@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. } }