stats (version 3.5.1)

FDist: The F Distribution

Description

Density, distribution function, quantile function and random generation for the F distribution with df1 and df2 degrees of freedom (and optional non-centrality parameter ncp).

Usage

df(x, df1, df2, ncp, log = FALSE)
pf(q, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE)
qf(p, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE)
rf(n, df1, df2, ncp)

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.

df1, df2

degrees of freedom. Inf is allowed.

ncp

non-centrality parameter. If omitted the central F is assumed.

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

df gives the density, pf gives the distribution function qf gives the quantile function, and rf generates random deviates.

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

The length of the result is determined by n for rf, 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 F distribution with df1 = $n_1$ and df2 = $n_2$ degrees of freedom has density $$f(x)=\frac{\mathrm{\Gamma }(n_1/2+n_2/2)}{\mathrm{\Gamma }(n_1/2)\mathrm{\Gamma }(n_2/2)}{\left(\frac{n_1}{n_2}\right)}^{n_1/2}{x}^{n_1/2-1}{(1+\frac{n_{1}x}{n_2})}^{-(n_1+n_2)/2}$$ for $x>0$.

It is the distribution of the ratio of the mean squares of $n_1$ and $n_2$ independent standard normals, and hence of the ratio of two independent chi-squared variates each divided by its degrees of freedom. Since the ratio of a normal and the root mean-square of $m$ independent normals has a Student's $t_m$ distribution, the square of a $t_m$ variate has a F distribution on 1 and $m$ degrees of freedom.

The non-central F distribution is again the ratio of mean squares of independent normals of unit variance, but those in the numerator are allowed to have non-zero means and ncp is the sum of squares of the means. See Chisquare for further details on non-central distributions.

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, volume 2, chapters 27 and 30. Wiley, New York.

See Also

Distributions for other standard distributions, including dchisq for chi-squared and dt for Student's t distributions.

Examples

# NOT RUN {
## Equivalence of pt(.,nu) with pf(.^2, 1,nu):
x <- seq(0.001, 5, len = 100)
nu <- 4
stopifnot(all.equal(2*pt(x,nu) - 1, pf(x^2, 1,nu)),
          ## upper tails:
 	  all.equal(2*pt(x,     nu, lower=FALSE),
		      pf(x^2, 1,nu, lower=FALSE)))

## the density of the square of a t_m is 2*dt(x, m)/(2*x)
# check this is the same as the density of F_{1,m}
all.equal(df(x^2, 1, 5), dt(x, 5)/x)

## Identity:  qf(2*p - 1, 1, df) == qt(p, df)^2  for  p >= 1/2
p <- seq(1/2, .99, length = 50); df <- 10
rel.err <- function(x, y) ifelse(x == y, 0, abs(x-y)/mean(abs(c(x,y))))
# }
# NOT RUN {
quantile(rel.err(qf(2*p - 1, df1 = 1, df2 = df), qt(p, df)^2), .90)  # ~= 7e-9
# }