sadists (version 0.2.5)

dnt: The doubly non-central t distribution.

Description

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

Usage

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)

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.

Arguments

x, q

vector of quantiles.

df

the degrees of freedom for the denominator, \(\nu\). 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 \(\mbox{log}(f)\).

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

Let \(Z \sim \mathcal{N}\left(\mu,1\right)\) independently of \(X \sim \chi^2\left(\theta,\nu\right)\). The random variable $$T = \frac{Z}{\sqrt{X/\nu}}$$ takes a doubly non-central t distribution with parameters \(\nu, \mu, \theta\).

References

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

See Also

t distribution functions, dt, pt, qt, rt

Examples

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)
# \donttest{
plot(ecdf(pvs.H0))
plot(ecdf(pvs.HA))
# }
# 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)