sadists (version 0.2.5)

dneta: The doubly non-central Eta distribution.

Description

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

Usage

ddneta(x, df, ncp1, ncp2, log = FALSE, order.max=6)
pdneta(q, df, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)
qdneta(p, df, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6)
rdneta(n, df, ncp1, ncp2)

Value

ddneta gives the density, pdneta gives the distribution function, qdneta gives the quantile function, and rdneta generates random deviates.

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

Arguments

x, q

vector of quantiles.

df

the degrees of freedom for the denominator chi square. We do not recycle this 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. Note that the sign of ncp1 is important, while ncp2 must be non-negative.

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, \(P[X > x]\).

Author

Steven E. Pav

Details

Suppose \(Z\) is a normal with mean \(\delta_1\), and standard deviation 1, independent of \(X \sim \chi^2\left(\delta_2,\nu_2\right)\), a non-central chi-square with \(\nu_2\) degrees of freedom and non-centrality parameter \(\delta_2\). Then $$Y = \frac{Z}{\sqrt{Z^2 + X}}$$ takes a doubly non-central Eta distribution with \(\nu_2\) degrees of freedom and non-centrality parameters \(\delta_1,\delta_2\). The square of a doubly non-central Eta is a doubly non-central Beta variate.

See Also

(doubly non-central) t distribution functions, ddnt, pdnt, qdnt, rdnt.

(doubly non-central) Beta distribution functions, ddnbeta, pdnbeta, qdnbeta, rdnbeta.

Examples

rv <- rdneta(500, df=100,ncp1=1.5,ncp2=12)
d1 <- ddneta(rv, df=100,ncp1=1.5,ncp2=12)
# \donttest{
plot(rv,d1)
# }
p1 <- ddneta(rv, df=100,ncp1=1.5,ncp2=12)
# should be nearly uniform:
# \donttest{
plot(ecdf(p1))
# }
q1 <- qdneta(ppoints(length(rv)), df=100,ncp1=1.5,ncp2=12)
# \donttest{
qqplot(x=rv,y=q1)
# }