@ARTICLE{18444395_1952,
author = {Anderson, Theodore W. and Darling , Donald A.},
keywords = {asymptotic theory, stochastic processes, непрерывное распределение, одинаково распределенные случайные величины, статистическая задача, статистическая модель},
title = {Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on
Stochastic Processes },
journal = {Annals of Mathematical Statistics},
year = {1952},
month = {},
volume = {23},
number = {2},
pages = {193-212},
url = {http://ecsocman.hse.ru/text/18444395/},
publisher = {},
language = {ru},
abstract = {The statistical problem treated is that of testing the hypothesis
that $n$ independent, identically distributed random variables have a
specified continuous distribution function F(x). If F_n(x) is the
empirical cumulative distribution function and \psi(t) is some
nonnegative weight function (0 \leqq t \leqq 1), we consider
n^{\frac{1}{2}} \sup_{-\infty and n\int^\infty_{-\infty}\lbrack F(x)
- F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x). A general method
for calculating the limiting distributions of these criteria is
developed by reducing them to corresponding problems in stochastic
processes, which in turn lead to more or less classical eigenvalue
and boundary value problems for special classes of differential
equations. For certain weight functions including \psi = 1 and \psi =
1/\lbrack t(1 - t) \rbrack we give explicit limiting distributions. A
table of the asymptotic distribution of the von Mises \omega^2
criterion is given. },
annote = {The statistical problem treated is that of testing the hypothesis
that $n$ independent, identically distributed random variables have a
specified continuous distribution function F(x). If F_n(x) is the
empirical cumulative distribution function and \psi(t) is some
nonnegative weight function (0 \leqq t \leqq 1), we consider
n^{\frac{1}{2}} \sup_{-\infty and n\int^\infty_{-\infty}\lbrack F(x)
- F_n(x) \rbrack^2 \psi\lbrack F(x)\rbrack dF(x). A general method
for calculating the limiting distributions of these criteria is
developed by reducing them to corresponding problems in stochastic
processes, which in turn lead to more or less classical eigenvalue
and boundary value problems for special classes of differential
equations. For certain weight functions including \psi = 1 and \psi =
1/\lbrack t(1 - t) \rbrack we give explicit limiting distributions. A
table of the asymptotic distribution of the von Mises \omega^2
criterion is given. }
}