latter expression by Dn itself, i.e. we have obtained Theorem 2.1. 0 Note that Theorem 2.1 immediately yields the LIL for Dn. In fact the LIL for Dn and even the functional LIL for {<>n(t),TJ :$ t:$ 1- TJ} (cf. Finkelstein (1971» follow direct from Theorem 1.1 in conjunction with the corresponding results for the Kiefer process. Theorem 1.1 also easily yields a result on the strong local behaviour of On, provided the bandwidth is tending to ° slowly enough. Define the oscillation modulus of On by Theorem 2.2 Let {an}~=l be a sequence of numbers satisfying an.l. 0, nan t,log (! )jloglog n -+ c E [0,(0) and an (njlogn)1/3 -+ 00, then I' WOn (an) - 1 1m sup (2 (1 ..L+I 1 »1/2 - a.s., n-+oo an og en og og n ( )1/2 I, . f WOn (an) c 1m In = -n-+oo (2an (!og ..L+loglog n})1/2 c+l an a.s. When c = 00, should be read as 1 and hence we then have Proof These results are well-known for the Kiefer process (or for the uniform empirical process); see, e.g., Chapter 14 of Shorack and Wellner (1986). Hence, using an{nj logn)1/3 -+ 00, Theorem 2.2 immediately follows from Theorem 1.1. 0 3 3 Proof of Theorem 1.1 In order to prove Theorem 1.1 we first need a few auxiliary results. One of these results, Proposition 3.1, may be of independent interest. We begin with introducing more notation. We write P and Pn for the probability measures corresponding to F and Fn respectively. Also for a function z: lR. ~ lR and a closed interval I = [a, b] c lR. we set z(I) = z(b) - z(a-), provided z(a-) exists. The class of intervals Ie lR. is denoted with TI and the length of such an I is written as III. The empirical process of Xl,'" ,Xn is denoted by and we will briefly write an (x) for (¥n«(-oo,x]), X E lR.. Recall that the Kom16s-Major-Tusnady (1975) Kiefer process approximation of the em-pirical process yields the existence of a Kiefer process k such that (1) I () - K(n,F(x)) 1_ 0 (IOgn)2) sup an X r;:: - r;:: a.s. xElR yn v n Write BP,n(x) = K:(njJx)). Then (1) immediately yields (2) Let Ix be the unique interval with IIxl = X, U(1]) :$ x :$ U(l -1]) , and maximal probability P. Denote with In,x a sample analogue of lx, i.e. an interval with length x and maximal empirical measure Pn• Denote the left endpoints of Ix and In,';1; by ix and Ln,x, respectively. The next lemma on the distance between ix and Ln,x is a key ingredient for the proof of Theorem 1.1. Lemma 3.1 There exists aGE (0,00) such that with probability one sup lix U(lI)~X~U(l-lI) eventually. (iOgn) 1/3 Ln,xl:$ C --n Proof Define h,n = In,x \ Ix and 12,n = Ix \ In,x. Then IIl,nl = II2,nl = lix - Ln,xl =: Bn· Observe that P(In,x) - PClx) = P(Il,n) - P(h,n), Pn(In,x) - Pn(lx) - Pn{Il,n) - Pn(h,n). Write b = f'(ix ) and d = - f'(ix + x). Then by a Taylor series expansion and the Glivenko-Cantelli theorem 4 Hence with probability one 1 2B~(b + d) + o(B;) = P(lx) - P(ln,x;} (3) We have = Pn(In,x) - Pn(Ix) - (P(ln,x) - P(lx» = Pn(Il,n) - Pn(Iz,n) - (P(II,n) - P(lz,n» 1 = n 1/ 2 {On(ll,n) - on(I2,n)} . IOn (Ij,n) I :::; lan(Ij,n)ll{IIi,nl<n-1/2} V I~I~(I({i;11{IIj,nl~n-1/2}IIj,nI1/2 ~ M (_1_) - (_l_)BI/Z . - 1 2 :::; Wn n 1/ 2 V Wn ni/Z n' J - , , where Wn and wn are the oscillation modulus and the Lipschitz-~ modulus of On respectively, see Mason, Shorack and Wellner (1983). Using the results in that paper it follows that the right-hand-side of (4) is almost surely bounded by C (n!/2 V Bn) (logn)1/2 eventually for some {J E (0,00). Combining this with (3), (4) and the fact that band d are bounded away from 0, Lemma 3.1 follows. 0 Note that Set and Proposition 3.1 H(x) = sup pel}. IE][,III:5x Hn(x) = sup Pn(I) lER,III:5x 'Yn{X) = v'fi,{Hn(x) - H(x», x;::: 0. There exists a Kiefer process K such that sup h'n{x) +n-1/ 2K(n,H(x»1 U('1):5x:5U(I-'1) (log n )2/3) o n i / 6 a.s. Proof We will take K as follows: K(n,H(x)) = - {K(n,F(£x +x» - K(n,F(£x»} = -n1/ 2BP,n(Ix)' Note that K(n, t), n E N, 1}:::; t :::; 1 -1}, is a Kiefer process. Now we have sup l'Yn(x) +n-1/zK(n,H(x»1 U('1):5 x:5U(I-'1) = sup Iv'n (sup Pn(I) - H(X») - Bp,n(lx) I U(I'/):5x:5U(I-I'/) III:5x = sup {v'n (sup Pn(I) - H(X») - BP,n(Ix)} U(I'/):5x:5u(1-'1) 111=x (5) V sup {BP,n(Ix) -.,fii (sup Pn(I) - H(X») } . U(17)::5x:5U(l-l'/) 111:::::x 5 The second term in the right-hand-side of (5) is bounded from above by sup {Bp,n(la;) - Vn (Pn(lx) - P(lx»} U(1/)$x$U(l-1/) - sup (Bp,n(l:l;) - an(lx)) U{1/)$x$U(l-1/) :s; sup IBP,n(l) - an(I)1 lei ((logn)2) = 0 ..;n a.s., where the last 'equality' follows from (2). Using Lemma 3.1, we see that with probability one the first term on the right in (5) can be bounded from above for large n by = 0 ((Iogn)2) 0 ((lOgn)2/3) ..;n + n I/6 ' where again (2) is applied to bound the first term; the bound on the second term follows from the well-known oscillation behaviour of the Kiefer process (see, e.g., Theorem 1.15.2 in Csorgo and Revesz (1981)). 