sadists (version 0.2.0)

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)

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.

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, $P[X > x]$.

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.

Details

Suppose $Z$ is a normal with mean $\delta_1$, 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)
plot(rv,d1)
p1 <- ddneta(rv, df=100,ncp1=1.5,ncp2=12)
# should be nearly uniform:
plot(ecdf(p1))
q1 <- qdneta(ppoints(length(rv)), df=100,ncp1=1.5,ncp2=12)
qqplot(x=rv,y=q1)