On the edgeworth expansion for the sum of a function of uniform spacings

On the edgeworth expansion for the sum of a function of uniform spacings
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  Journal of Statistical Planning and Inference 17 (1987) 149-157 North-Holland 149 ON THE EDGEWORTH EXPANSION FOR THE SUM OF A FUNCTION OF UNIFORM SPACINGS R.J.M.M. DOES Department of Medical lnformatics and Statistics University of Limburg P.O. Box 616 6200 MD Maastricht The Netherlands R. HELMERS Centre for Mathematics and Computer Science P.O. Box 4079 1009 AB Amsterdam The Netherlands C.A.J. KLAASSEN Department of Mathematics University of Leiden P.O. Box 9512 2300 RA Leiden The Netherlands Received 18 March 1986; revised manuscript received 17 November 1986 Recommended by R. Bhattacharya Abstract: An Edgeworth expansion for the sum of a fixed function g of normed uniform spacings is established under a natural moment assumption and an appropriate version of Cram6r's condi- tion. This condition is shown to hold under an easily verifiable and mild assumption on the func- tion g. This is done by proving Cram6r's condition for statistics of the general typef(X) under quite weak assumptions on the random variable X and the function f: ~,,1~ ~k. AMS Subject Classification: Primary 62E20; Secondary 62G30, 60F05. Key words and phrases: Edgeworth expansions; Uniform spacings; Cram&'s condition. 1. Introduction Let U1, U2, ... be a sequence of independent uniform (0,1) random variables. For n = 1, 2,..., UI:,--- U2:,---'-" < U,:, denote the ordered UI, U2, ..., U,. Let U0.., = 0 and U,+l:, = 1. Uniform spacings are defined by Djn=Uj:,-Uj_I:,, j= 1, 2,...,n+ 1. (1.1) Let g" [0, oo)---} IR be a fixed nonlinear measurable function and define statistics T,, by n+l Tn = E g n + 1)D).), n = 1, 2, .... (1.2) j=l 0378-3758/87/ 3.50 © 1987, Elsevier Science Publishers B.V. (North-Holland)  150 R.J.M.M. Does et al. / Uniform spacings Statistics of this form can be used for testing uniformity. There has been considerable interest into the asymptotic distribution theory for statistics of type (1.2). An excellent survey of first-order limit theory for statistics of the form (1.2) was given by Pyke (1972) according to whom a study of the rate of convergence for sums of functions of uniform spacings is of interest. We will use the following well-known characterization, which has been applied by Le Cam (1958) in order to prove first-order limit theorems. Let Yj, j= 1, 2,..., be independent exponential random variables with expectation 1. Let, for n= 1, 2,..., and Then n+l Wn= ~ g(Yj) (1.3) j=l n+l Sn= ~ (Yj-1). (1.4) j=l IS.=O), (1.5) i.e. T n has the same distribution as a sum of independent random variables given another sum of independent random variables. With the aid of (1.5) Does and Klaassen (1984a,b) proved Berry-Esseen bounds of the order n-1/2 for the normal approximation for statistics based on uniform spacings under natural moment assumptions. In Does and Helmers (1982) Edgeworth expansions were established for statistics of the form (1.2) under a natural moment assumption and an integrability condition on the simultaneous characteristic function of (Y- 1, g(Y)). In the present paper it is shown that the lat- ter integrability condition can be replaced by a much weaker and more natural Cram6r-type condition. This condition holds under an easily verifiable and mild assumption on the function g, as is stated in Theorem 2.1. Cram6r's condition for statistics of the general type f(X), where X is a random variable taking values in R" and f: IR m-, [Rk is a measurable function, is studied in Lemma 3.1. 