# ddnt: The doubly non-central t distribution. In shabbychef/sadists: Some Additional Distributions

## Description

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

## Usage

 ```1 2 3 4 5 6 7``` ```ddnt(x, df, ncp1, ncp2, log = FALSE, order.max=6) pdnt(q, df, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6) qdnt(p, df, ncp1, ncp2, lower.tail = TRUE, log.p = FALSE, order.max=6) rdnt(n, df, ncp1, ncp2) ```

## Arguments

 `x, q` vector of quantiles. `df` the degrees of freedom for the denominator, v. We do not recycle these versus the `x,q,p,n`. `ncp1, ncp2` the non-centrality parameters for the numerator and denominator, respectively, mu and theta 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

Let Z ~ N(u,1) independently of X ~ x^2(theta,v). The random variable

T = Z / sqrt(X/v)

takes a doubly non-central t distribution with parameters v, mu, theta.

## Value

`ddnt` gives the density, `pdnt` gives the distribution function, `qdnt` gives the quantile function, and `rdnt` 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]

## References

Krishnan, Marakatha. "Series Representations of the Doubly Noncentral t-Distribution." Journal of the American Statistical Association 63, no. 323 (1968): 1004-1012.

t distribution functions, `dt, pt, qt, rt`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17``` ```rvs <- rdnt(128, 20, 1, 1) dvs <- ddnt(rvs, 20, 1, 1) pvs.H0 <- pdnt(rvs, 20, 0, 1) pvs.HA <- pdnt(rvs, 20, 1, 1) ## Not run: plot(ecdf(pvs.H0)) plot(ecdf(pvs.HA)) ## End(Not run) # compare to singly non-central dv1 <- ddnt(1, df=10, ncp1=5, ncp2=0, log=FALSE) dv2 <- dt(1, df=10, ncp=5, log=FALSE) pv1 <- pdnt(1, df=10, ncp1=5, ncp2=0, log.p=FALSE) pv11 <- pdnt(1, df=10, ncp1=5, ncp2=0.001, log.p=FALSE) v2 <- pt(1, df=10, ncp=5, log.p=FALSE) q1 <- qdnt(pv1, df=10, ncp1=5, ncp2=0, log.p=FALSE) ```