(N + 1) TN = ainh(xi,n), (2.1) (N + 1) TN = E at,nh(uj,n), (2.2) denote a linear combination of ordered observations in the sample, while proposing

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1 ASYMPTOTIC ORMALITY OF LIEAR COMBIATIOS OF FUCTIOS OF ORDER STATISTICS, H1* BY ZAKKULA GOVIDARAJULU CASE ISTITUTE OF TECHOLOGY, CLEVELAD, OHIO Communicated by Jerzy eyman, January 15, 1968 (1) Introduction.-Mosteller6 introduced the term "systematic statistic" to denote a linear combination of ordered observations in the sample, while proposing some simple and useful estimates of location and scale parameters. Bennett' derived asymptotically optimal weight functions for estimating location and scale parameters on the basis of complete and multicensored samples. Quite independently, Jung5 derived these results for the uncensored case under rather restrictive conditions on the underlying distribution. Placket7 considered estimates based on a,4-trimmed samples (that is, observations below ath and above #3th sample percentiles are missing or discarded) and characterized the asymptotic optimal weights. Tukey8 advocated the use of a-trimmed and Winsorized means in robust estimation of the location parameter of symmetric populations. Weiss,9 while proposing an estimate for scale parameter based on a,f3-trimmed sample, studied the asymptotic normality of those estimates for which the weight function is of bounded variation. However, the asymptotic normality of systematic statistics based on complete samples has not been considered by the above authors. Recently, a systematic study of the asymptotic normality of linear combinations of functions of order statistics has been made by Govindarajulu,3 imposing sufficient conditions on the weight-functions and practically no restrictions on the underlying distribution functions. Chernoff, Gastwirth and Johns, Jr.,2 have also studied the same problem, imposing restrictions on the weight function as well as smoothness condition on the tails of the underlying distribution. Although their results are quite interesting, the sufficient conditions are too restrictive and the normalizing constants are complicated to evaluate (see Theorem 1 of Chernoff et al.2). Hence, it is of interest to unify, simplify, strengthen, and generalize the results of Chernoff et al.2 and to extend these results to situations the weights are expected values of certain order statistics, to vector-valued statistics, and to statistics based on random samples. (2) Assumptions and otation. Let X1,X2,...,X be a random sample of size drawn from a continuous population having F(x) for its cumulative distribution function (cdf). Let F-' be an inverse (not necessarily unique) of the cdf F. Then define the class of statistics of interest by ( + 1) T = aih(xi,), (2.1) i = 1 at are some given constants, XI, are the ordered X's, and h is a given continuous function. Let hf-' = H. Then, alternatively, one can rewrite T as ( + 1) T = E at,h(uj,), (2.2) i 1 713

2 714 MATHEMATICS: Z. GOVIDARAJULU PROC.. A. S. Ui(i = 1, 2,..., ) denote the order statistics in a random sample of size drawn from the uniform distribution on [0,1]. When h (x) = x, T generates the systematic statistics and by specializing the ai,, one can derive many well-known statistics. Let g2(u) > 1 be U-shaped and Lebesgue-square integrable and gl(u) be Lebesgue integrable for 0 < u < 1. Throughout, K denotes a generic constant independent of F and. Also, let t, = (i/ + 1), i = 1,2,...,. We shall assume that H is absolutely continuous and shall say: (i) H(or(a,H)) 8 if Ial,H'(t) < Kgl(ti)92(ti), (ii) H(or(a,H)) E S* if Ial{H(ti + ts) -H(ti)} < Kj ijgl(ti)92(ti) for ijj < K/g2(ti), (iii) H(or(a,H)) E 8o if Iaj,I < K[t(l - ti)] + ' and IH'(t1)I < K[ti(l _ t1)] /2 + a+sforsomeo< a < 3/2andO <8< 1/2, and (iv) H(or(a,H)) 81 if either a K 92(t,) and H'(t1) < K gi(t,) or IaiI < K g1(t,) and 'H'(ti)I < K92(ti), for i = 1,2,...,. Clearly, 8o C 8,8* C 8 and 81 C S. Also, a subset S of 8, 8*, 8, or 81 will be called relatively compact if every double sequence { ak, Hk} c S admits a convergent subsequence. (3) Certain Properties of Functions (a,h).-let ai, = a(i/ + 1), i = 1,2,.... Then we have the following lemma. LEMMA 3.1. Let A(u) = (1 - u)1fjol-u th'(t) a(t)dt, 0 < u < 1. (3.1) Then, there exists a number bo such that sup {r 12 (u)du} < bo. (3.2) (a,h) es O Furthermore, for every e > 0, there is a number b such that fw - B2(u)du < E, for every (ah) e 8 (3.3) JB(uj > b ~ Remark 3.1.1: If a, is not a continuous weight function, Lemma 3.1 remains valid provided the integrals are replaced by the appropriate summations. The function spaces { a and I H'} will be topologized by the topology of convergence in Lebesgue measure. These topologies can be induced by the metric C' fl - f21 dist. (f1(x),f2(x)) = I1 I f f dx, (34) f = a or H'. Let OD denote the class of distributions on the real line. One can topologize O by requiring F, F at every continuity point of F. This topology can also be induced by the bounded Lipschitz norm defined as follows. DEFIITIO 3.1. If P and Q are two finite signed measures on the real line, then IP - QI BL = suplfhdp-r hdqj, (3.5) the supremum is taken over all functions h such that h < 1 and h (x) - h(y)i < Ix - yl

