qnchisqAppr {DPQ}R Documentation

Compute Approximate Quantiles of Noncentral Chi-Squared Distribution

Description

Compute quantiles (inverse distribution values) for the non-central chi-squared distribution.

....... using Johnson,Kotz, and other approximations ..............

Usage

qchisqAppr.0 (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qchisqAppr.1 (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qchisqAppr.2 (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qchisqAppr.3 (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qchisqApprCF1(p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qchisqApprCF2(p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)

qchisqCappr.2 (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qchisqN       (p, df, ncp = 0, qIni = qchisqAppr.0, ...)

qnchisqAbdelAty  (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qnchisqBolKuz    (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qnchisqPatnaik   (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qnchisqPearson   (p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qnchisqSankaran_d(p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)

Arguments

p

vector of probabilities.

df

degrees of freedom > 0, maybe non-integer.

ncp

non-centrality parameter \delta; ....

lower.tail, log.p

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

qIni

a function that computes an approximate noncentral chi-squared quantile as starting value x0 for the Newton algorithm newton().

...

further arguments to newton(), notably eps or maxiter.

Details

Compute (approximate) quantiles, using approximations analogous to those for the probabilities, see pnchisqPearson.

qchisqAppr.0():

...TODO...

qchisqAppr.1():

...TODO...

qchisqAppr.2():

...TODO...

qchisqAppr.3():

...TODO...

qchisqApprCF1():

...TODO...

qchisqApprCF2():

...TODO...

qchisqCappr.2():

...TODO...

qchisqN():

Uses Newton iterations with pchisq() and dchisq() to determine qchisq(.) values.

qnchisqAbdelAty():

...TODO...

qnchisqBolKuz():

...TODO...

qnchisqPatnaik():

...TODO...

qnchisqPearson():

...TODO...

qnchisqSankaran_d():

...TODO...

Value

numeric vectors of (noncentral) chi-squared quantiles, corresponding to probabilities p.

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 qchisqAppr.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.

See Also

qchisq.

Examples

 pp <- c(.001, .005, .01, .05, (1:9)/10, .95, .99, .995, .999)
 pkg <- "package:DPQ"
 qnchNms <- c(paste0("qchisqAppr.",0:3), paste0("qchisqApprCF",1:2),
              "qchisqN", "qchisqCappr.2", ls(pkg, pattern = "^qnchisq"))
 qnchF <- sapply(qnchNms, get, envir = as.environment(pkg))
 for(ncp in c(0, 1/8, 1/2)) {
   cat("\n~~~~~~~~~~~~~\nncp: ", ncp,"\n=======\n")
   print(sapply(qnchF, function(F) Vectorize(F, "p")(pp, df = 3, ncp=ncp)))
 }

## Bug: qnchisqSankaran_d() has numeric overflow problems for large df:
qnchisqSankaran_d(pp, df=1e200, ncp = 100)

## One current (2019-08) R bug: Noncentral chi-squared quantiles on *LOG SCALE*

## a)  left/lower tail : -------------------------------------------------------
qs <- 2^seq(0,11, by=1/16)
pqL <- pchisq(qs, df=5, ncp=1, log.p=TRUE)
plot(qs, -pqL, type="l", log="xy") # + expected warning on log(0) -- all fine
qpqL <- qchisq(pqL, df=5, ncp=1, log.p=TRUE) # severe overflow :
qm <- cbind(qs, pqL, qchisq=qpqL
	, qchA.0 = qchisqAppr.0 (pqL, df=5, ncp=1, log.p=TRUE)
	, qchA.1 = qchisqAppr.1 (pqL, df=5, ncp=1, log.p=TRUE)
	, qchA.2 = qchisqAppr.2 (pqL, df=5, ncp=1, log.p=TRUE)
	, qchA.3 = qchisqAppr.3 (pqL, df=5, ncp=1, log.p=TRUE)
	, qchACF1= qchisqApprCF1(pqL, df=5, ncp=1, log.p=TRUE)
	, qchACF2= qchisqApprCF2(pqL, df=5, ncp=1, log.p=TRUE)
	, qchCa.2= qchisqCappr.2(pqL, df=5, ncp=1, log.p=TRUE)
	, qnPatnaik   = qnchisqPatnaik   (pqL, df=5, ncp=1, log.p=TRUE)
	, qnAbdelAty  = qnchisqAbdelAty  (pqL, df=5, ncp=1, log.p=TRUE)
	, qnBolKuz    = qnchisqBolKuz    (pqL, df=5, ncp=1, log.p=TRUE)
	, qnPearson   = qnchisqPearson   (pqL, df=5, ncp=1, log.p=TRUE)
	, qnSankaran_d= qnchisqSankaran_d(pqL, df=5, ncp=1, log.p=TRUE)
)

round(qm[ qs %in% 2^(0:11) , -2])
#=> Approximations don't overflow but are not good enough

## b)  right/upper tail , larger ncp -------------------------------------------
qS <- 2^seq(-3, 3, by=1/8)
pqLu <- pchisq(qS, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
## using "the alternative" (here is currently identical):
identical(pqLu, (pqLu.<- log1p(-pchisq(qS, df=5, ncp=100)))) # here TRUE
plot (qS, -pqLu, type="l", log="xy") # fine
qpqLu <- qchisq(pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
cbind(qS, pqLu, pqLu, qpqLu)# # severe underflow
qchMat <- cbind(qchisq = qpqLu
	, qchA.0 = qchisqAppr.0 (pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qchA.1 = qchisqAppr.1 (pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qchA.2 = qchisqAppr.2 (pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qchA.3 = qchisqAppr.3 (pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qchACF1= qchisqApprCF1(pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qchACF2= qchisqApprCF2(pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qchCa.2= qchisqCappr.2(pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qnPatnaik   = qnchisqPatnaik   (pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qnAbdelAty  = qnchisqAbdelAty  (pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qnBolKuz    = qnchisqBolKuz    (pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qnPearson   = qnchisqPearson   (pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	, qnSankaran_d= qnchisqSankaran_d(pqLu, df=5, ncp=100, log.p=TRUE, lower.tail=FALSE)
	)
cbind(L2err <- sort(sqrt(colSums((qchMat - qS)^2))))
##--> "Sankaran_d", "CF1" and "CF2" are good here

plot (qS, qpqLu, type = "b", log="x", lwd=2)
lines(qS, qS, col="gray", lty=2, lwd=3)
top3 <- names(L2err)[1:3]
use <- c("qchisq", top3)
matlines(qS, qchMat[, use]) # 3 of the approximations are "somewhat ok"
legend("topleft", c(use,"True"), bty="n", col=c(palette()[1:4], "gray"),
                  lty = c(1:4,2), lwd = c(2, 1,1,1, 3))
[Package DPQ version 0.5-8 Index]