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\eqn{B(a,b) = B_1(a,b)} is the Beta function (\code{\link{beta}}).
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\eqn{B(a,b) = B_1(a,b)} is the Beta function (\code{\link{beta}}).
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\eqn{I_x(a,b)} is \code{pbeta(x, a, b)}.
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\eqn{I_x(a,b)} is \code{pbeta(x, a, b)}.
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The noncentral Beta distribution (with \code{ncp} \eqn{ = \lambda})
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The noncentral Beta distribution (with \code{ncp} \eqn{ = \lambda})
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is defined
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is defined (Johnson \abbr{et al.}, 1995, pp.\sspace{}502) as the distribution of
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\bibcitep{|R:Johnson+Kotz+Balakrishnan:1995|pp.\sspace{}502}
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as the distribution of
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\eqn{X/(X+Y)} where \eqn{X \sim \chi^2_{2a}(\lambda)}{X ~ chi^2_2a(\lambda)}
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\eqn{X/(X+Y)} where \eqn{X \sim \chi^2_{2a}(\lambda)}{X ~ chi^2_2a(\lambda)}
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and \eqn{Y \sim \chi^2_{2b}}{Y ~ chi^2_2b}.
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and \eqn{Y \sim \chi^2_{2b}}{Y ~ chi^2_2b}.
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There, \eqn{\chi^2_n(\lambda)}{chi^2_n(\lambda)} is the noncentral
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There, \eqn{\chi^2_n(\lambda)}{chi^2_n(\lambda)} is the noncentral
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chi-squared distribution with \eqn{n} degrees of freedom and
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chi-squared distribution with \eqn{n} degrees of freedom and
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non-centrality parameter \eqn{\lambda}, see \link{Chisquare}.
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non-centrality parameter \eqn{\lambda}, see \link{Chisquare}.
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\itemize{
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\itemize{
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\item The central \code{dbeta} is based on a binomial probability, using code
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\item The central \code{dbeta} is based on a binomial probability, using code
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contributed by Catherine Loader (see \code{\link{dbinom}}) if either
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contributed by Catherine Loader (see \code{\link{dbinom}}) if either
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shape parameter is larger than one, otherwise directly from the definition.
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shape parameter is larger than one, otherwise directly from the definition.
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The non-central case is based on the derivation as a Poisson
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The non-central case is based on the derivation as a Poisson
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mixture of betas
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mixture of betas (Johnson \abbr{et al.}, 1995, pp.\sspace{}502--3).
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\bibcitep{|R:Johnson+Kotz+Balakrishnan:1995|pp.\sspace{}502--3}.
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\item The central \code{pbeta} for the default (\code{log_p = FALSE})
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\item The central \code{pbeta} for the default (\code{log_p = FALSE})
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uses a C translation based on \bibcitep{R:Didonato+Morris:1992}.
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uses a C translation based on \bibcitep{R:Didonato+Morris:1992}.
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(See also \bibcitet{R:Brown+Levy:1994}.)
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(See also \bibcitet{R:Brown+Levy:1994}.)
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