3 VOL. 59, 1968 MATHEMATICS: Z. GOVIDARAJULU 715 (4) Asymptotic ormality of a Certain Sum of Random Variables.-In this section we will find a sufficient condition for the asymptotic normality of a random term of interest to us. Let having 1 W = ( + 1)11' E Bj( j)-16* (4.1) j=1 = Sj- 1, 5, being independent negative exponential random variables e-z, x > 0 for their cdf and + 1 -j Bj = E tih'(ti)ai,, t, = (i/ + 1), (4.2) H is an absolutely continuous function and aj, are some constants. Clearly, EW = 0 and var W = Z Bj2( j)-2. That is, j= 1 var W = ( + 1) 'E E titkhh(tj)h(tk)ajak,{ E j -2}. (4.3) i = 1 k = 1 j=max (i,k) Also, when aft = a(i/ + 1), we have O = var W -- 2 ff u(1 - v)a(u)a(v)h'(u)h'(v)dudv. (4.4) O< U < v < 1 From Lemma 3.1, it follows that var W is finite for all (a,h) S. Then we have the following lemma. LEMMA 4.1. Let W be as defined in (4.1) and a2 be given by either (4.3) or (4.4). If (ah) E 8, then, for every e > 0 a-2 Z f x2dgj(x)-o(-- ), (4.5) j =1 X > ef Gj(x) is the distribution function of ( + 1) - '12Bj( + 1 _ j) -- and Bj is given by (4.2). Remark 4.1.1: Condition (4.5) is the Lindeberg-Feller condition for the asymptotic normality of W. (5) Main Results.-In this section we shall state the main results of the paper and briefly sketch the proof of the basic result, namely, Theorem 5.1. THEOREM 5.1. Let T = ( + 1)-1/2 aj{h(ui) - H(ti)}, (5.1) H = hf-1, Uj denotes the ith smallest order statistic in a random sample of size drawn from the uniform distribution [0, 1], ax are some given constants (in particular, a,, could be a ((i/ + 1))), and tj = (i/ + 1), i = 1,2,....,. Let P be the distribution of T and Q be the normal distribution having zero expectation and variance (T2 [a,h] given by (4.3) (or (4.4)). If S is a relatively compact subset of S*, So, or 81, then, for every e > 0 there exists an (e) such that > (e) implies j P - QIIBL < 6 for every H(or (a,h)) E S. Further, there is an

