pnchisqAppr {DPQ}R Documentation

(Approximate) Probabilities of Non-Central Chi-squared Distribution

Description

Compute (approximate) probabilities for the non-central chi-squared distribution.

The non-central chi-squared distribution with df= n degrees of freedom and non-centrality parameter ncp = \lambda has density

f(x) = f_{n,\lambda}(x) = e^{-\lambda / 2} \sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)

for x \ge 0; for more, see R's help page for pchisq.

Historical relatively simple approximations listed in Johnson, Kotz, and Balakrishnan (1995):

pnchisq():

an R implementation of R's own C pnchisq_raw(), but almost only up to Feb.27, 2004, long before the log.p=TRUE addition there, including logspace arithmetic in April 2014, its finish on 2015-09-01. Currently for historical reference only.

pnchisqV():

a Vectorize()d pnchisq.

pnchisqRC():

R's C implementation as of Aug.2019; but with many more options. Currently extreme cases tend to hang on Winbuilder (?)

pnchisqIT:

....

pnchisqTerms:

....

pnchisqT93:

pure R implementations of approximations when both q and ncp are large, by Temme(1993), from Johnson et al., p.467, formulas (29.71 a), and (29.71 b), using auxiliary functions pnchisqT93a() and pnchisqT93b() respectively, with adapted formulas for the log.p=TRUE cases.

pnchisq_ss():

....

ss:

....

ss2:

....

ss2.:

....

Usage

pnchisq          (q, df, ncp = 0, lower.tail = TRUE, 
                  cutOffncp = 80, itSimple = 110, errmax = 1e-12, reltol = 1e-11,
                  maxit = 10* 10000, verbose = 0, xLrg.sigma = 5)
pnchisqV(x, ..., verbose = 0)

pnchisqRC        (q, df, ncp = 0, lower.tail = TRUE, log.p = FALSE,
                  no2nd.call = FALSE,
                  cutOffncp = 80, small.ncp.logspace = small.ncp.logspaceR2015,
                  itSimple = 110, errmax = 1e-12,
                  reltol = 8 * .Machine$double.eps, epsS = reltol/2, maxit = 1e6,
                  verbose = FALSE)
pnchisqAbdelAty  (q, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
pnchisqBolKuz    (q, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
pnchisqPatnaik   (q, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
pnchisqPearson   (q, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
pnchisqSankaran_d(q, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
pnchisq_ss       (x, df, ncp = 0, lower.tail = TRUE, log.p = FALSE, i.max = 10000)
pnchisqTerms     (x, df, ncp,     lower.tail = TRUE, i.max = 1000)

pnchisqT93  (q, df, ncp, lower.tail = TRUE, log.p = FALSE, use.a = q > ncp)
pnchisqT93.a(q, df, ncp, lower.tail = TRUE, log.p = FALSE)
pnchisqT93.b(q, df, ncp, lower.tail = TRUE, log.p = FALSE)

ss   (x, df, ncp, i.max = 10000, useLv = !(expMin < -lambda && 1/lambda < expMax))
ss2  (x, df, ncp, i.max = 10000, eps = .Machine$double.eps)
ss2. (q, df, ncp = 0, errmax = 1e-12, reltol = 2 * .Machine$double.eps,
      maxit = 1e+05, eps = reltol, verbose = FALSE)

Arguments

x

numeric vector (of ‘quantiles’, i.e., abscissa values).

q

number ( ‘quantile’, i.e., abscissa value.)

df

degrees of freedom > 0, maybe non-integer.

ncp

non-centrality parameter \delta; ....

lower.tail, log.p

logical, see, e.g., pchisq().

i.max

number of terms in evaluation ...

use.a

logical vector for Temme pnchisqT93*() formulas, indicating to use formula ‘a’ over ‘b’. The default is as recommended in the references, but they did not take into account log.p = TRUE situations.

cutOffncp

a positive number, the cutoff value for ncp...

itSimple

...

errmax

absolute error tolerance.

reltol

convergence tolerance for relative error.

maxit

maximal number of iterations.

xLrg.sigma

positive number ...

no2nd.call

logical indicating if a 2nd call is made to the internal function ....

small.ncp.logspace

logical vector or function, indicating if the logspace computations for “small” ncp (defined to fulfill ncp < cutOffncp !).

epsS

small positive number, the convergence tolerance of the ‘simple’ iterations...

verbose

logical or integer specifying if or how much the algorithm progress should be monitored.

...

further arguments passed from pnchisqV() to pnchisq().

useLv

logical indicating if logarithmic scale should be used for \lambda computations.

eps

convergence tolerance, a positive number.

Details

pnchisq_ss()

uses si <- ss(x, df, ..) to get the series terms, and returns 2*dchisq(x, df = df +2) * sum(si$s).

ss()

computes the terms needed for the expansion used in pnchisq_ss().

ss2()

computes some simple “statistics” about ss(..).

Value

ss()

returns a list with 3 components

s

the series

i1

location (in s[]) of the first change from 0 to positive.

max

(first) location of the maximal value in the series (i.e., which.max(s)).

Author(s)

Martin Maechler, from May 1999; starting from a post to the S-news mailing list by Ranjan Maitra (@ math.umbc.edu) who showed a version of our pchisqAppr.0() thanking Jim Stapleton for providing it.

