Распределение Колмогорова

Распределе́ние Колмого́рова в теории вероятностей — это абсолютно непрерывное распределение, широко используемое в математической статистике для оценки распределения выборки.

Распределение Колмогорова
Носитель	$[0,\infty )$
Функция распределения	$\sum _{k=-\infty }^{\infty }(-1)^{k}\exp(-2k^{2}x^{2}),\,x>0$

Определение

Случайная величина $X$ имеет распределение Колмогорова, если её функция распределения $F_{X}$ имеет вид:

F_{X}(x)=K(x)={\begin{cases}\sum \limits _{k=-\infty }^{\infty }(-1)^{k}e^{-2k^{2}x^{2}},&x>0;\\0,&x\leqslant 0.\end{cases}}

Обозначается: $X\sim \mathrm {K}$ .

Связь с другими объектами теории вероятностей

Случайная величина

X=\sup _{t\in [0,1]}|B(t)|,

где $B(t)$ — броуновский мост, имеет распределение Колмогорова.

Пусть $X_{1},\ldots ,X_{n}$ — выборка объёма $n$ , порождённая случайной величиной с непрерывной функцией распределения $F(x)$ . Пусть $F_{n}(x)$ — выборочная функция распределения. Если ввести обозначение

D_{n}=\sup \limits _{x\in \mathbb {R} }|F_{n}(x)-F(x)|,

то выполняется

\lim _{n\to \infty }P({\sqrt {n}}D_{n}\leqslant t)=K(t)

Утверждение носит название теоремы Колмогорова. Его можно использовать для нахождения доверительного интервала функции распределения $F(x)$ .

См. также

Литература

J. Durbin, Regional Conf. Series on Applied Math. 9 (SIAM, 1973)).

Вероятностные распределения

Дискретные

Абсолютно
непрерывные

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