GammaDist               package:stats               R Documentation

The Gamma Distribution

Description:

     Density, distribution function, quantile function and random
     generation for the Gamma distribution with parameters 'shape' and
     'scale'.

Usage:

     dgamma(x, shape, rate = 1, scale = 1/rate, log = FALSE)
     pgamma(q, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)
     qgamma(p, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)
     rgamma(n, shape, rate = 1, scale = 1/rate)

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.

    rate: an alternative way to specify the scale.

shape, scale: shape and scale parameters.

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:

     If 'scale' is omitted, it assumes the default value of '1'.

     The Gamma distribution with parameters 'shape' = a and 'scale' = s
     has density

               f(x)= 1/(s^a Gamma(a)) x^(a-1) e^-(x/s)

     for x > 0, a > 0 and s > 0. The mean and variance are E(X) = a*s
     and Var(X) = a*s^2.

     'pgamma()' uses algorithm AS 239, see the references.

Value:

     'dgamma' gives the density, 'pgamma' gives the distribution
     function 'qgamma' gives the quantile function, and 'rgamma'
     generates random deviates.

Note:

     The S parametrization is via 'shape' and 'rate': S has no 'scale'
     parameter.

     The cumulative hazard H(t) = - log(1 - F(t)) is '-pgamma(t, ...,
     lower = FALSE, log = TRUE)'.

     'pgamma' is closely related to the incomplete gamma function.  As
     defined by Abramowitz and Stegun 6.5.1

         P(a,x) = 1/Gamma(a) integral_0^x t^(a-1) exp(-t) dt

     P(a, x) is 'pgamma(x, a)'.  Other authors (for example Karl
     Pearson in his 1922 tables) omit the normalizing factor, defining
     the incomplete gamma function as 'pgamma(x, a) * gamma(a)'.

References:

     Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) _The New S
     Language_. Wadsworth & Brooks/Cole.

     Shea, B. L. (1988) Algorithm AS 239,  Chi-squared and Incomplete
     Gamma Integral, _Applied Statistics (JRSS C)_ *37*, 466-473.

     Abramowitz, M. and Stegun, I. A. (1972) _Handbook of Mathematical
     Functions._ New York: Dover. Chapter 6: Gamma and Related
     Functions.

See Also:

     'gamma' for the Gamma function, 'dbeta' for the Beta distribution
     and 'dchisq' for the chi-squared distribution which is a special
     case of the Gamma distribution.

Examples:

     -log(dgamma(1:4, shape=1))
     p <- (1:9)/10
     pgamma(qgamma(p,shape=2), shape=2)
     1 - 1/exp(qgamma(p, shape=1))

