DPQ (version 0.5-9)

b_chi: Compute \(E[\chi_\nu] / \sqrt{\nu}\) useful for t- and chi-Distributions

Description

$$b_\chi(\nu) := E[\chi(\nu)] / \sqrt{\nu} = \frac{\sqrt{2/\nu}\Gamma((\nu+1)/2)}{\Gamma(\nu/2)},$$ where \(\chi(\nu)\) denotes a chi-distributed random variable, i.e., the square of a chi-squared variable, and \(\Gamma(z)\) is the Gamma function, gamma() in R.

This is a relatively important auxiliary function when computing with non-central t distribution functions and approximations, specifically see Johnson et al.(1994), p.520, after (31.26a), e.g., our pntJW39().

Its logarithm, $$lb_\chi(\nu) := log\bigl(\frac{\sqrt{2/\nu}\Gamma((\nu+1)/2)}{\Gamma(\nu/2)}\bigr),$$ is even easier to compute via lgamma and log, and I have used Maple to derive an asymptotic expansion in \(\frac{1}{\nu}\) as well.

Note that \(lb_\chi(\nu)\) also appears in the formula for the t-density (dt) and distribution (tail) functions.

Usage

b_chi     (nu, one.minus = FALSE, c1 = 341, c2 = 1000)
b_chiAsymp(nu, order = 2, one.minus = FALSE)
#lb_chi    (nu, ......) # not yet
lb_chiAsymp(nu, order)
c_dt(nu)       # warning("FIXME: current c_dt() is poor -- base it on lb_chi(nu) !")
c_dtAsymp(nu)  # deprecated in favour of lb_chi(nu)
c_pt(nu)       # warning("use better c_dt()") %---> FIXME deprecate even stronger ?

Value

a numeric vector of the same length as nu.

Arguments

nu

non-negative numeric vector of degrees of freedom.

one.minus

logical indicating if \(1 - b()\) should be returned instead of \(b()\).

c1, c2

boundaries for different approximation intervals used:
for 0 < nu <= c1, internal b1() is used,
for c1 < nu <= c2, internal b2() is used, and
for c2 < nu, the b_chiAsymp() function is used, (and you can use that explicitly, also for smaller nu).

FIXME: c1 and c2 were defined when the only asymptotic expansion known to me was the order = 2 one. A future version of b_chi will very likely use b_chiAsymp(*, order) for higher orders, and the c1 and c2 arguments will change, possibly be abolished.

order

the polynomial order in \(\frac{1}{\nu}\) of the asymptotic expansion of \(b_\chi(\nu)\) for \(\nu\to\infty\).

The default, order = 2 corresponds to the order you can get out of the Abramowitz and Stegun (6.1.47) formula. Higher order expansions were derived using Maple by Martin Maechler in 2002, see below, but implemented in b_chiAsymp() only in 2018.

Author

Martin Maechler

Details

One can see that b_chi() has the properties of a CDF of a continuous positive random variable: It grows monotonely from \(b_\chi(0) = 0\) to (asymptotically) one. Specifically, for large nu, b_chi(nu) = b_chiAsymp(nu) and $$1 - b_\chi(\nu) \sim \frac{1}{4\nu}.$$

More accurately, derived from Abramowitz and Stegun, 6.1.47 (p.257) for a= 1/2, b=0, $$\Gamma(z + 1/2) / \Gamma(z) \sim \sqrt(z)*(1 - 1/(8z) + 1/(128 z^2) + O(1/z^3)),$$ and applied for \(b_\chi(\nu)\) with \(z = \nu/2\), we get $$b_\chi(\nu) \sim 1 - (1/(4\nu) * (1 - 1/(8\nu)) + O(\nu^{-3})),$$ which has been implemented in b_chiAsymp(*, order=2) in 1999.

Even more accurately, Martin Maechler, used Maple to derive an asymptotic expansion up to order 15, here reported up to order 5, namely with \(r := \frac{1}{4\nu}\), $$b_\chi(\nu) = c_\chi(r) = 1 - r + \frac{1}{2}r^2 + \frac{5}{2}r^3 - \frac{21}{8}r^4 - \frac{399}{8}r^5 + O(r^6).$$

References

Johnson, Kotz, Balakrishnan (1995) Continuous Univariate Distributions, Vol 2, 2nd Edition; Wiley; Formula on page 520, after (31.26a)

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

The t-distribution (base R) page pt; our pntJW39().

Examples

curve(b_chi, 0, 20); abline(h=0:1, v=0, lty=3)
r <- curve(b_chi, 1e-10, 1e5, log="x")
with(r, lines(x, b_chi(x, one.minus=TRUE), col = 2))

## Zoom in to c1-region
rc1 <- curve(b_chi, 340.5, 341.5, n=1001)# nothing to see
e <- 1e-3; curve(b_chi, 341-e, 341+e, n=1001) # nothing
e <- 1e-5; curve(b_chi, 341-e, 341+e, n=1001) # see noise, but no jump
e <- 1e-7; curve(b_chi, 341-e, 341+e, n=1001) # see float "granularity"+"jump"

## Zoom in to c2-region
rc2 <- curve(b_chi, 999.5, 1001.5, n=1001) # nothing visible
e <- 1e-3; curve(b_chi, 1000-e, 1000+e, n=1001) # clear small jump
c2 <- 1500
e <- 1e-3; curve(b_chi(x,c2=c2), c2-e, c2+e, n=1001)# still
## - - - -
c2 <- 3000
e <- 1e-3; curve(b_chi(x,c2=c2), c2-e, c2+e, n=1001)# ok asymp clearly better!!
curve(b_chiAsymp, add=TRUE, col=adjustcolor("red", 1/3), lwd=3)
if(requireNamespace("Rmpfr")) {
 xm <- Rmpfr::seqMpfr(c2-e, c2+e, length.out=1000)

}
## - - - -
c2 <- 4000
e <- 1e-3; curve(b_chi(x,c2=c2), c2-e, c2+e, n=1001)# ok asymp clearly better!!
curve(b_chiAsymp, add=TRUE, col=adjustcolor("red", 1/3), lwd=3)

grCol <- adjustcolor("forest green", 1/2)
curve(b_chi,                    1/2, 1e11, log="x")
curve(b_chiAsymp, add = TRUE, col = grCol, lwd = 3)
## 1-b(nu) ~= 1/(4 nu) a power function <==> linear in log-log scale:
curve(b_chi(x, one.minus=TRUE), 1/2, 1e11, log="xy")
curve(b_chiAsymp(x, one.minus=TRUE), add = TRUE, col = grCol, lwd = 3)