Chisquare {stats} R Documentation

## 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 <= 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) 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` = λ has density

f(x) = exp(-lambda/2) SUM_{r=0}^infty ((lambda/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).

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.

### 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` based on 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).

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.

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

### 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)

## 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)
## End(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)))
```

[Package stats version 2.9.0 Index]