Chi-squared Random Samples
Example Data | 244 | 238 |
The (non-central) $\chi^2$ 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
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
df |
degrees of freedom (non-negative, but can be non-integer). |
ncp |
non-centrality parameter (non-negative). |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. |
Details
The chi-squared distribution with df
= n ≥ 0
degrees of freedom has density
f_n(x) = 1 / (2^(n/2) Γ(n/2)) x^(n/2-1) e^(-x/2)
for x > 0. 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
= λ has density
f(x) = exp(-λ/2) SUM_{r=0}^∞ ((λ/2)^r / r!) dchisq(x, df + 2r)
for x ≥ 0. For integer n, this is the distribution of
the sum of squares of n normals each with variance one,
λ being the sum of squares of the normal means; further,
E(X) = n + λ, Var(X) = 2(n + 2*λ), and
E((X - E(X))^3) = 8(n + 3*λ).
Note that the degrees of freedom df
= n, can be
non-integer, and also n = 0 which is relevant for
non-centrality λ > 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 -> 0.
Note that ncp
values larger than about 1e5 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 central of 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. Appl.Statist., 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.
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)))