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Teoriya Veroyatnostei i ee Primeneniya, 1970, Volume 15, Issue 4, Pages 745–749
(Mi tvp1942)
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This article is cited in 5 scientific papers (total in 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
Citation:
Yu. N. Tyurin, “On parametric hypotheses testing with nonparametric tests”, Teor. Veroyatnost. i Primenen., 15:4 (1970), 745–749; Theory Probab. Appl., 15:4 (1970), 722–726
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