The Asymptotic Sufficiency of a Relatively Small Number of Order Statistics in Tests of Fit

Abstract

For each $n, X_n(1), \cdots, X_n(n)$ are independent and identically distributed continuous random variables over $(0, 1)$, with common density function equal to $1 + r(x)/n^{\frac{1}{2}}, r(x)$ unknown but satisfying certain regularity conditions. The problem is to test the hypothesis that $r(x) = 0$ for all $x$ in (0, 1). $Y_n(1) < \cdots < Y_n(n)$ are the ordered values of $X_n(1), \cdots, X_n(n). \delta$ is a fixed value in the open interval $(\frac{3}{4}, 1)$. It is shown that $Y_n(\lbrack n^\delta\rbrack), Y_n(2\lbrack n^\delta\rbrack), \cdots$ are asymptotically sufficient, and can be assumed to have a joint normal distribution for all asymptotic purposes. Using these facts, a test of the hypothesis is constructed with a good asymptotic power curve.