dnbeta {sadists}

The doubly non-central Beta distribution.

Description

Density, distribution function, quantile function and random generation for the doubly non-central Beta distribution.

Usage

ddnbeta(x, df1, df2, ncp1, ncp2, log = FALSE, order.max=6)

pdnbeta(q, df1, df2, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)

qdnbeta(p, df1, df2, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)

rdnbeta(n, df1, df2, ncp1, ncp2)

Arguments

x, q

vector of quantiles.

df1, df2

the degrees of freedom for the numerator and denominator. We do not recycle these versus the x,q,p,n.

ncp1, ncp2

the non-centrality parameters for the numerator and denominator. We do not recycle these versus the x,q,p,n.

log

logical; if TRUE, densities f are given as \mbox{log}(f).

order.max

the order to use in the approximate density, distribution, and quantile computations, via the Gram-Charlier, Edeworth, or Cornish-Fisher expansion.

p

vector of probabilities.

n

number of observations.

log.p

logical; if TRUE, probabilities p are given as \mbox{log}(p).

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x].

Details

Suppose x_i \sim \chi^2\left(\delta_i,\nu_i\right) be independent non-central chi-squares for i=1,2. Then

Y = \frac{x_1}{x_1 + x_2}

takes a doubly non-central Beta distribution with degrees of freedom \nu_1, \nu_2 and non-centrality parameters \delta_1,\delta_2.

Value

ddnbeta gives the density, pdnbeta gives the distribution function, qdnbeta gives the quantile function, and rdnbeta generates random deviates.

Invalid arguments will result in return value NaN with a warning.

Note

The PDF, CDF, and quantile function are approximated, via the Edgeworth or Cornish Fisher approximations, which may not be terribly accurate in the tails of the distribution. You are warned.

The distribution parameters are not recycled with respect to the x, p, q or n parameters, for, respectively, the density, distribution, quantile and generation functions. This is for simplicity of implementation and performance. It is, however, in contrast to the usual R idiom for dpqr functions.

Author(s)

Steven E. Pav [e-mail]

See Also

(doubly non-central) F distribution functions, ddnf, pdnf, qdnf, rdnf.

Examples

rv <- rdnbeta(500, df1=100,df2=500,ncp1=1.5,ncp2=12)
d1 <- ddnbeta(rv, df1=100,df2=500,ncp1=1.5,ncp2=12)

plot(rv,d1)

p1 <- ddnbeta(rv, df1=100,df2=500,ncp1=1.5,ncp2=12)
# should be nearly uniform:

plot(ecdf(p1))

q1 <- qdnbeta(ppoints(length(rv)), df1=100,df2=500,ncp1=1.5,ncp2=12)

qqplot(x=rv,y=q1)