shape1, shape2 and scale.
dtrgamma(x, shape1, shape2, rate = 1, scale = 1/rate, log = FALSE)
ptrgamma(q, shape1, shape2, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)
qtrgamma(p, shape1, shape2, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)
rtrgamma(n, shape1, shape2, rate = 1, scale = 1/rate)
mtrgamma(order, shape1, shape2, rate = 1, scale = 1/rate)
levtrgamma(limit, shape1, shape2, rate = 1, scale = 1/rate, order = 1)length(n) > 1, the length is
taken to be the number required.TRUE, probabilities/densities
$p$ are returned as $log(p)$.TRUE (default), probabilities are
$P[X <= x]$,="" otherwise,="" $p[x=""> x]$.dtrgamma gives the density,
ptrgamma gives the distribution function,
qtrgamma gives the quantile function,
rtrgamma generates random deviates,
mtrgamma gives the $k$th raw moment, and
levtrgamma gives the $k$th moment of the limited loss
variable.Invalid arguments will result in return value NaN, with a warning.
shape1 $= a$, shape2 $= b$ and scale
$= s$ has density:
$$f(x) = \frac{\tau u^\alpha e^{-u}}{x \Gamma(\alpha)}, %
\quad u = (x/\theta)^\tau$$
for $x > 0$, $a > 0$, $b > 0$
and $s > 0$.
(Here $Gamma(a)$ is the function implemented
by R's gamma() and defined in its help.)The Transformed Gamma is the distribution of the random variable $s X^(1/b),$ where $X$ has a Gamma distribution with shape parameter $a$ and scale parameter $1$ or, equivalently, of the random variable $Y^(1/b)$ with $Y$ a Gamma distribution with shape parameter $a$ and scale parameter $s^b$.
The Transformed Gamma probability distribution defines a family of distributions with the following special cases:
shape2 == 1;
shape1 ==
1;
shape2 ==
shape1 == 1.
The $k$th raw moment of the random variable $X$ is $E[X^k]$ and the $k$th limited moment at some limit $d$ is $E[min(X, d)^k]$.
exp(dtrgamma(2, 3, 4, 5, log = TRUE))
p <- (1:10)/10
ptrgamma(qtrgamma(p, 2, 3, 4), 2, 3, 4)
mtrgamma(2, 3, 4, 5) - mtrgamma(1, 3, 4, 5) ^ 2
levtrgamma(10, 3, 4, 5, order = 2)
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