Approximations of the (Noncentral) Chi-Squared Density

Compute the density function \(f(x, *)\) of the (noncentral) chi-squared distribution.

distribution, math

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))

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 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().


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.


numeric vector similar to x, containing the (logged if log=TRUE) values of the density \(f(x,*)\).


R's dchisq() is typically more uniformly accurate than the approximations nowadays, apart from dnchisqR() which should behave the same. There may occasionally exist small differences between dnchisqR(x, *) and dchisq(x, *) for the same parameters.


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.

See Also

R's own dchisq().

  • dnchisqR
  • dchisqAsym
  • dnchisqBessel
  • dnoncentchisq
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))
# }
Documentation reproduced from package DPQ, version 0.4-2, License: GPL (>= 2)

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