shape and
scale.
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)length(n) > 1, the length
is taken to be the number required.scale strictly.TRUE, probabilities/densities $p$
are returned as $log(p)$.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.The length of the result is determined by n for
rgamma, 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.
dgamma is computed via the Poisson density, using code contributed
by Catherine Loader (see dbinom). pgamma uses an unpublished (and not otherwise documented)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.scale is omitted, it assumes the default value of 1. The Gamma distribution with parameters shape $= a$
and scale $= s$ has density
$$
f(x)= \frac{1}{{\sigma}^{\alpha}\Gamma(\alpha)} {x}^{\alpha-1} e^{-x/\sigma}%
$$
for $x \ge 0$, $a > 0$ and $s > 0$.
(Here $Gamma(a)$ is the function implemented by R's
gamma() and defined in its help. Note that $a = 0$
corresponds to the trivial distribution with all mass at point 0.)
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)
Note that for smallish values of shape (and moderate
scale) a large parts of the mass of the Gamma distribution is
on values of $x$ so near zero that they will be represented as
zero in computer arithmetic. So rgamma may well return values
which will be represented as zero. (This will also happen for very
large values of scale since the actual generation is done for
scale = 1.)
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.
NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, section 8.2.
gamma for the gamma function. Distributions for other standard distributions, including
dbeta for the Beta distribution and dchisq
for the chi-squared distribution which is a special case of the Gamma
distribution.
-log(dgamma(1:4, shape = 1))
p <- (1:9)/10
pgamma(qgamma(p, shape = 2), shape = 2)
1 - 1/exp(qgamma(p, shape = 1))
# even for shape = 0.001 about half the mass is on numbers
# that cannot be represented accurately (and most of those as zero)
pgamma(.Machine$double.xmin, 0.001)
pgamma(5e-324, 0.001) # on most machines 5e-324 is the smallest
# representable non-zero number
table(rgamma(1e4, 0.001) == 0)/1e4
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