Chisquare
The (non-central) Chi-Squared Distribution
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.
- Keywords
- distribution
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
length(n) > 1, the length is taken to be the number required.- 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 \(P[X \le x]\), otherwise, \(P[X > x]\).
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 \(x > 0\). 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
$$
f(x) = e^{-\lambda / 2}
\sum_{r=0}^\infty \frac{(\lambda/2)^r}{r!}\, f_{n + 2r}(x)$$
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 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.
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.
Examples
library(stats)
# NOT RUN {
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)
# }
# NOT RUN {
## 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)
# }
# NOT RUN {
## "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)))
# }