0 The shortt process 8n can be seen as a quantile process version of the empirical-type process In. Therefore we will obtain Theorem 1.1 by 'inverting' Proposition 3.1. We will begin with proving a version of Theorem 1.1 for the 'uniformized' version 5n of 8n · Set l\(t) = sup Pn(l), 0 < t < 1, Pn(O) = 0, fin(1) = 1, III$U{t) 6 and Then set Proposition 3.2 For the Kiefer process of Proposition 3.1 we have _ (IOgn)2/3) sup l<5n (t)-n-1/2K(n,t)I=O 1/6 ~~t~l-~ n a.s. Proof Since Hn and hence Pn make only jumps of size ~ almost surely, we obtain that (6) So by Proposition 3.1 and the LIL for the Kiefer process it follows that (7) sup IP-1(t) - tl = 0 (JIOglOg n) a.s. ~99-71 n Using (7) in conjunction with the oscillation modulus results for a Kiefer process, again Proposition 3.1, and (6), we have sup 15n(t) - n-1/2 K(n, t)! 7199-11 ~ sup hln (Pn(P;l(t» - P;l(t)) + n-1/2K(n,P;1(t»1 1199-'11 + sup n-1/2 IK (n,P;1(t» - K(n, t)1 + 7n '1199-'11 = 0 ((IOgn)2/3) + 0 ((IOg10g n)1/4(IOgn)I/2) + _1 = 0 ((logn)2 /3) 11.1/6 11.1/4 v'ii 11.1/6 a.s. Note that actually Proposition 3.1 has to be used on an interval slightly larger than [U(11), U(I-11)J. This causes no problems however, since It is continuous. 0 Now we have in hands the necessary ingredients to give a short proof of our main result. Proof of Theorem 1.1 It is shown in Einmahl and Mason (1992) that almost surely So by the mean value theorem, we have for some tn between t and p;l(t), that (8) n-1/2K(n t)) + (h(U(t» -1) n-1/2K(n t) 'h(U(tn» , a.s. 7 The following facts, which can be shown by elementary analysis, will be needed: for U('I]) :5 x :5 U(1 - '1]) we have hex) = f(ix) = f(ix + x), h'( ) == f'(ix)f'(ix + x) < 0 x f'(£X) - f'(ix + x) . Hence we have that hand !h'! are bounded away from 0 and 00 on [U('I]) , U(1 '1])]; in fact this even holds on a slightly larger interval. Observe that by the mean value theorem h(U(t) - h(U(tn)) == (t - tn)h'(u(in»/h(U(in)), with in between t and tn. So Proposition 3.2, (7), (8) and the LIL for the Kiefer process yield sup l<5n(t) - n-1/2 K(n, t)! 1199-11 = 0 ( (lo!~t3) + 0 ( Jl~ n) 0 (loglog n)'/2) _ (IOgn)2/3) - 0 n I/6 a.s. o References [1] K.L. Chung. An estimate concerning the Kolmogorov limit distribution. Trans. Amer. Math. Soc., 67:36-50, 1949. [2] C. Croux and P. Rousseeuw. A class of high-breakdown scale estimators based on sub-ranges. Comm. Statist. Theory Methods, 21:1935-1951, 1992. [3] M. Csorgo and P. Revesz. Strong Approximations in Probability and Statistics. Academic, New York, 1981. [4] P. Deheuvels. Strong laws for local quantile processes. Ann. Probab., 25:2007-2054, 1997. [5] J.H.J. Einmahl and D.M. Mason. Generalized quantile processes. Ann. Statist., 20:1062-1078,1992. [6] H. Finkelstein. The law of the iterated logarithm for empirical distributions. Ann. Math. Statist., 42:607-615, 1971. [7] R. Grube!. The length of the shorth. Ann. Statist., 16:191-219, 1988. [8] J. Kom16s, P. Major, and G. TusnMy. An approximation of partial sums of independent rv's and the sample distribution function, 1. Z. Wahrsch. Verw. Geb., 32:111-131, 1975. [9] R.D. Martin and R.H. Zamar. Bias robust estimation of scale. Ann. Statist., 21:991-1017, 1993. 8 [10] D.M. Mason, G.R. Shorack, and J.A. Wellner. Strong limit theorems for oscillation moduli of the uniform empirical process. Z. Wahrsch. Verw. Gebiete, 65:83-97, 1983. [11] P. Rousseeuw and A. Leroy. A robust scale estimator based on the shortest half. Statist. Neerlandica, 42:103-116, 1988. [12] G.R. Shorack and J.A. Wellner. Empirical Processes with Applications to Statistics. Wiley, New York, 1986. Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands Brabantia Nederland P.O. Box 25 5580 AA Waalre The Netherlands 9 

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A strong approximation of the shortt process

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