Pareto: The Pareto Distribution

In cmottet/DistributionPty: Properties of some distribution functions

Description

Density, distribution function, distribution function derivatives, quantile function and random generation for the Pareto distribution function with shape shape and scale scale.

Usage

rpareto(n = 1, scale = 1, shape = 1)

ppareto(x, scale = 1, shape = 1)

dpareto(x, scale = 1, shape = 1)

Dpareto(x, d, scale = 1, shape = 1)

qpareto(p, scale = 1, shape = 1)

Arguments

n	Number of observations. If length(n) > 1, the length is taken to be the number required. Default value is 1
scale, shape	Positive real values respectively defining the shape and scale parameter of the Pareto distributon. Default value is 1 for both of them
x, q	Vector of quantiles
d	An non-negative integer giving the order of the derivative
p	vector of probabilities

Details

The Pareto distribution has the following cumultative distribution function

F(x) = 1 - (x/scale)^(-α)

for all x > scale

Value

dpareto gives the density, ppareto gives the distribution function, Dpareto gives the distribution function derivative, qpareto gives the quantile function, and rpareto generates random observations.

Examples

# Simulate a pareto sample
n <- 1e2 ; scale <- 1 ; shape <- 1
set.seed(100) ; X <- sort(rpareto(n, scale , shape))

# Compare the ECDF and the distribution function
plot(X, ecdf(X)(X))
lines(X,ppareto(X,shape,scale) , col = "red")

# Compare kernel density and density function
with(density(X, from = min(X)), plot(x, y))
lines(X,dpareto(X,shape,scale), col = "red")

# Visualize the distribution derivatives for 0 <= d <= 3
D <- 0:3

derivatives <- sapply(D, function(d) Dpareto(X,d,scale, shape))
dataPlot <- data.frame(
             x = rep(X,length(D)),
             y = c(derivatives),
             D = rep(paste0("d = ",D ), each = length(X))
             )

library(ggplot2)
ggplot(dataPlot, aes(x = x, y = y)) +
geom_line() +
facet_grid(D~., scales = "free_y")

cmottet/DistributionPty documentation built on May 13, 2019, 8:44 p.m.