InverseBurr | R Documentation |
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
=
\tau
, shape2
= \gamma
and scale
= \theta
, has density:
f(x) = \frac{\tau \gamma (x/\theta)^{\gamma \tau}}{%
x [1 + (x/\theta)^\gamma]^{\tau + 1}}
for x > 0
, \tau > 0
, \gamma > 0
and
\theta > 0
.
The inverse Burr is the distribution of the random variable
\theta \left(\frac{X}{1 - X}\right)^{1/\gamma},
where X
has a beta distribution with parameters \tau
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 k
th raw moment of the random variable X
is
E[X^k]
, -\tau\gamma < k < \gamma
.
The k
th limited moment at some limit d
is E[\min(X,
d)^k]
, k > -\tau\gamma
and 1 - k/\gamma
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 k
th raw moment, and
levinvburr
gives the k
th 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)
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.