The noncentral beta distribution (Type I) is the distribution of the ratio
where is a
noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter , and is a central
chi-squared random variable with degrees of freedom n, independent of .[1]
In this case,
A Type II noncentral beta distribution is the distribution
of the ratio
where the noncentral chi-squared variable is in the denominator only.[1] If follows
the type II distribution, then follows a type I distribution.
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 is the
incomplete beta function. That is,
where is the
beta function, and are the shape parameters, and is the
noncentrality parameter. The density of Y is the same as that of 1-X with the degrees of freedom reversed.[1]
Related distributions
Transformations
If , then follows a
noncentral F-distribution with degrees of freedom, and non-centrality parameter .
If follows a
noncentral F-distribution with numerator degrees of freedom and denominator degrees of freedom, then
follows a noncentral Beta distribution:
.
This is derived from making a straightforward transformation.
Special cases
When , the noncentral beta distribution is equivalent to the (central)
beta distribution.
^
abcdeChattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234.
doi:
10.1080/00031305.1995.10476151.
^Posten, H.O. (1993). "An Effective Algorithm for the Noncentral Beta Distribution Function". The American Statistician. 47 (2): 129–131.
doi:
10.1080/00031305.1993.10475957.
JSTOR2685195.