GammaDist               package:stats               R Documentation

_T_h_e _G_a_m_m_a _D_i_s_t_r_i_b_u_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

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

_U_s_a_g_e:

     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)

_A_r_g_u_m_e_n_t_s:

    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.  Must be strictly positive.

log, log.p: logical; if 'TRUE', probabilities/densities p are returned
          as log(p).

lower.tail: logical; if TRUE (default), probabilities are P[X <= x],
          otherwise, P[X > x].

_D_e_t_a_i_l_s:

     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. (Here Gamma(a) is the function
     implemented by R's 'gamma()' and defined in its help.)

     The mean and variance are E(X) = a*s and Var(X) = a*s^2.

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

_V_a_l_u_e:

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

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

_N_o_t_e:

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

     '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)'.

_S_o_u_r_c_e:

     'dgamma' is computed via the Poisson density, using code
     contributed by Catherine Loader (see 'dbinom').

     Prior to R 2.1.0 'pgamma' used AS239 (Shea, 1988).  It currently
     uses an unpublished (and not otherwise documented) algorithm
     `mainly by Morten Welinder'.

     'qgamma' is based on a C translation of

     Best, D. J. and D. E. Roberts (1975). Algorithm AS91. Percentage
     points of the chi-squared distribution. _Applied Statistics_, 
     *24*, 385-388.

     plus a final Newton step to improve the approximation.

     'rgamma' for 'shape >= 1' uses

     Ahrens, J. H. and Dieter, U. (1982). Generating gamma variates by
     a modified rejection technique. _Communications of the ACM_, *25*,
     47-54,

     and for '0 < shape < 1' uses

     Ahrens, J. H. and Dieter, U. (1974). Computer methods for sampling
     from gamma, beta, Poisson and binomial distributions. _Computing_,
     *12*, 223-246.

_R_e_f_e_r_e_n_c_e_s:

     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.

_S_e_e _A_l_s_o:

     '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.

_E_x_a_m_p_l_e_s:

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