2. An Edgeworth expansion Let Y be an exponential random variable with expectation 1 and let g be a fixed real-valued measurable function defined on [R + . Introduce, whenever well-defined, a function ~ by g(y) = (g(y) -tu- r(y- 1))(o 2 -- rE) - 1/2, y>0, (2.1) where and lu=Eg(Y), tr2 = Var g(Y) (2.2) r = Cov(g(Y), Y). (2.3)  R.J.M.M. Does et al. / Uniform spacings 151 Note that 0 -2= 2 2 iff g is linear and hence T~ is degenerate. We shall establish an asymptotic expansion with uniform remainder o(n-1) for the distribution function Fn(x ) =p((n(0-2_ 2-2))-I/2(T n- (n + 1)p)_x), xe ~. (2.4) Let ~ denote the characteristic function of (Y- 1, g(Y)), i.e. ~o(s, t) =Ee is(r- l)+itg(r), (S, t) e [~2. (2.5) Let q~ and ~ denote the distribution function and density of the standard normal distribution and let 1[. [] denote the Euclidian norm in ~2, i.e. I[(s, t)[[ = (s2+ t2) 1/2, for (s, t) e IR 2. Theorem 2.1. Let F~ be as in (2.4) (cf. (1.2), (2.2) and (2.3)) and let g [0, oo)~ IR be a measurable function such that Eg4(y) < oo (2.6) and for an interval (c, d)C (0, oo) on which g is almost everywhere differentiable with derivative g', then where g' is not essentially constant on (c, d); lira n sup {Fn(x)-Pn(x)[ =0, n "-'* oo XE~ with (2.7) (2.8) l~n(X ) = (~)(X) -- ~)(x){n - 1/2{ I K3(X 2 1) + a) + n -1 l X4(X 3 -- 3X) + 7~xZ(x 5- 10x 3 + 15x) + ~-(- 4ax 3 + b)x + +aK3x3)} (2.9) K3=E~,3(Y), x4=E~4(y)-3-3{(E~2(Y)(Y-1)} , a= -½E~(Y)(Y- 1) 2, b= 3{E~(Y)(Y- 1)2} 2- 2E~2(Y)(Y - 1) 2 +4E~2(Y)(Y - 1)+6. (2.10) We note that Condition (2.6) is obviously necessary for the expansion (2.9) to be well-defined. Instead of Condition (2.7) on the function g, Does and Helmers (1982) used an integrability condition on the simultaneous characteristic function of (Y- 1, g(Y)), i.e. f °°-o~ ~-oo ]Q(s't)lPdsdt<°°' for some p_> 1, (2.11) to validate the expansion (2.9). Integrability conditions like (2.11) are commonly en- countered in problems of establishing asymptotic expansions for conditional  152 R.J.M.M. Does et al. / Uniform spacings distributions (see e.g. Michel (1979)). An assumption equivalent to (2.11) is that there exists an integer k such that the k-th convolution of (Y- 1, g(Y)) has a bound- ed density (cf. Bhattacharya and Rao (1976), Theorem 19.1). According to the Riemann-Lebesgue lemma (cf. Theorem 4.1 in Bhattacharya and Rao (1976)) this implies that [p(s, t)lk~0 as II(s, t)[I ~ oo. Consequently (2.11) is much stronger than lim sup I~o(s, t) I < 1. (2.12) II(s, t)n -* From the proof of Theorem 2.1 it is clear that (2.6) and Cram6r's condition (2.12) suffice for (2.8) to hold. Lemma 3.1 shows that (2.7) implies (2.12). The fact is that this lemma gives conditions for the validity of Cram6r's condition for statistics of the general type f(X), where X is a random m-vector and f: [R"---, ~k is a measurable function. Lemma 3.1 extends Lemma 1.4 of Bhattacharya (1977) with a new and