qnchisqAppr: Compute Approximate Quantiles of Noncentral Chi-Squared...
In DPQ: Density, Probability, Quantile ('DPQ') Computations
qnchisqAppr 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))
DPQ documentation built on Dec. 5, 2023, 3:05 a.m.
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| qnchisqAppr | 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 |
ncp |
non-centrality parameter |
lower.tail, log.p |
logical, see, e.g., |
qIni |
a |
... |
further arguments to |
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()anddchisq()to determineqchisq(.)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))
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