# dPPS: The Pareto Positive Stable (PPS) distribution In ParetoPosStable: Computing, Fitting and Validating the PPS Distribution

## Description

Density, distribution function, hazard function, quantile function and random generation for the Pareto Positive Stable (PPS) distribution with parameters `lam`, `sc` and `v`.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```dPPS(x, lam, sc, v, log = FALSE) hPPS(x, lam, sc, v) pPPS(x, lam, sc, v, lower.tail = TRUE, log.p = FALSE) qPPS(p, lam, sc, v, lower.tail = TRUE, log.p = FALSE) rPPS(n, lam, sc, v) ```

## Arguments

 `x` vector of quantiles. `lam` vector of (non-negative) first shape parameters. `sc` vector of (non-negative) scale parameters. `v` vector of (non-negative) second shape parameters. `log` logical; if TRUE, probabilities/densities p are returned as log(p). `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. `log.p` logical; if TRUE, probabilities/densities p are returned as log(p). `p` vector of probabilities. `n` number of random values to return.

## Details

The PPS distribution has density

f(x) = λ ν [log(x / σ)] ^ (ν-1) exp(- λ [log(x / σ)] ^ ν) / x,

cumulative distribution function

F(x) = 1 - exp(- λ [log(x / σ) ^ ν]),

quantile function

Q(p) = σ exp([- (1 / λ) log(1 - p)] ^ (1 / ν))

and hazard function

λ ν (log(x / σ)) ^ (ν - 1) x ^ (-1).

See Sarabia and Prieto (2009) for the details about the numbers random generation.

## Value

`dPPS` gives the (log) density, `pPPS` gives the (log) distribution function, `qPPS` gives the quantile function, and `rpois` generates random samples. Invalid parameters will result in return value `NaN`, with a warning. The length of the result is determined by `n` for `rPPS`, and is the common length of the numerical arguments for the other functions.

## References

Sarabia, J.M and Prieto, F. (2009). The Pareto-positive stable distribution: A new descriptive model for city size data, Physica A: Statistical Mechanics and its Applications, 388(19), 4179-4191.

## Examples

 ```1 2 3 4 5``` ```print(x <- sort(rPPS(10, 1.2, 100, 2.3))) dPPS(x, 1.2, 100, 2.3) pPPS(x, 1.2, 100, 2.3) qPPS(pPPS(x, 1.2, 100, 2.3), 1.2, 100, 2.3) hPPS(x, 1.2, 100, 2.3) ```

ParetoPosStable documentation built on May 30, 2017, 8:26 a.m.