| GeneralizedBeta {actuar} | R Documentation |
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Generalized Beta distribution
with parameters shape1, shape2, shape3 and
scale.
dgenbeta(x, shape1, shape2, shape3, rate = 1, scale = 1/rate,
log = FALSE)
pgenbeta(q, shape1, shape2, shape3, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
qgenbeta(p, shape1, shape2, shape3, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
rgenbeta(n, shape1, shape2, shape3, rate = 1, scale = 1/rate)
mgenbeta(order, shape1, shape2, shape3, rate = 1, scale = 1/rate)
levgenbeta(limit, shape1, shape2, shape3, rate = 1, scale = 1/rate,
order = 1)
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. |
shape1, shape2, shape3, scale |
parameters. Must be strictly positive. |
rate |
an alternative way to specify the scale. |
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]. |
order |
order of the moment. |
limit |
limit of the loss variable. |
The Generalized Beta distribution with parameters shape1 = a, shape2 = b, shape3
= c and scale = s, has
density:
f(x) = Gamma(a + b)/(Gamma(a) * Gamma(b)) (c (x/s)^(ac) [1 - (x/s)^c]^(b - 1))/x
for 0 < x < s, a > 0,
b > 0, c > 0 and s >
0. (Here Gamma(a) is the function implemented
by R's gamma() and defined in its help.)
The Generalized Beta is the distribution of the random variable
s X^(1/c),
where X has a Beta distribution with parameters a and b.
The kth raw moment of the random variable X is E[X^k] and the kth limited moment at some limit d is E[min(X, d)].
dgenbeta gives the density,
pgenbeta gives the distribution function,
qgenbeta gives the quantile function,
rgenbeta generates random deviates,
mgenbeta gives the kth raw moment, and
levgenbeta gives the kth moment of the limited loss
variable.
Invalid arguments will result in return value NaN, with a warning.
This is not the generalized three-parameter beta distribution defined on page 251 of Johnson et al, 1995.
Vincent Goulet vincent.goulet@act.ulaval.ca
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004), Loss Models, From Data to Decisions, Second Edition, Wiley.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, Wiley.
exp(dgenbeta(2, 2, 3, 4, 0.2, log = TRUE)) p <- (1:10)/10 pgenbeta(qgenbeta(p, 2, 3, 4, 0.2), 2, 3, 4, 0.2) mgenbeta(2, 1, 2, 3, 0.25) - mgenbeta(1, 1, 2, 3, 0.25) ^ 2 levgenbeta(10, 1, 2, 3, 0.25, order = 2)