Order Statistics and Statistics of Structure $(d)$

Abstract

This note discusses the asymptotic independence of the $(d)$-structured order statistics $F(X^{(i)})$ and some of the more usual statistics of structure $(d)$. (For a discussion of structure $(d)$ and related concepts, see references [2], [3] and [4].) Asymptotic independence holds, in particular, in the case of the Kolmogoroff-Smirnoff statistic, and thus provides approximate significance levels for the simultaneous test of the hypothesis that the population c.d.f. has specified form and the hypothesis that the sample contains no outlying observations. If the Kolmogoroff-Smirnoff statistic is used in conjunction with the largest order statistic $X^{(n)}$, the acceptance region of the resulting test can be characterized as follows: For acceptance (of the hypothesis that the population c.d.f. has specified shape and that the sample contains no outliers), the $n$ risers of the sample c.d.f. must fall within a three-sided region. This region is the intersection of the usual Kolmogoroff-Smirnoff region with the half-plane to the left of the critical value for $X^{(n)}$. The example of the last section deals with this case. Section 1 contains the theory on which the subsequent development is based. It involves the multivariate probability integral transformation $T_G$ discussed in [10], which has the property that $T_G(X), X$ distributed according to $G$, is uniform over the unit cube, so that $T_H^{-1}T_G(X)$ has distribution $H$. The transformation enters the argument in essentially this way: If $Y$ is a vector distributed according to $H,f$ and $g$ are two functions of $Y$, and $H(t)$ is the conditional distribution of $Y$, given $g(Y) \leqq t$, then $f(Y)$ and $g(Y)$ are independent if and only, for essentially all $t, f(Y)$ and $f(T_H^{-1}T_{H(t)}(Y))$ have the same distribution conditionally on $g(Y) \leqq t$. In the present application, $Y$ is a random sample from the uniform distribution, $g$ is an order statistic of this sample, and $f$ is the function of $Y$ corresponding to a statistic of structure $(d)$ such as the Kolmogoroff-Smirnoff statistic or Sherman's statistic [12]. It may appear that the above method recommends itself primarily on grounds of novelty, rather than suitability. This is borne out by the fact that the relatively weak requirement that the distributions of $f(Y)$ and $f(T_H^{-1}T_{H(t)}, Y))$ agree asymptotically is verified below by showing that in fact $f(Y)$ and $f(T_H^{-1}T_{H(t)}(Y))$ themselves agree asymptotically, i.e., that their difference converges in probability to zero. This last is made to follow in turn from the very rapid covergence of $|Y - T_H^{-1}T_{H(t)}(Y)|$ to zero. One has the option of having $H$ or $g$ perform the ordering of the sample By this is meant that $H$ can be the uniform distribution over the unit cube, and $g$ the function $k$th largest coordinate of $Y$. Or $H$ can be the joint distribution of the order statistics of $Y$ and $g$ the function $k$th coordinate of $Y$. The second of these two approaches yields a form for $T^{-1}_HT_{H(t)}$ especially suitable (due to the common multiplier $C_n(x, t, l)$ appearing in Equations (1)) for establishing the convergence to zero of $f(Y) - f(T^{-1}_HT_{H(t)}(Y))$ for a fairly large class of statistics $f$. These include the statistics of Kolmogoroff-Smirnoff type, whose essential combining operation is sup, and also certain other statistics with less tractable combining operations, such as Sherman's statistic, where the combining operation is addition. Section 2 is devoted to this second approach, applied to Sherman's statistic. The first approach, though seemingly unnatural for statistics with combining operations more demanding than sup, has the advantage of yielding expressions as easy to manipulate for bivariate (e.g., two order statistics) as for univariate $g$. Section 3 is devoted to this first approach, applied to statistics of Kolmogoroff-Smirnoff type, with $g$ bivariate. The last section contains an application, together with a computation indicating that independence sets in quite rapidly.