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On some asymptotically optimal non-parametric criteria
A. A. Borovkov,
N. M. Sycheva Novosibirsk
Abstract:
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, $M(\theta_1,\theta_2)=\{t\colon\theta_1\le F^0(t)\le\theta_2\}$,
$0\le\theta_1<\theta_2<1$ 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.
Received: 10.10.1967