pnormAsymp: Asymptotic Approxmation of (Extreme Tail) 'pnorm()'

In DPQ: Density, Probability, Quantile ('DPQ') Computations

Asymptotic Approxmation of (Extreme Tail) 'pnorm()'

Description

Provide the first few terms of the asymptotic series approximation to pnorm()'s (extreme) tail, from Abramawitz and Stegun's 26.2.13 (p.932).

Usage

pnormAsymp(x, k, lower.tail = FALSE, log.p = FALSE)

Arguments

x	positive (at least non-negative) numeric vector.
lower.tail, log.p	logical, see, e.g., pnorm().
k	integer \ge 0 indicating how many terms the approximation should use; currently k \le 5.

Value

a numeric vector “as” x; see the examples, on how to use it with arbitrary precise mpfr-numbers from package Rmpfr.

Author(s)

Martin Maechler

References

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. https://en.wikipedia.org/wiki/Abramowitz_and_Stegun provides links to the full text which is in public domain.

See Also

pnormU_S53 for (also asymptotic) upper and lower bounds.

Examples

x <- c((2:10)*2, 25, (3:9)*10, (1:9)*100, (1:8)*1000, (2:4)*5000)
Px <- pnorm(x, lower.tail = FALSE, log.p=TRUE)
PxA <- sapply(setNames(0:5, paste("k =",0:5)),
              pnormAsymp, x=x, lower.tail = FALSE, log.p=TRUE)
## rel.errors :
signif(head( cbind(x, 1 - PxA/Px) , 20))

## Look more closely with high precision computations
if(requireNamespace("Rmpfr")) {
  ## ensure our function uses Rmpfr's dnorm(), etc:
  environment(pnormAsymp) <- asNamespace("Rmpfr")
  environment(pnormU_S53) <- asNamespace("Rmpfr")
  x. <- Rmpfr::mpfr(x, precBits=256)
  Px. <- Rmpfr::pnorm(x., lower.tail = FALSE, log.p=TRUE)
  ## manual, better sapplyMpfr():
  PxA. <- sapply(setNames(0:5, paste("k =",0:5)),
                 pnormAsymp, x=x., lower.tail = FALSE, log.p=TRUE)
  PxA. <- new("mpfrMatrix", unlist(PxA.), Dim=dim(PxA.), Dimnames=dimnames(PxA.))
  PxA2 <- Rmpfr::cbind(pn_dbl = Px, PxA.,
                       pnormU_S53 = pnormU_S53(x=x., lower.tail = FALSE, log.p=TRUE))
  ## rel.errors : note that pnormU_S53() is very slightly better than "k=2":
  print( Rmpfr::roundMpfr(Rmpfr::cbind(x., 1 - PxA2/Px.), precBits = 13), width = 111)
  pch <- c("R", 0:5, "U")
  matplot(x, abs(1 -PxA2/Px.), type="o", log="xy", pch=pch,
          main="pnorm(<tail>) approximations' relative errors - pnormAsymp(*, k=k)")
  legend("bottomleft", colnames(PxA2), col=1:6, pch=pch, lty=1:5, bty="n", inset=.01)
  at1 <- axTicks(1, axp = c(par("xaxp")[1:2], 3))
  axis(1, at=at1)
  abline(h = 1:2* 2^-53, v = at1, lty=3, col=adjustcolor("gray20", 1/2))
  axis(4, las=2, at= 2^-53, label = quote(epsilon[C]), col="gray20")
}

DPQ documentation built on July 18, 2025, 3:01 p.m.