extremefit (version 1.0.2)

Pareto Distribution: Pareto distribution

Description

Density, distribution function, quantile function and random generation for the Pareto distribution where \(a\), \(loc\) and \(scale\) are respectively the shape, the location and the scale parameters.

Usage

ppareto(q, a = 1, loc = 0, scale = 1)
dpareto(x, a = 1, loc = 0, scale = 1)
qpareto(p, a = 1, loc = 0, scale = 1)
rpareto(n, a = 1, loc = 0, scale = 1)

Arguments

q

a vector of quantiles.

a

a vector of shape parameter of the Pareto distribution.

loc

a vector of location parameter of the Pareto distribution.

scale

a vector of scale parameter of the Pareto distribution.

x

a vector of quantiles.

p

a vector of probabilities.

n

a number of observations. If length(n) > 1, the length is taken to be the number required.

Value

dpareto gives the density, ppareto gives the distribution function, qpareto gives the quantile function, 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.

The numerical arguments other than n are recycled to the length of the result. Only the first elements of the logical arguments are used.

Details

If \(shape\), \(loc\) or \(scale\) parameters are not specified, the respective default values are \(1\), \(0\) and \(1\).

The cumulative Pareto distribution is $$ F(x) = 1- ((x-loc)/scale) ^ {-a}, x > loc, a > 0, scale > 0 $$ where \(a\) is the shape of the distribution.

The density of the Pareto distribution is

$$ f(x) = (((x-loc)/scale)^( - a - 1) * a/scale) * (x-loc >= scale), x > loc, a > 0, scale > 0 $$

Examples

# NOT RUN {
par(mfrow = c(3,1))
plot(function(x) dpareto(x), 1, 5,ylab="density",
     main = " Pareto density ")

plot(function(x) ppareto(x), 1, 5,ylab="distribution function",
     main = " Pareto Cumulative ")

plot(function(x) qpareto(x), 0, 1,ylab="quantile",
     main = " Pareto Quantile ")

#generate a sample of pareto distribution of size n
n <- 100
x <- rpareto(n)

# }