# pnchi1sq

From DPQ v0.3-3
0th

Percentile

##### (Probabilities of Non-Central Chisquared Distribution for Special Cases

Computes probabilities for the non-central chi squared distribution, in special cases, currently for df = 1 and df = 3, using ‘exact’ formulas only involving the standard normal (Gaussian) cdf $\Phi()$ and its derivative $\phi()$, i.e., R's pnorm() and dnorm().

Keywords
distribution, math
##### Usage
pnchi1sq(q, ncp = 0, lower.tail = TRUE, log.p = FALSE, epsS = .01)
pnchi3sq(q, ncp = 0, lower.tail = TRUE, log.p = FALSE, epsS = .04)
##### Arguments
q

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

ncp

non-centrality parameter $\delta$; ....

lower.tail, log.p

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

epsS

small number, determining where to switch from the “small case” to the regular case, namely by defining small <- sqrt(q/ncp) <= epsS.

##### Details

In the “small case” (epsS above), the direct formulas suffer from cancellation, and we use Taylor series expansions in $s := \sqrt{q}$, which in turn use “probabilists'” Hermite polynomials $He_n(x)$.

The default values epsS have currently been determined by experiments as those in the ‘Examples’ below.

##### Value

a numeric vector “like” q+ncp, i.e., recycled to common length.

##### References

Johnson et al.(1995), see ‘References’ in pnchisqPearson.

https://en.wikipedia.org/wiki/Hermite_polynomials

pchisq, the (simple and R-like) approximations, such as pnchisqPearson and the wienergerm approximations, pchisqW() etc.

• pnchi1sq
• pnchi3sq
##### Examples
# NOT RUN {
qq <- seq(9500, 10500, length=1000)
m1 <- cbind(pch = pchisq  (qq, df=1, ncp = 10000),
p1  = pnchi1sq(qq,       ncp = 10000))
matplot(qq, m1, type = "l"); abline(h=0:1, v=10000+1, lty=3)
all.equal(m1[,"p1"], m1[,"pch"], tol=0) # for now,  2.37e-12

m3 <- cbind(pch = pchisq  (qq, df=3, ncp = 10000),
p3 = pnchi3sq(qq,       ncp = 10000))
matplot(qq, m3, type = "l"); abline(h=0:1, v=10000+3, lty=3)
all.equal(m3[,"p3"], m3[,"pch"], tol=0) # for now,  1.88e-12

stopifnot(exprs = {
all.equal(m1[,"p1"], m1[,"pch"], tol=1e-10)
all.equal(m3[,"p3"], m3[,"pch"], tol=1e-10)
})

### Very small 'x' i.e., 'q' would lead to cancellation: -----------

##  df = 1 --------------

qS <- c(0, 2^seq(-40,4, by=1/16))
m1s <- cbind(pch = pchisq  (qS, df=1, ncp = 1)
, p1.0= pnchi1sq(qS,       ncp = 1, epsS = 0)
, p1.4= pnchi1sq(qS,       ncp = 1, epsS = 1e-4)
, p1.3= pnchi1sq(qS,       ncp = 1, epsS = 1e-3)
, p1.2= pnchi1sq(qS,       ncp = 1, epsS = 1e-2)
)
cols <- adjustcolor(1:5, 1/2); lws <- seq(4,2, by = -1/2)
abl.leg <- function(x.leg = "topright", epsS = 10^-(4:2), legend = NULL)
{
abline(h = .Machine\$double.eps, v = epsS^2,
lty = c(2,3,3,3), col= adjustcolor(1, 1/2))
if(is.null(legend))
legend <- c(quote(epsS == 0), as.expression(lapply(epsS,
function(K) substitute(epsS == KK,
list(KK = formatC(K, w=1))))))
legend(x.leg, legend, lty=1:4, col=cols, lwd=lws, bty="n")
}
matplot(qS, m1s, type = "l", log="y" , col=cols, lwd=lws)
matplot(qS, m1s, type = "l", log="xy", col=cols, lwd=lws) ; abl.leg("right")
## ====  "Errors" ===================================================
## Absolute: -------------------------
matplot(qS,     m1s[,1] - m1s[,-1] , type = "l", log="x" , col=cols, lwd=lws)
matplot(qS, abs(m1s[,1] - m1s[,-1]), type = "l", log="xy", col=cols, lwd=lws)
abl.leg("bottomright")
## Relative: -------------------------
matplot(qS,     1 - m1s[,-1]/m1s[,1] , type = "l", log="x",  col=cols, lwd=lws)
abl.leg()
matplot(qS, abs(1 - m1s[,-1]/m1s[,1]), type = "l", log="xy", col=cols, lwd=lws)
abl.leg()
# }
# NOT RUN {
<!-- %% all.equal(m1s[,"p1"], m1s[,"pch"], tol=0) # for now,  2.37e-12 -->
# }
# NOT RUN {
##  df = 3 --------------  %% FIXME:  the 'small' case is clearly wrong <<<

qS <- c(0, 2^seq(-40,4, by=1/16))
ee <- c(1e-3, 1e-2, .04)
m3s <- cbind(pch = pchisq  (qS, df=3, ncp = 1)
, p1.0= pnchi3sq(qS,       ncp = 1, epsS = 0)
, p1.3= pnchi3sq(qS,       ncp = 1, epsS = ee)
, p1.2= pnchi3sq(qS,       ncp = 1, epsS = ee)
, p1.1= pnchi3sq(qS,       ncp = 1, epsS = ee)
)
matplot(qS, m3s, type = "l", log="y" , col=cols, lwd=lws)
matplot(qS, m3s, type = "l", log="xy", col=cols, lwd=lws); abl.leg("right", ee)
## ====  "Errors" ===================================================
## Absolute: -------------------------
matplot(qS,     m3s[,1] - m3s[,-1] , type = "l", log="x" , col=cols, lwd=lws)
matplot(qS, abs(m3s[,1] - m3s[,-1]), type = "l", log="xy", col=cols, lwd=lws)
abl.leg("right", ee)
## Relative: -------------------------
matplot(qS,     1 - m3s[,-1]/m3s[,1] , type = "l", log="x",  col=cols, lwd=lws)
abl.leg(, ee)
matplot(qS, abs(1 - m3s[,-1]/m3s[,1]), type = "l", log="xy", col=cols, lwd=lws)
abl.leg(, ee)
# }

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

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