# Burr: The Burr Distribution In actuar: Actuarial Functions and Heavy Tailed Distributions

 Burr R Documentation

## The Burr Distribution

### Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Burr distribution with parameters `shape1`, `shape2` and `scale`.

### Usage

```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)
```

### Arguments

 `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, 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.

### Details

The Burr distribution with parameters `shape1` = a, `shape2` = b and `scale` = s has density:

f(x) = (a b (x/s)^b)/(x [1 + (x/s)^b]^(a + 1))

for x > 0, a > 0, b > 0 and s > 0.

The Burr is the distribution of the random variable

s (X/(1 - X))^(1/b),

where X has a beta distribution with parameters 1 and a.

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 kth raw moment of the random variable X is E[X^k], -shape2 < k < shape1 * shape2.

The kth limited moment at some limit d is E[min(X, d)^k], k > -shape2 and shape1 - k/shape2 not a negative integer.

### Value

`dburr` gives the density, `pburr` gives the distribution function, `qburr` gives the quantile function, `rburr` generates random deviates, `mburr` gives the kth raw moment, and `levburr` gives the kth moment of the limited loss variable.

Invalid arguments will result in return value `NaN`, with a warning.

### Note

`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.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

### References

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.

### Examples

```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)
```

actuar documentation built on July 16, 2022, 9:05 a.m.