shape1, shape2 and
scale.
dinvtrgamma(x, shape1, shape2, rate = 1, scale = 1/rate, log = FALSE)
pinvtrgamma(q, shape1, shape2, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)
qinvtrgamma(p, shape1, shape2, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE)
rinvtrgamma(n, shape1, shape2, rate = 1, scale = 1/rate)
minvtrgamma(order, shape1, shape2, rate = 1, scale = 1/rate)
levinvtrgamma(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,="" $p[x=""> x]$.dinvtrgamma gives the density,
pinvtrgamma gives the distribution function,
qinvtrgamma gives the quantile function,
rinvtrgamma generates random deviates,
minvtrgamma gives the $k$th raw moment, and
levinvtrgamma 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:
$$f(x) = \frac{\tau u^\alpha e^{-u}}{x \Gamma(\alpha)}, %
\quad u = (\theta/x)^\tau$$
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 Inverse 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 Inverse Transformed Gamma distribution defines a family of distributions with the following special cases:
shape2 == 1;
shape1 == 1;
shape1 == shape2 == 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(dinvtrgamma(2, 3, 4, 5, log = TRUE))
p <- (1:10)/10
pinvtrgamma(qinvtrgamma(p, 2, 3, 4), 2, 3, 4)
minvtrgamma(2, 3, 4, 5)
levinvtrgamma(200, 3, 4, 5, order = 2)
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