pareto {distributionsrd}

The Pareto distribution

Description

Density, distribution function, quantile function, raw moments and random generation for the Pareto distribution.

Usage

dpareto(x, k = 2, xmin = 1, log = FALSE, na.rm = FALSE)

ppareto(q, k = 2, xmin = 1, lower.tail = TRUE, log.p = FALSE, na.rm = FALSE)

qpareto(p, k = 2, xmin = 1, lower.tail = TRUE, log.p = FALSE)

mpareto(r = 0, truncation = xmin, k = 2, xmin = 1, lower.tail = TRUE)

rpareto(n, k = 2, xmin = 1)

Arguments

x, q

vector of quantiles

xmin, k

Scale and shape of the Pareto distribution, defaults to 1 and 2 respectively.

log, log.p

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

na.rm

Removes values that fall outside the support of the distribution

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 Pareto distribution

truncation

lower truncation parameter, defaults to xmin

n

number of observations

Details

Probability and Cumulative Distribution Function:

f(x) = \frac{kx_{min}^{k}}{x^{k+1}}, \qquad F_X(x) = 1-(\frac{x_{min} }{x})^{k}

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

\mu^{r}_{y} = k x_{min}^k \frac{- y^{r-k} }{r-k}, \qquad k>r

Value

dpareto returns the density, ppareto the distribution function, qpareto the quantile function, mpareto the rth moment of the distribution and rpareto generates random deviates.

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

Examples

## Pareto density
plot(x = seq(1, 5, length.out = 100), y = dpareto(x = seq(1, 5, length.out = 100), k = 2, xmin = 1))

## Pareto relates to the exponential distribution available in the stats package
ppareto(q = 5, k = 2, xmin = 3)
pexp(q = log(5 / 3), rate = 2)

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

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

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

mean(x)
mpareto(r = 1, lower.tail = FALSE)

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