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5 papers
Short Communications
On parametric hypotheses testing with nonparametric tests
Yu. N. Tyurin Moscow
Abstract:
Let
$\xi_1,\dots,\xi_n$ be a sample with a theoretical distribution function s
$F(t,\theta)$. Here
$\theta$ is an unknown
$r$-dimensional parameter. We consider the “regular” case when its maximum likelihood estimator
$\theta^*$ has usual asymptotical properties. We denote the empirical distribution function by
$F_n(t)$.
In this paper, limiting properties of
$\sqrt n[F_n(t)-F(t,\theta^*)]$ are discussed. It is proved that
$\lim\sqrt n[F_n(t)-F(t,\theta^*)]$ is a conditioned Gaussian process. After a natural change of the time variable
$s=F(t,\theta^*)$ we obtain a conditioned Wiener process
$v(s)$ on
$[0,1]$ satisfying
$r$ linear conditions
$$
\int_0^1m_i(s,\theta)\,dv(s)=0,\quad i=1,\dots,r,
$$
and
$v(1)=0$. If
$\theta$ is the location-scale parameter the conditions are free of
$\theta$. A linear transformation
$v\to\tilde v$ is constructed, where the Wiener process
$\tilde v(s)$ satisfies
$r+1$ conditions:
$$
\tilde v(0)=\tilde v(t_1)=\dots=\tilde v(t_r)=\tilde v(1)=0.
$$
Quantities
$0<t_1<\dots<t_r<1$ can be chosen arbitrarily.
Now it is possible to use for the process
$\tilde v$ such well-known goodness-of-fit tests as Kolmogorov's or Cramer–von Mises' ones.
Received: 02.08.1969