References

Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions Vol 2, 2nd ed.; Wiley; chapter 29 Noncentral \chi^2-Distributions; notably Section 8 Approximations, p.461 ff.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. https://en.wikipedia.org/wiki/Abramowitz_and_Stegun

See Also

pchisq and the wienergerm approximations for it: pchisqW() etc.

r_pois() and its plot function, for an aspect of the series approximations we use in pnchisq_ss().

Examples

## set of quantiles to use :
qq <- c(.001, .005, .01, .05, (1:9)/10, 2^seq(0, 10, by= 0.5))
## Take "all interesting" pchisq-approximation from our pkg :
pkg <- "package:DPQ"
pnchNms <- c(paste0("pchisq", c("V", "W", "W.", "W.R")),
             ls(pkg, pattern = "^pnchisq"))
pnchNms <- pnchNms[!grepl("Terms$", pnchNms)]
pnchF <- sapply(pnchNms, get, envir = as.environment(pkg))
str(pnchF)
ncps <- c(0, 1/8, 1/2)
pnchR <- as.list(setNames(ncps, paste("ncp",ncps, sep="=")))
for(i.n in seq_along(ncps)) {
  ncp <- ncps[i.n]
  pnF <- if(ncp == 0) pnchF[!grepl("chisqT93", pnchNms)] else pnchF
  pnchR[[i.n]] <- sapply(pnF, function(F)
            Vectorize(F, names(formals(F))[[1]])(qq, df = 3, ncp=ncp))
}
str(pnchR, max=2)
		 

## A case where the non-central P[] should be improved :
## First, the central P[] which is close to exact -- choosing df=2 allows
## truly exact values: chi^2 = Exp(1) !
opal <- palette()
palette(c("black", "red", "green3", "blue", "cyan", "magenta", "gold3", "gray44"))
cR  <- curve(pchisq   (x, df=2,        lower.tail=FALSE, log.p=TRUE), 0, 4000, n=2001)
cRC <- curve(pnchisqRC(x, df=2, ncp=0, lower.tail=FALSE, log.p=TRUE),
             add=TRUE, col=adjustcolor(2,1/2), lwd=3, lty=2, n=2001)
cR0 <- curve(pchisq   (x, df=2, ncp=0, lower.tail=FALSE, log.p=TRUE),
             add=TRUE, col=adjustcolor(3,1/2), lwd=4,        n=2001)
## smart "named list" constructur :
list_ <- function(...)
   `names<-`(list(...), vapply(sys.call()[-1L], as.character, ""))
JKBfn <-list_(pnchisqPatnaik,
              pnchisqPearson,
              pnchisqAbdelAty,
              pnchisqBolKuz,
              pnchisqSankaran_d)
cl. <- setNames(adjustcolor(3+seq_along(JKBfn), 1/2), names(JKBfn))
lw. <- setNames(2+seq_along(JKBfn),                   names(JKBfn))
cR.JKB <- sapply(names(JKBfn), function(nmf) {
  curve(JKBfn[[nmf]](x, df=2, ncp=0, lower.tail=FALSE, log.p=TRUE),
        add=TRUE, col=cl.[[nmf]], lwd=lw.[[nmf]], lty=lw.[[nmf]], n=2001)
})
legend("bottomleft", c("pchisq", "pchisq.ncp=0", "pnchisqRC", names(JKBfn)),
       col=c(palette()[1], adjustcolor(2:3,1/2), cl.),
       lwd=c(1,3,4, lw.), lty=c(1,2,1, lw.))
palette(opal)# revert

all.equal(cRC, cR0, tol = 1e-15) # TRUE [for now]
## the problematic "jump" :
as.data.frame(cRC)[744:750,]
if(.Platform$OS.type == "unix")
  ## verbose=TRUE  may reveal which branches of the algorithm are taken:
  pnchisqRC(1500, df=2, ncp=0, lower.tail=FALSE, log.p=TRUE, verbose=TRUE) #
  ## |-->  -Inf currently

## The *two*  principal cases (both lower.tail = {TRUE,FALSE} !), where
##  "2nd call"  happens *and* is currently beneficial :
dfs <- c(1:2, 5, 10, 20)
pL. <- pnchisqRC(.00001, df=dfs, ncp=0, log.p=TRUE, lower.tail=FALSE, verbose = TRUE)
pR. <- pnchisqRC(   100, df=dfs, ncp=0, log.p=TRUE,                   verbose = TRUE)
## R's own non-central version (specifying 'ncp'):
pL0 <- pchisq   (.00001, df=dfs, ncp=0, log.p=TRUE, lower.tail=FALSE)
pR0 <- pchisq   (   100, df=dfs, ncp=0, log.p=TRUE)
## R's *central* version, i.e., *not* specifying 'ncp' :
pL  <- pchisq   (.00001, df=dfs,        log.p=TRUE, lower.tail=FALSE)
pR  <- pchisq   (   100, df=dfs,        log.p=TRUE)
cbind(pL., pL, relEc = signif(1-pL./pL, 3), relE0 = signif(1-pL./pL0, 3))
cbind(pR., pR, relEc = signif(1-pR./pR, 3), relE0 = signif(1-pR./pR0, 3))

[Package DPQ version 0.5-8 Index]