Pareto distribution

Description

Density, distribution function, quantile function and random generation 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)
rpareto(n, shape, scale = 1)
```

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. `n` number of random values to return. `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].

Details

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

f(x) = shape * scale^shape / x^(shape+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).

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.

Note

These functions implement the Pareto distribution of the first kind sensu Johnson et al. (1995, pp.574).

The pdf and cdf are defined as zero for all `x < scale`, but the functions `[dp]pareto` currently return an error if `scale > min(x)`, to avoid some fitting and plotting problems.

References

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

`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)
```