stats (version 3.6.2)

# Chisquare: 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

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, $$P[X > x]$$.

## 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.

## 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.

## 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

# 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)))
# }