The Inverse Burr Distribution
Density function, distribution function, quantile function, random
generation, raw moments and limited moments for the Inverse Burr
distribution with parameters shape1, shape2 and
scale.
dinvburr(x, shape1, shape2, rate = 1, scale = 1/rate,
log = FALSE)
pinvburr(q, shape1, shape2, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
qinvburr(p, shape1, shape2, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
rinvburr(n, shape1, shape2, rate = 1, scale = 1/rate)
minvburr(order, shape1, shape2, rate = 1, scale = 1/rate)
levinvburr(limit, shape1, shape2, rate = 1, scale = 1/rate,
order = 1)x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
shape1, shape2, 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 inverse Burr distribution with parameters shape1 = a, shape2 = b and scale
= s, has density:
f(x) = a b (x/s)^(ba)/(x [1 + (x/s)^b]^(a + 1))
for x > 0, a > 0, b > 0 and s > 0.
The inverse Burr is the distribution of the random variable
s (X/(1 - X))^(1/b),
where X has a beta distribution with parameters a and 1.
The inverse Burr distribution has the following special cases:
A Loglogistic distribution when shape1
== 1;
An Inverse Pareto distribution when
shape2 == 1;
An Inverse Paralogistic distribution
when shape1 == shape2.
The kth raw moment of the random variable X is E[X^k], -shape1 * shape2 < k < shape2.
The kth limited moment at some limit d is E[min(X, d)^k], k > -shape1 * shape2 and 1 - k/shape2 not a negative integer.
dinvburr gives the density,
invburr gives the distribution function,
qinvburr gives the quantile function,
rinvburr generates random deviates,
minvburr gives the kth raw moment, and
levinvburr gives the kth moment of the limited loss
variable.
Invalid arguments will result in return value NaN, with a warning.
levinvburr computes the limited expected value using
betaint.
Also known as the Dagum distribution. 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.
exp(dinvburr(2, 2, 3, 1, log = TRUE)) p <- (1:10)/10 pinvburr(qinvburr(p, 2, 3, 1), 2, 3, 1) ## variance minvburr(2, 2, 3, 1) - minvburr(1, 2, 3, 1) ^ 2 ## case with 1 - order/shape2 > 0 levinvburr(10, 2, 3, 1, order = 2) ## case with 1 - order/shape2 < 0 levinvburr(10, 2, 1.5, 1, order = 2)
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.