On the Completeness of Order Statistics

Abstract

Let $X_1, X_2, \cdots, X_n$ be a sample of a one-dimensional random variable $X$; let the order statistic $T(X_1, X_2, \cdots, X_n)$ be defined in such a manner that $T(x_1, x_2, \cdots, x_n) = (x^{(1)}, x^{(2)}, \cdots, x^{(n)})$ where $x^{(1)} \leqq x^{(2)} \leqq \cdots \leqq x^{(n)}$ denote the ordered $x's$; and let $\Omega$ be a class of one-dimensional cpf's, i.e., cumulative probability functions. The order statistic, $T$, is said to be a complete statistic with respect to the class, $\{P^{(n)} \mid P \epsilon \Omega\}$, of $n$-fold power probability distributions if $E_p^{(n)}\{h\lbrack T(X_1, \cdots, X_n)\rbrack\} = 0$ for all $P \epsilon \Omega$ implies $h\lbrack T(x_1, \cdots, x_n)\rbrack = 0, a.e., P^{(n)}$, for all $F \epsilon \Omega$. The class $\Omega$ is said to be symmetrically complete whenever the latter condition holds. Since the completeness of the order statistic plays an essential role in nonparametric estimation and hypothesis testing, e.g., Fraser [2] and Bell [1], it is of interest to determine those classes of cpf's for which the order statistic is complete. Many of the traditionally studied classes of cpf's on the real line are known to be symmetrically complete, e.g., all continuous cpf's ([4], pp. 131-134, 152-153); all cpf's absolutely continuous with respect to Lebesgue measure ([3], pp. 23-31); and all exponentials of a certain form ([4], pp. 131-134). The object of this note is to present a different ([4], pp. 131-134, 152-153) demonstration of the symmetric completeness of the class of all continuous cpf's; and to extend this and other known completeness results to probability spaces other than the real line, e.g., Fraser [2], and Lehmann and Scheffe [5], [6]. The paper is divided into four sections. Section 1 contains the introduction and summary. In Section 2 the notation and terminology are introduced. The main theorem is presented in Section 3, and some consequences of the proof of the main theorem and known results are indicated in Section 4.