Burr | R Documentation |
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Burr distribution with
parameters shape1
, shape2
and scale
.
dburr(x, shape1, shape2, rate = 1, scale = 1/rate,
log = FALSE)
pburr(q, shape1, shape2, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
qburr(p, shape1, shape2, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
rburr(n, shape1, shape2, rate = 1, scale = 1/rate)
mburr(order, shape1, shape2, rate = 1, scale = 1/rate)
levburr(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 Burr distribution with parameters shape1
=
\alpha
, shape2
= \gamma
and scale
= \theta
has density:
f(x) = \frac{\alpha \gamma (x/\theta)^\gamma}{%
x [1 + (x/\theta)^\gamma]^{\alpha + 1}}
for x > 0
, \alpha > 0
, \gamma > 0
and \theta > 0
.
The 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 1
and \alpha
.
The Burr distribution has the following special cases:
A Loglogistic distribution when shape1
== 1
;
A Paralogistic distribution when
shape2 == shape1
;
A Pareto distribution when shape2 ==
1
.
The k
th raw moment of the random variable X
is
E[X^k]
, -\gamma < k < \alpha\gamma
.
The k
th limited moment at some limit d
is E[\min(X,
d)^k]
, k > -\gamma
and \alpha - k/\gamma
not a negative integer.
dburr
gives the density,
pburr
gives the distribution function,
qburr
gives the quantile function,
rburr
generates random deviates,
mburr
gives the k
th raw moment, and
levburr
gives the k
th moment of the limited loss
variable.
Invalid arguments will result in return value NaN
, with a warning.
levburr
computes the limited expected value using
betaint
.
Distribution also known as the Burr Type XII or Singh-Maddala 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.
dpareto4
for an equivalent distribution with a location
parameter.
exp(dburr(1, 2, 3, log = TRUE))
p <- (1:10)/10
pburr(qburr(p, 2, 3, 2), 2, 3, 2)
## variance
mburr(2, 2, 3, 1) - mburr(1, 2, 3, 1) ^ 2
## case with shape1 - order/shape2 > 0
levburr(10, 2, 3, 1, order = 2)
## case with shape1 - order/shape2 < 0
levburr(10, 1.5, 0.5, 1, order = 2)
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