4 716 MATHEMATICS: Z. GOVIDARAJULU PROc.. A. S. (e, b) such that > (eb) and a2 > b implies that the bounded Lipschitz norm may be replaced by the Kolmogorov vertical distance. Brief sketch of the proof: Write H(Ui.) - H(tj) = (Ui - tj)h'(tj) + [H(Ui) -H(ti) -(Use-ts)H'(tj)]. (5.2) Also, one can write Ui - ti = -S*+1-i,ti + { Ui + S*+1-,t- tj. (5.3) Also, S*+1-i, = S+J-i - E J = S+1-, + Ints - fl, (5.4) 6 =iz + 1 -i S*+I-.= Ej 6/( + 1 -j), 6 = 6-1, 6, are independent exponential random variables with the common cdf 1 -ek X >, 7Al' E j-1+lnt and 6=1 S+11, = In UI,. (5.5) Combining equations (5.1)-(5.5), we obtain 3 T = W + E Ck. (5.6) k= 1 W is given by (4.1), and C1, = ( + 1) E ai,h (ti){ UI, - tj - tj In Ui + tf In ti}, (5.7) and C2, = -( + 1)/2 E tlfl,al,h (tl), (5.8) C3, = ( + 1)-1/2 I {H(Ui.) - H(t) - (UI, - tl)h'(tj)}af,. (5.9) One can show (which is nontrivial, of course) that the higher-order random terms Ck,(k = 1,2,3) converge to zero in probability for every (a,h) E 8*, 8g, or 81. ow, W is precisely the random term studied in Section (4). So, because of Lemma 4.1, one can readily apply Lindeberg-Feller form of the central limit theorem to infer the first part of Theorem 5.1. The second statement of Theorem 5.1 follows from the first by simply considering W/T instead of W. This completes the brief sketch of the proof of Theorem 5.1. Remark 5.1.1: In equation (5.1) one can replace ( + 1)-1/2 a,h(t,) by ( + 1) 1/2 a(t)h(t)dt when a,, = a(i/ + 1) provided

5 VOL. 59, 1968 MATHEMATICS: Z. GOVIDARAJULU ( + 1)-'/2 2 a,h(ti) - ( + 1)1/2 j' a(t)h(t)dtj -* 0 as a. Remark 5.1.2: Theorem 5.1 unifies, simplifies, strengthens, and generalizes Theorems 1-3 of Chernoff et al.2 In some instances, the weights aj, may be expectations of suitable order statistics. In this regard we need the following further notation and state some results of interest. For any function a, let a be the function defined on (0,1) as follows. If y = k( + 1)-1, k = 1,2,...,, let a(y) = Ea(Uk) = fa(x)1(x,k)dx, (5.10) 13(xk) =![(k - 1)!( - k)!]-xk--j(1 -X)-k 0 < x < 1 (5.11) is the density of Uk* Complete the definition of a by interpolating linearly between successive values, { k/ ( + 1), (k + 1/ + 1) }, and leaving a constant below (1/ + 1) and above (/ + 1). Then, it is well known that a(y) a(y), 0 < y < 1 (see Hoeffding4), and we have the following theorem. THEOREM 5.2. Let ai, = a(ti) and T = ( + 1) -'{ 2a(ti)H(Ui,) - 2 a(ti)h(ti)}. (5.12) Let P be the distribution of T and Q be the distribution of a normal random variable having mean zero and variance given by (4.4). Then the conclusion of Theorem 6.1 holds for every (a,h) C S*, So, or 81 and satisfying either (i) a'(u)i < K[u(1 - u)]i1 + 8 IH(u)I < K[u(1 - u)f1i + a + or (ii) IH(u){a(u + t) -a(u)} < K W [u(1 -u)]- '+ for some < K[u(1-u)]l/2-8,K<,0<ai<,al2< 3/2and <6 < 1/2, or (iii) IHI and Ia'l are U-shaped and IH(u)a'(u) I K[u(1 -u)]-'/2 + Proof: One can write - T = ( + 1) 1/2{ a(tj)h(ui) - 2 a(tj)h(tj)} + ( + 1) - '/21{[(t) - a(ti)]h(tj)l. (5.13) One can show that the second term converges to zero as )o and also, Theorem 5.1 is applicable to the first term, because (a,h) C S*, SO, or 81 implies (&H) C S*3, Soy or 81. Further, one can replace a(t) by a(t) in the asymptotic variance of the first term (5.13). COROLLARY Let k be a fixed integer and let bj, j = 1,2,...,k be bounded constants. Let k a(i/ + 1) = Z bje[wj,), i = 1,2,...,, (5.14) Wi is the ith smallest order statistic in a random sample of size drawn from a population whose distribution function is the inverse of the function V. If (V',H) satisfies either (i) or (ii) of Theorem 5.2forj = 1,2,...,k, then the conclusion of Theorem 6.1 holds for T a(tj) are given by (5.14).