simpler proof. Another way to prove Cram6r's condition for a statistic of type f(X) is to show the stronger property that its distribution has a nonzero absolutely continuous component with respect to Lebesgue measure. This problem is treated in Sadikova (1966), Yurinskii (1972) and Bhattacharya and Ghosh (1978). The conditions of Lemma 2.2 of Bhattacharya and Ghosh (1978) are a little bit more restrictive than the conditions of Lemma 3.1, but the conclusion of their Lemma 2.2 is much stronger. In Section 3 of Pyke (1965) some examples of functions g are given. These func- tions are related to gl(x)=x', r>0, r~:l, g2(x)=lx-1 [, g3(x)=logx and g4(x) = X -1. The functions gl, g2 and g3 are all included in Theorem 2.1. Note that g4 does not satisfy (2.6). If we standardize the statistic Tn (cf. (1.2)) exactly then it should be possible to verify that under the assumptions of Theorem 2.1 relation (2.8) holds, with F,, replaced by the distribution function of (T~- ET~)(Var Tn)-1/2 and/~ by the right- hand side of (2.9) with a = b = 0. One may prove this by a refinement of Lemma 3.4 from Does and Klaassen (1984a). We note that, although we have proved our results for a fixed function g it seems to be possible to generalize Theorem 2.1 to functions gjn; i.e. functions depending on the j-th spacing and sample size n. A Berry-Esseen theorem for this more general case was proved in Does and Klaassen (1984b). 3. Proof of Theorem 2.1 Without loss of generality we may replace g by ~ (cf. (2.1)), because this does not affect F n and the assumptions of the theorem. In other words we assume that/~ = 0, tr2= 1 and r=0 (cf. (2.2) and (2.3)). Let Xn denote the characteristic function of n- 1/2 Tn; i.e.  R.J.M.M. Does et al. / Uniform spacings 153 N o 2:,(0 = e i'x dFn(x), (3.1) --GO with F,, as in (2.4). By Esseen's smoothing lemma (see e.g. Feller (1971), Lemma XVI 3.2) it suffices to prove that ' [X.(t)-)~.(t)] dt=o(n_ 1) It[<_nlogn [tl where )~, is the Fourier-Stieltjes transform of Fn (cf. (2.9)); i.e. (3.2) Since ,~n(t) = e itx d x) --oo e -`2/2 l+n--i~it-- K,3..it3 bt2+(K4+4aK3) 6n"~ 8n 24n t4-72n 32 t6 1" (3.3) ]t-l(,~,(t) - 1)]<_Eln -1/2 Tnt<n-l/2(n+ 1)Elg((n+ 1)Dln)l =hi/2 Ig(y)l 1--- dy<e 2//1/2 ]g(y)[e -y dy, 0 n+l 0 for any t, 25,(0)=)~n(0)= 1 and )~n has a bounded continuous derivative with respect to t, it is easily verified (see also (2.50) of Bickel and Van Zwet (1978)) that i It1-11x,(t)-~,(t)l t=O n-3/2). (3.4) itl<_n 2 According to Lemma 3.1 of Does and Klaassen (1984a) we can choose a regular version of the conditional distribution of n-1/2W~ given n-1/2Sn=x (cf. (1.3)-(1.5)), such that for this version Xn (t) = Ee itn ~/2 T,, = E(eitn ~/2 w,, I n - 1/2 Sn = 0). (3.5) Let ~n be the characteristic function of (n- 1/2S~, n- 1/2 Wn); i.e. q/n(s, t) = [o(sn-1/2, tn- 1/2)]n+ 1, (3.6) with 0 as in (2.5). With the aid of Plancherel's identity (see e.g. Theorem 4.1 of Bhattacharya and Rao (1976)) we check that for all t (cf. Does and Klaassen (1984a), formulas (3.15) and (3.16)), f qJ.(s, t)l ds= Io(sn-1/2, n-l/z)l +l ds --OO --OO ~ //1/2 ,t Go oo [O(s, tn- 1/2,,2~1 ds = 2nn 1/2 e- 2y_..dv = zrn ,10 /2 (3.7) Let h n be the density of//-1/2S n. In view of Lemma 3.1 of Does and Klaassen
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