pnt {DPQ}

Non-central t Probability Distribution - Algorithms and Approximations

Description

Compute different approximations for the non-central t-Distribution cumulative probability distribution function.

Usage

pntR      (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
           use.pnorm = (df > 4e5 ||
                        ncp^2 > 2*log(2)*(-.Machine$double.min.exp)),
                                          itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntR1     (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
           use.pnorm = (df > 4e5 ||
                        ncp^2 > 2*log(2)*(-.Machine$double.min.exp)),
                                          itrmax = 1000, errmax = 1e-12, verbose = TRUE)

pntP94    (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
                                          itrmax = 1000, errmax = 1e-12, verbose = TRUE)
pntP94.1  (t, df, ncp, lower.tail = TRUE, log.p = FALSE,
                                          itrmax = 1000, errmax = 1e-12, verbose = TRUE)

pnt3150   (t, df, ncp, lower.tail = TRUE, log.p = FALSE, M = 1000, verbose = TRUE)
pnt3150.1 (t, df, ncp, lower.tail = TRUE, log.p = FALSE, M = 1000, verbose = TRUE)

pntLrg    (t, df, ncp, lower.tail = TRUE, log.p = FALSE)

pntJW39   (t, df, ncp, lower.tail = TRUE, log.p = FALSE)
pntJW39.0 (t, df, ncp, lower.tail = TRUE, log.p = FALSE)

Arguments

t

vector of quantiles (called q in pt(..)).

df

degrees of freedom (> 0, maybe non-integer). df = Inf is allowed.

ncp

non-centrality parameter \delta \ge 0; If omitted, use the central t distribution.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X \le x], otherwise, P[X > x].

use.pnorm

logical indicating if the pnorm() approximation of Abramowitz and Stegun (26.7.10) should be used, which is available as pntLrg().

The default corresponds to R pt()'s own behaviour (which is most probably suboptimal).

itrmax

number of iterations / terms.

errmax

convergence bound for the iterations.

verbose

logical or integer determining the amount of diagnostic print out to the console.

M

positive integer specifying the number of terms to use in the series.

Details

pntR1():

a pure R version of the (C level) code of R's own pt(), additionally giving more flexibility (via arguments use.pnorm, itrmax, errmax whose defaults here have been hard-coded in R's C code).

This implements an improved version of the AS 243 algorithm from Lenth(1989);

R's help on non-central pt() says:

This computes the lower tail only, so the upper tail suffers from cancellation and a warning will be given when this is likely to be significant.

and (in ‘Note:’)

The code for non-zero ncp is principally intended to be used for moderate values of ncp: it will not be highly accurate, especially in the tails, for large values.

pntR():

the Vectorize()d version of pntR1().

pntP94(), pntP94.1():

New versions of pntR1(), pntR(); using the Posten (1994) algorithm. pntP94() is the Vectorize()d version of pntP94.1().

pnt3150(), pnt3150.1():

Simple inefficient but hopefully correct version of pntP94..() This is really a direct implementation of formula (31.50), p.532 of Johnson, Kotz and Balakrishnan (1995)

pntLrg():

provides the pnorm() approximation (to the non-central t) from Abramowitz and Stegun (26.7.10), p.949; which should be employed only for large df and/or ncp.

pntJW39.0():

use the Jennett & Welch (1939) approximation see Johnson et al. (1995), p. 520, after (31.26a). This is still fast for huge ncp but has wrong asymptotic tail for |t| \to \infty. Crucially needs b=b_chi(df).

pntJW39():

is an improved version of pntJW39.0(), using 1-b =b_chi(df, one.minus=TRUE) to avoid cancellation when computing 1 - b^2.

Value

a number for pntJKBf1() and .pntJKBch1().

a numeric vector of the same length as the maximum of the lengths of x, df, ncp for pntJKBf() and .pntJKBch().

Author(s)

Martin Maechler

References

Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions Vol~2, 2nd ed.; Wiley; chapter 31, Section 5 Distribution Function, p.514 ff

Lenth, R. V. (1989). Algorithm AS 243 — Cumulative distribution function of the non-central t distribution, JRSS C (Applied Statistics) 38, 185–189.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover; formula (26.7.10), p.949

See Also

pt, for R's version of non-central t probabilities.

Examples

tt <- seq(0, 10, len = 21)
ncp <- seq(0, 6, len = 31)
dt3R   <- outer(tt, ncp, pt, , df = 3)
dt3JKB <- outer(tt, ncp, pntR, df = 3)# currently verbose
stopifnot(all.equal(dt3R, dt3JKB, tolerance = 4e-15))# 64-bit Lnx: 2.78e-16