The Transformed Beta Distribution
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Transformed Beta distribution
with parameters shape1, shape2, shape3 and
scale.
dtrbeta(x, shape1, shape2, shape3, rate = 1, scale = 1/rate,
log = FALSE)
ptrbeta(q, shape1, shape2, shape3, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
qtrbeta(p, shape1, shape2, shape3, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
rtrbeta(n, shape1, shape2, shape3, rate = 1, scale = 1/rate)
mtrbeta(order, shape1, shape2, shape3, rate = 1, scale = 1/rate)
levtrbeta(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 |
shape1, shape2, shape3, scale |
parameters. Must be strictly positive. |
rate |
an alternative way to specify the scale. |
log, log.p |
logical; if |
lower.tail |
logical; if |
order |
order of the moment. |
limit |
limit of the loss variable. |
The transformed beta distribution with parameters shape1 = a, shape2 = b, shape3
= c and scale = s, has
density:
f(x) = Gamma(a + c)/(Gamma(a) * Gamma(c)) (b (x/s)^(bc))/ (x [1 + (x/s)^b]^(a + c))
for x > 0, 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 transformed beta is the distribution of the random variable
s (X/(1 - X))^(1/b),
where X has a beta distribution with parameters c and a.
The transformed beta distribution defines a family of distributions with the following special cases:
A Burr distribution when shape3 == 1;
A loglogistic distribution when shape1
== shape3 == 1;
A paralogistic distribution when
shape3 == 1 and shape2 == shape1;
A generalized Pareto distribution when
shape2 == 1;
A Pareto distribution when shape2 ==
shape3 == 1;
An inverse Burr distribution when
shape1 == 1;
An inverse Pareto distribution when
shape2 == shape1 == 1;
An inverse paralogistic distribution
when shape1 == 1 and shape3 == shape2.
The kth raw moment of the random variable X is E[X^k], -shape3 * shape2 < k < shape1 * shape2.
The kth limited moment at some limit d is E[min(X, d)^k], k > -shape3 * shape2 and shape1 - k/shape2 not a negative integer.
dtrbeta gives the density,
ptrbeta gives the distribution function,
qtrbeta gives the quantile function,
rtrbeta generates random deviates,
mtrbeta gives the kth raw moment, and
levtrbeta gives the kth moment of the limited loss
variable.
Invalid arguments will result in return value NaN, with a warning.
levtrbeta computes the limited expected value using
betaint.
Distribution also known as the Generalized Beta of the Second Kind and Pearson Type VI. See also Kleiber and Kotz (2003) for alternative names and parametrizations.
The "distributions" package vignette provides the
interrelations between the continuous size distributions in
actuar and the complete formulas underlying the above functions.
Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon
Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.
Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.
dfpareto for an equivalent distribution with a location
parameter.
exp(dtrbeta(2, 2, 3, 4, 5, log = TRUE)) p <- (1:10)/10 ptrbeta(qtrbeta(p, 2, 3, 4, 5), 2, 3, 4, 5) qpearson6(0.3, 2, 3, 4, 5, lower.tail = FALSE) ## variance mtrbeta(2, 2, 3, 4, 5) - mtrbeta(1, 2, 3, 4, 5)^2 ## case with shape1 - order/shape2 > 0 levtrbeta(10, 2, 3, 4, scale = 1, order = 2) ## case with shape1 - order/shape2 < 0 levtrbeta(10, 1/3, 0.75, 4, scale = 0.5, order = 2)
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