weibull {distributionsrd}

The Weibull distribution

Description

Raw moments for the Weibull distribution.

Usage

mweibull(r = 0, truncation = 0, shape = 2, scale = 1, lower.tail = TRUE)

Arguments

r

rth raw moment of the distribution, defaults to 1.

truncation

lower truncation parameter, defaults to 0.

shape, scale

shape and scale of the distribution with default values of 2 and 1 respectively.

lower.tail

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

Details

Probability and Cumulative Distribution Function:

f(x) = \frac{shape}{scale}(\frac{\omega}{scale})^{shape-1}e^{-(\frac{\omega}{scale})^shape} , \qquad F_X(x) = 1-e^{-(\frac{\omega}{scale})^shape}

The y-bounded r-th raw moment of the distribution equals:

\mu^r_y = scale^{r} \Gamma(\frac{r}{shape} +1, (\frac{y}{scale})^shape )

where \Gamma(,) denotes the upper incomplete gamma function.

Value

returns the truncated rth raw moment of the distribution.

Examples

## The zeroth truncated moment is equivalent to the probability function
pweibull(2, shape = 2, scale = 1)
mweibull(truncation = 2)

## The (truncated) first moment is equivalent to the mean of a (truncated) random sample,
#for large enough samples.
x <- rweibull(1e5, shape = 2, scale = 1)
mean(x)
mweibull(r = 1, lower.tail = FALSE)

sum(x[x > quantile(x, 0.1)]) / length(x)
mweibull(r = 1, truncation = quantile(x, 0.1), lower.tail = FALSE)