# qnchisqAppr

From DPQ v0.3-3
0th

Percentile

##### Compute Approximate Quantiles of Noncentral Chi-Squared Distribution

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

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

Keywords
distribution
##### 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.

##### 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.

qchisq.

##### Aliases
• qnchisqAppr
• qchisqAppr.0
• qchisqAppr.1
• qchisqAppr.2
• qchisqAppr.3
• qchisqApprCF1
• qchisqApprCF2
• qchisqCappr.2
• qchisqN
• qnchisqAbdelAty
• qnchisqBolKuz
• qnchisqPatnaik
• qnchisqPearson
• qnchisqSankaran_d
##### Examples
# NOT RUN {
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))
# }

Documentation reproduced from package DPQ, version 0.3-3, License: GPL (>= 2)

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