Martingale approach in the theory of goodness-of-fit tests
Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 246-265
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Let us consider the parametric hypothesis that the distribution function $F$ of i. i. d. random variables belongs to the given parametric family of distribution functions $\mathbf F=\{F(x,\theta),\,\theta\in\Theta\}$. It is well-known that the limiting distribution of the parametric empirical process
$\sqrt n[F_n(x)-F(x,\widehat\theta)]$ depends on $F$, and therefore the «usual» testing procedures become inconvenient.
In the paper we consider the parametric empirical process as a semimartingale. By means of its Doob–Meyer decomposition we construct some martingale and show that this martingale converges weakly to the Wiener process. This fact enables us to use the distribution-free asymptotic theory of testing parametric hypotheses.
@article{TVP_1981_26_2_a1,
author = {E. V. Hmaladze},
title = {Martingale approach in the theory of goodness-of-fit tests},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {246--265},
publisher = {mathdoc},
volume = {26},
number = {2},
year = {1981},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a1/}
}
E. V. Hmaladze. Martingale approach in the theory of goodness-of-fit tests. Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 2, pp. 246-265. http://geodesic.mathdoc.fr/item/TVP_1981_26_2_a1/