# dsumchisqpow: The sum of (non-central) chi-squares raised to powers... In sadists: Some Additional Distributions

## Description

Density, distribution function, quantile function and random generation for the distribution of the weighted sum of non-central chi-squares taken to powers.

## Usage

 ```1 2 3 4 5 6 7``` ```dsumchisqpow(x, wts, df, ncp=0, pow=1, log = FALSE, order.max=6) psumchisqpow(q, wts, df, ncp=0, pow=1, lower.tail = TRUE, log.p = FALSE, order.max=6) qsumchisqpow(p, wts, df, ncp=0, pow=1, lower.tail = TRUE, log.p = FALSE, order.max=6) rsumchisqpow(n, wts, df, ncp=0, pow=1) ```

## Arguments

 `x, q` vector of quantiles. `wts` the vector of weights. This is recycled against the `df, ncp, pow`, but not against the `x,q,p,n`. `df` the vector of degrees of freedom. This is recycled against the `wts, ncp, pow`, but not against the `x,q,p,n`. `ncp` the vector of non-centrality parameters. This is recycled against the `wts, df, pow`, but not against the `x,q,p,n`. `pow` the vector of the power parameters. This is recycled against the `wts, df, ncp`, but not against 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

Let X_i ~ chi^2(delta_i, v_i) be independently distributed non-central chi-squares, where v_i are the degrees of freedom, and delta_i are the non-centrality parameters. Let w_i and p_i be given constants. Suppose

Y = sum w_i (X_i)^(p_i).

Then Y follows a weighted sum of chi-squares to power distribution.

## Value

`dsumchisqpow` gives the density, `psumchisqpow` gives the distribution function, `qsumchisqpow` gives the quantile function, and `rsumchisqpow` 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.

The 'sum of chisquare power' distribution does not generalize the 'chi-bar-square' distribution, whose density is the sum of chi-square densities.

## Author(s)

Steven E. Pav shabbychef@gmail.com

The upsilon distribution, `dupsilon,pupsilon,qupsilon,rupsilon`.
 ```1 2 3 4 5 6 7 8``` ```wts <- c(1,-3,4) df <- c(100,20,10) ncp <- c(5,3,1) pow <- c(1,0.5,1) rvs <- rsumchisqpow(128, wts, df, ncp, pow) dvs <- dsumchisqpow(rvs, wts, df, ncp, pow) qvs <- psumchisqpow(rvs, wts, df, ncp, pow) pvs <- qsumchisqpow(ppoints(length(rvs)), wts, df, ncp, pow) ```