6 718 MATHEMATICS: Z. GOVIDARAJULU PROC.. A. S. (6) Case of Vector Random Variables.-We may be interested in the study of the joint asymptotic normality of the vector T = (T()...,T()), (6.1) - ( + 1) 1/ a(') [Hi(Uj,) - Hi(j/ + 1)], i = 1,2,...,c and Hi(u) = hif-1(u), i = 1,2,...,c. (6.2) In some cases = ni + n n,, n, is the size of the subsample drawn from G, (i = 1,2,... c), F = z (n1/)g, denotes the combined population cdf, and XI, are the order statistics in the pooled sample of size. In a straightforward manner the results of Section (5) are generalized to the vector case. (7) Case ofrandom Sample Size.-If there is an increasing sequence of positive numbers {* tending to infinity, and {,} is a sequence of random variables taking positive integer values such that r/* 1 in probability as r -, then the results of Sections (4) and (5) are extended to the case when is random. (8) Applications.-The statistics T defined in (2.1) includes many statistics that have appeared in the literature, for example, the sample quantile, the sample median, a-trimmed mean, and the a-winsorized mean. Jung's asymptotically optimal estimate of the scale parameter a- of a normal distribution is given by = ( + 1)-1 (P-1(i/ + 1)Xi,. (8.1) i = 1 The asymptotic normality of f follows from Theorem 5.1 with a = qp-1 and H = F-1 since (a,h) E S* or 8o. As a small sample modification, consider = ( + 1) '1 ixl, (8.2) /1i, denotes the expected value of the ith smallest standard normalorder statistic in a random sample of size. The asymptotic normality of ' followed from Theorem 5.2. Also, consider the following function of the sample spacings T = ( + 1) E ( i = 2 i)(xi Xi-1,) (8.3) = (X - X,,)/( + 1) (8.4) X denotes the sample mean. The asymptotic normality of T as given in (8.4) follows from Theorem 5.1 provided F-1 satisfies certain restrictions. Summary.-A class of linear combinations of functions of order statistics in a random sample drawn from an arbitrary population is defined and sufficient conditions for their asymptotic normality are given. The basic result of this paper unifies, strengthens, and generalizes the results of Chernoff, Gastwirth, and John, Jr.2 The cases when the weights are expected values of suitable order

7 VOL. 59, 1968 MATHEMATICS: Z. GOVIDARAJULU 719 statistics and when the sample size is random are also investigated. In a natural way, these results are extended to the vector-valued linear combinations of functions of order statistics. Some applications are also considered. * This research was supported in part by the Mathematics Division of the Air Force Office of Scientific Research under grant no. AF-AFOSR This research is also supported by the ational Science Foundation grant SF-GP Bennett, C. A., Asymptotic Properties of Ideal Linear Estimators, unpublished dissertation, University of Michigan (1952). 2Chernoff, H., J. L. Gastwirth, and M. V. Johns, Jr., "Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation," Ann. Math. Statist., 38, (1967). 3 Govindarajulu, Z., "Asymptotic normality of linear combinations of functions of order statistics in one and several samples," submitted to Ann. Math. Statist. 4Hoeffding, W., "On the distribution of the expected values of the order statistics," Ann. Math. Statist., 24, (1953). 5 Jung, J., "On linear estimates defined by a continuous weight function," Ark. Mat., 3, (1955). 6 Mosteller, F., "On some useful inefficient statistics," Ann. Math. Statist., 17, (1946). 7Plackett, R. L., "Linear estimation from censored data," Ann. Math. Statist., 29, (1958). 8 Tukey, J. W., "The future of data analysis," Ann. Math. Statist., 33, 1-67 (1962). 9 Weiss, L., "On the asymptotic distribution of an estimate of a scale parameter," aval Res. Logist. Quart., 10, 1-11 (1963).