The (non-central) Chi-Squared Distribution

Description

Density, distribution function, quantile function and random generation for the chi-squared (\chi^2) distribution with df degrees of freedom and optional non-centrality parameter ncp.

Usage

dchisq(x, df, ncp = 0, log = FALSE)
pchisq(q, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
qchisq(p, df, ncp = 0, lower.tail = TRUE, log.p = FALSE)
rchisq(n, df, ncp = 0)

Arguments

Details

The chi-squared distribution with df= n \ge 0 degrees of freedom has density

f_n(x) = \frac{1}{{2}^{n/2} \Gamma (n/2)} {x}^{n/2-1} {e}^{-x/2}

for x > 0, where f_0(x) := \lim_{n \to 0} f_n(x) = \delta_0(x), a point mass at zero, is not a density function proper, but a “\delta distribution”.
The mean and variance are n and 2n.

The non-central chi-squared distribution with df= n degrees of freedom and non-centrality parameter ncp = \lambda has density

f(x) = f_{n,\lambda}(x) = e^{-\lambda / 2} \sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)

for x \ge 0. For integer n, this is the distribution of the sum of squares of n normals each with variance one, \lambda being the sum of squares of the normal means; further,
E(X) = n + \lambda, Var(X) = 2(n + 2*\lambda), and E((X - E(X))^3) = 8(n + 3*\lambda).

Note that the degrees of freedom df= n, can be non-integer, and also n = 0 which is relevant for non-centrality \lambda > 0, see Johnson et al. (1995, chapter 29). In that (noncentral, zero df) case, the distribution is a mixture of a point mass at x = 0 (of size pchisq(0, df=0, ncp=ncp)) and a continuous part, and dchisq() is not a density with respect to that mixture measure but rather the limit of the density for df \to 0.

Note that ncp values larger than about 1e5 (and even smaller) may give inaccurate results with many warnings for pchisq and qchisq.

Value

dchisq gives the density, pchisq gives the distribution function, qchisq gives the quantile function, and rchisq generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

The length of the result is determined by n for rchisq, and is the maximum of the lengths of the numerical arguments for the other functions.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Note

Supplying ncp = 0 uses the algorithm for the non-central distribution, which is not the same algorithm used if ncp is omitted. This is to give consistent behaviour in extreme cases with values of ncp very near zero.

The code for non-zero ncp is principally intended to be used for moderate values of ncp: it will not be highly accurate, especially in the tails, for large values.

Source

The central cases are computed via the gamma distribution.

The non-central dchisq and rchisq are computed as a Poisson mixture of central chi-squares (Johnson et al., 1995, p.436).

The non-central pchisq is for ncp < 80 computed from the Poisson mixture of central chi-squares and for larger ncp via a C translation of

Ding, C. G. (1992) Algorithm AS275: Computing the non-central chi-squared distribution function. Applied Statistics, 41 478–482.

which computes the lower tail only (so the upper tail suffers from cancellation and a warning will be given when this is likely to be significant).

The non-central qchisq is based on inversion of pchisq.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, chapters 18 (volume 1) and 29 (volume 2). Wiley, New York.

See Also

Distributions for other standard distributions.

A central chi-squared distribution with n degrees of freedom is the same as a Gamma distribution with shape \alpha = n/2 and scale \sigma = 2. Hence, see dgamma for the Gamma distribution.

The central chi-squared distribution with 2 d.f. is identical to the exponential distribution with rate 1/2: \chi^2_2 = Exp(1/2), see dexp.

Examples

require(graphics)

dchisq(1, df = 1:3)
pchisq(1, df =  3)
pchisq(1, df =  3, ncp = 0:4)  # includes the above

x <- 1:10
## Chi-squared(df = 2) is a special exponential distribution
all.equal(dchisq(x, df = 2), dexp(x, 1/2))
all.equal(pchisq(x, df = 2), pexp(x, 1/2))

## non-central RNG -- df = 0 with ncp > 0:  Z0 has point mass at 0!
Z0 <- rchisq(100, df = 0, ncp = 2.)
graphics::stem(Z0)

## visual testing
## do P-P plots for 1000 points at various degrees of freedom
L <- 1.2; n <- 1000; pp <- ppoints(n)
op <- par(mfrow = c(3,3), mar = c(3,3,1,1)+.1, mgp = c(1.5,.6,0),
          oma = c(0,0,3,0))
for(df in 2^(4*rnorm(9))) {
  plot(pp, sort(pchisq(rr <- rchisq(n, df = df, ncp = L), df = df, ncp = L)),
       ylab = "pchisq(rchisq(.),.)", pch = ".")
  mtext(paste("df = ", formatC(df, digits = 4)), line =  -2, adj = 0.05)
  abline(0, 1, col = 2)
}
mtext(expression("P-P plots : Noncentral  "*
                 chi^2 *"(n=1000, df=X, ncp= 1.2)"),
      cex = 1.5, font = 2, outer = TRUE)
par(op)

## "analytical" test
lam <- seq(0, 100, by = .25)
p00 <- pchisq(0,      df = 0, ncp = lam)
p.0 <- pchisq(1e-300, df = 0, ncp = lam)
stopifnot(all.equal(p00, exp(-lam/2)),
          all.equal(p.0, exp(-lam/2)))