sads (version 0.2.2)

dpareto: Pareto distribution

Description

Density, distribution function and quantile function for the Pareto distribution with parameters shape and scale.

Usage

dpareto(x, shape, scale = min(x), log = FALSE)
ppareto(q, shape, scale = min(q), lower.tail = TRUE, log.p = FALSE)
qpareto(p, shape, scale = min(p), lower.tail = TRUE, log.p = FALSE)

Arguments

x

vector of (non-negative integer) quantiles. In the context of species abundance distributions, this is a vector of abundances of species in a sample.

q

vector of (non-negative integer) quantiles. In the context of species abundance distributions, a vector of abundances of species in a sample.

p

vector of probabilities.

shape

positive real; shape parameter, a.k.a Pareto's index or tail index.

scale

positive real, scale >= min(x); scale parameter.

log, log.p

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

lower.tail

logical; if TRUE (default), probabilities are P[X <= x],="" otherwise,="" p[x=""> x].

Value

• dpareto gives the (log) density, ppareto gives the (log) distribution function, qpareto gives the quantile function.

  Invalid values for parameters shape or scale will result in return values NaN, with a warning.

Details

The Pareto distribution is a continuous power-law density distribution with scale (a) and shape (b) parameters with the form:

$$f(x) = \frac{b a^b} {x^{b+1}}$$

For all x >= scale, and

f(x) = 0 otherwise.

The shape parameter is known as Pareto's index or tail index, and increases the decay of f(x). This distribution was originally used to describe the allocation of wealth or income among individuals in human societies. As a continuous counterpart of Zipf Law, Pareto distribution describes well many other variables that follow a power-law.

In ecology the Pareto distribution can be used to describe the distribution of abundances among species in a biological assemblage (a.k.a. biological community) or in a sample taken from such an assemblage. Though much less used than the lognormal to fit SADs, it can fit better the extremities of the empirical distributions to which the lognormal applies (Johnson et al. 1995, p.608).

References

Johnson, N.L., Kotz, S. and Balakrishnan, N. 1995. Continuous Univariate Distributions, volume 2, chapter 20. Wiley, New York.

See Also

Pareto in packages VGAM and actuar for more general and flexible implementations; fitpareto for maximum likelihood estimation in the context of species abundance distributions.

Examples

par(mfrow=c(1,2))
curve(dpareto(x, shape=3, scale=1), 1,8, ylab="Density",
      main="Pareto PDF")
curve(ppareto(x, shape=3, scale=1), 1,8, ylab="Probability",
      main="Pareto CDF")
par(mfrow=c(1,1))


## Quantile is the inverse function of probability:
p.123 <-ppareto(1:3,shape=3,scale=0.99) 
all.equal(qpareto(p.123, shape=3, scale=0.99), 1:3)