distributionsrd (version 0.0.6)

burr: The Burr distribution

Description

Density, distribution function, quantile function, raw moments and random generation for the Burr distribution, also known as the Burr Type XII distribution or the Singh-Maddala distribution.

Usage

dburr(x, shape1 = 2, shape2 = 1, scale = 0.5, log = FALSE)
pburr(q, shape1 = 2, shape2 = 1, scale = 0.5, log.p = FALSE, lower.tail = TRUE)
qburr(p, shape1 = 2, shape2 = 1, scale = 0.5, log.p = FALSE, lower.tail = TRUE)
mburr(
  r = 0,
  truncation = 0,
  shape1 = 2,
  shape2 = 1,
  scale = 0.5,
  lower.tail = TRUE
)
rburr(n, shape1 = 2, shape2 = 1, scale = 0.5)

Arguments

x, q

vector of quantiles

shape1, shape2, scale

Shape1, shape2 and scale of the Burr distribution, defaults to 2, 1 and 0.5.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities (moments) are \(P[X \le x]\) \((E[x^r|X \le y])\), otherwise, \(P[X > x]\) \((E[x^r|X > y])\)

p

vector of probabilities

r

rth raw moment of the distribution

truncation

lower truncation parameter

n

number of observations

Value

dburr returns the density, pburr the distribution function, qburr the quantile function, mburr the rth moment of the distribution and rburr generates random deviates.

The length of the result is determined by n for rburr, and is the maximum of the lengths of the numerical arguments for the other functions.

Details

Probability and Cumulative Distribution Function:

$$f(x) =\frac{\frac{kc}{scale}(\frac{\omega}{scale})^{shape2-1}}{(1+(\frac{\omega}{scale})^shape2)^{shape1+1}}, \qquad F_X(x) = 1-\frac{1}{(1+(\frac{\omega}{scale})^shape2)^shape1}$$

The y-bounded r-th raw moment of the Fr<U+00E9>chet distribution equals:

$$scale^{r} shape1 [\boldsymbol{B}(\frac{r}{shape2} +1,shape1-\frac{r}{shape2}) - \boldsymbol{B}(\frac{(\frac{y}{scale})^{shape2}}{1+(\frac{y}{scale})^{shape2}};\frac{r}{shape2} +1,shape1-\frac{r}{shape2})], \qquad shape2>r, kc>r$$

Examples

# NOT RUN {
## Burr density
plot(x = seq(0, 5, length.out = 100), y = dburr(x = seq(0, 5, length.out = 100)))
plot(x = seq(0, 5, length.out = 100), y = dburr(x = seq(0, 5, length.out = 100), shape2 = 3))

## Demonstration of log functionality for probability and quantile function
qburr(pburr(2, log.p = TRUE), log.p = TRUE)

## The zeroth truncated moment is equivalent to the probability function
pburr(2)
mburr(truncation = 2)

## The (truncated) first moment is equivalent to the mean of a
#(truncated) random sample, for large enough samples.
x <- rburr(1e5, shape2 = 3)

mean(x)
mburr(r = 1, shape2 = 3, lower.tail = FALSE)

sum(x[x > quantile(x, 0.1)]) / length(x)
mburr(r = 1, shape2 = 3, truncation = quantile(x, 0.1), lower.tail = FALSE)
# }