Compute the density function \(f(x, *)\) of the (noncentral) chi-squared distribution.
dnchisqR (x, df, ncp, log = FALSE,
eps = 5e-15, termSml = 1e-10, ncpLarge = 1000)
dnchisqBessel(x, df, ncp, log = FALSE)
dchisqAsym (x, df, ncp, log = FALSE)
dnoncentchisq(x, df, ncp, kmax = floor(ncp/2 + 5 * (ncp/2)^0.5))numeric vector similar to x, containing the (logged if
log=TRUE) values of the density \(f(x,*)\).
non-negative numeric vector.
degrees of freedom (parameter), a positive number.
non-centrality parameter \(\delta\); ....
logical indicating if the result is desired on the log scale.
positive convergence tolerance for the series expansion: Terms
are added while term * q > (1-q)*eps, where q is the term's
multiplication factor.
positive tolerance: in the series expansion, terms are
added to the sum as long as they are not smaller than termSml *
sum even when convergence according to eps had occured. This
was not part of the original C code, but was added later for
safeguarding against infinite loops, from tools:::Rd_expr_PR(14105), e.g., for
dchisq(2000, 2, 1000).
in the case where mid underflows to 0, when
log is true, or ncp >= ncpLarge, use a central
approximation. In theory, an optimal choice of ncpLarge would
not be arbitrarily set at 1000 (hardwired in R's
dchisq() here), but possibly also depend on x or
df.
the number of terms in the sum for dnoncentchisq().
Martin Maechler, April 2008
dnchisqR() is a pure R implementation of R's own C implementation
in the sources, R/src/nmath/dnchisq.c, additionally exposing the
three “tuning parameters” eps, termSml, and ncpLarge.
dnchisqBessel() implements Fisher(1928)'s exact closed form formula
based on the Bessel function \(I_{nu}\), i.e., R's
besselI() function;
specifically formula (29.4) in Johnson et al. (1995).
dchisqAsym() is the simple asymptotic approximation from
Abramowitz and Stegun's formula 26.4.27, p. 942.
dnoncentchisq() uses the (typically defining) infinite series expansion
directly, with truncation at kmax, and terms \(t_k\) which
are products of a Poisson probability and a central chi-square density, i.e.,
terms t.k := dpois(k, lambda = ncp/2) * dchisq(x, df = 2*k + df)
for k = 0, 1, ..., kmax.
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.
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995)
Continuous Univariate Distributions Vol~2, 2nd ed.; Wiley.
Chapter 29, Section 3 Distribution, (29.4), p. 436.
R's own dchisq().
x <- sort(outer(c(1,2,5), 2^(-4:5)))
fRR <- dchisq (x, 10, 2)
f.R <- dnchisqR(x, 10, 2)
all.equal(fRR, f.R, tol = 0) # 64bit Lnx (F 30): 1.723897e-16
stopifnot(all.equal(fRR, f.R, tol = 4e-15))
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