Statistics & Probability Letters 6 (1988) 159-162 February 1988 North-Holland WHEN ARE INTERMEDIATE PROCESSES OF THE SAME STOCHASTIC ORDER? Bruce COOIL Owen Graduate School of Management, Vanderbilt University, Nashville, TN, USA Received April 1987 Revised June 1987 Abstract: Let Z(~ n) represent the ruth largest order statistic in a random sample of size n. Here we study the process Z[~,~I, t > 0, where re(n) is an intermediate sequence such that m ---,oo, m/n ~ 0 as n ~ oo. Keywords: intermediate order statistics, differentiable domains of attraction, extremal distribution. Let Z~~) >/Z: t") >I --- >/Z~ ~) represent the de- scending order statistics formed from a random sample of size n taken from a common distribu- tion function F. When re(n) is an intermediate sequence such that m~ and m/n~O asn~oo, (1) intermediate order statistics of the form 7(~) L'[mt], t > 0, have been used in a variety of ways to make statistical inferences about the upper tail of F (here [x] refers to the integer part of x, and t is fixed with respect to n). For examples of such estimators see Pickands (1975), Davis and Resnick (1984), Hall and Welsh (1984, 1985), Csbrg~5 et al. (1985), Smith and Weissman (1985), and Smith (1986). One can study the asymptotic joint distri- bution of such estimators by considering the sto- chastic process of the form to~n)(t ) =- k [ L'[ mt]y(n) -- Q( mt/n ) )/a~ ~) (2) where a~ ") > 0 does not depend on t and where Q(.) is the quantile function of the upper tail of F, Q(p)=inf{y:l-F(y)<~p}, 0<p~<l. (3) Research supported by the 1987 Dean's Fund of the Owen Graduate School of Management and the Vanderbih Univer- sity Research Council. 0167-7152/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland) Of particular interest are conditions under which there exists a scaling sequence a~ ") > 0 such that to~")(t) has a limiting stochastic process to(t) as n --* oo. In this paper we find a necessary condi- tion for the existence of sequences m(n) and a~ n) such that to~")(t), t > 0, has a nondegenerate limit- ing process to(t), which applies whenever F is continuously differentiable in the upper tail. (This condition has already been shown to be sufficient.) We will say to~")(t) has a limiting stochastic process to(t) and write to~")(t) ~D to(t), t > 0, whenever all finite dimensional distributions of the process to~n)(t) converge to those of to(t): (to n) ( tl ) ..... to(ran) ( tk )) d( to(tl) ..... to( tk )) (4) as n---,oo for any values 0<t 1< ... <t k<oo. The weak convergence of the stochastic process to~n)(t) to a nondegenerate limit to(t) is of particu- lar interest because it implies that the centered intermediate order statistics Z (n) - I-,,1 Q(mt/n) are asymptotically of the same stochastic order for all t. When this is true, it becomes possible to esti- mate, among other things, functions of extreme quantiles of the form Q(mt/n), t > O, or the limits of such functions as n-~ oo, by using the same functions of the corresponding intermediate order statistics even when these functions do not rescale 159