On some asymptotically optimal non-parametric criteria

A. A. Borovkov ; N. M. Sycheva

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 3, pp. 385-417

Résumé

Let $X$ be a simple sample of size $n$ from a continuous distribution function $F(x)$, $F_n(x)$ be the empirical distribution function determined by this sample. Let $$ G_s(X)=\sqrt n\sup_{M(\theta_1,\theta_2)}\frac{F^0(t)-F_n(t)}{g(F^0(t))}, $$ where $F^0$ is a continuous distribution function, $M(\theta_1,\theta_2)=\{t\colon\theta_1\le F^0(t)\le\theta_2\}$, $0\le\theta_1\theta_21$ are fixed, $g(t)$ belongs to the set of analytic on $[\theta_1,\theta_2]$ nonvanishing functions. A class of tests $\{G_g(X)\ge x\}$ ($g\in K(\theta_1,\theta_2)$) based on the statistics $G_g(X)$ for testing of the hypothesis $F=F^0$ against some set of a alternatives $F^1$ separafed from $F^0$ by a fixed distance $$ 0\delta\le\sup|F^0(t)-F^1(t)| $$ is considered. On this set some probabilistic measure $\mu$ is given. If a sequence of errorsof the first kind $\varepsilon=\mathbf P\{G_g(X)\ge x\mid F^0\}=\varepsilon(n)\to0$ as $n\to\infty$ is fixed, then it turns out to be possible to find a function $\psi$ independent of $\mu$ that realizes the asymptotically/ most powerful test. The form of the function $\psi$ and the asymptotical formulas for the distribution $G_\psi(X)$ as $n\to\infty$ are given in the paper. Also the tables of quantiles of $G_\psi(X)$ for different $n$'s, a number of significance levels and intervals $[\theta_1,\theta_2]$ are given.