noncentral beta distribution について

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In probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a generalization of the (central) beta distribution.
The noncentral beta distribution (Type I) is the distribution of the ratio
:

X = \frac,

where

\chi^2_m(\lambda)

is a
noncentral chi-squared random variable with degrees of freedom ''m'' and noncentrality parameter

\lambda

, and

\chi^2_n

is a central chi-squared random variable with degrees of freedom ''n'', independent of

\chi^2_m(\lambda)

.
In this case,

X \sim \mbox\left(\frac,\frac,\lambda\right)

A Type II noncentral beta distribution is the distribution
of the ratio
:

Y = \frac,

where the noncentral chi-squared variable is in the denominator only.〔 If

Y

follows
the type II distribution, then

X = 1 - Y

follows a type I distribution.
== Cumulative distribution function ==

The Type I cumulative distribution function is usually represented as a Poisson mixture of central beta random variables:〔
:

F(x) = \sum_^\infty P(j) I_x(\alpha+j,\beta),

where λ is the noncentrality parameter, ''P''(.) is the Poisson(λ/2) probability mass function, ''\alpha=m/2'' and ''\beta=n/2'' are shape parameters, and

I_x(a,b)

is the incomplete beta function. That is,
:

F(x) = \sum_^\infty \frac\left(\frac\right)^je^I_x(\alpha+j,\beta).

The Type II cumulative distribution function in mixture form is
:

F(x) = \sum_^\infty P(j) I_x(\alpha,\beta+j).

Algorithms for evaluating the noncentral beta distribution functions are given by Posten and Chattamvelli.〔

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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