# ddneta: The doubly non-central Eta distribution. In sadists: Some Additional Distributions

## Description

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

## Usage

 ```1 2 3 4 5 6 7``` ```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 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 log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

## Details

Suppose Z is a normal with mean delta_1, independent of X ~ X^2(delta_2,v_2), a non-central chi-square with v_2 degrees of freedom and non-centrality parameter delta_2. Then

Y = Z/sqrt(Z^2 + X)

takes a doubly non-central Eta distribution with v_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.

## 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.

## 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 [email protected]

(doubly non-central) t distribution functions, `ddnt, pdnt, qdnt, rdnt`.
(doubly non-central) Beta distribution functions, `ddnbeta, pdnbeta, qdnbeta, rdnbeta`.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```rv <- rdneta(500, df=100,ncp1=1.5,ncp2=12) d1 <- ddneta(rv, df=100,ncp1=1.5,ncp2=12) ## Not run: plot(rv,d1) ## End(Not run) p1 <- ddneta(rv, df=100,ncp1=1.5,ncp2=12) # should be nearly uniform: ## Not run: plot(ecdf(p1)) ## End(Not run) q1 <- qdneta(ppoints(length(rv)), df=100,ncp1=1.5,ncp2=12) ## Not run: qqplot(x=rv,y=q1) ## End(Not run) ```