LaplacesDemon: dist.Pareto – R documentation

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dist.Pareto

Pareto Distribution

Description

These functions provide the density, distribution function, quantile function, and random generation for the pareto distribution.

Usage

dpareto(x, alpha, log=FALSE)
ppareto(q, alpha)
qpareto(p, alpha)
rpareto(n, alpha)

Arguments

`x,q`	These are each a vector of quantiles.
`p`	This is a vector of probabilities.
`n`	This is the number of observations, which must be a positive integer that has length 1.
`alpha`	This is the shape parameter alpha, which must be positive.
`log`	Logical. If `log=TRUE`, then the logarithm of the density or result is returned.

Details

Application: Continuous Univariate
Density: p(theta) = alpha / theta^(alpha + 1), theta >= 1
Inventor: Vilfredo Pareto (1848-1923)
Notation 1: theta ~ PA(alpha)
Notation 2: p(theta) = PA(theta | alpha)
Parameter 1: shape parameter alpha > 0
Mean: E(theta) = alpha / (alpha - 1)
Variance: var(theta) = alpha / (alpha - 1)^2 (alpha - 2), alpha > 2
Mode: mode(theta) = 1

The Pareto distribution, sometimes called the Bradford distribution, is related to the exponential distribution. The gamma distribution is the conjugate prior distribution for the shape parameter alpha in the Pareto distribution. The Pareto distribution is the conjugate prior distribution for the range parameters of a uniform distribution. An extension, elsewhere, is the symmetric Pareto distribution.

Value

dpareto gives the density, ppareto gives the distribution function, qpareto gives the quantile function, and rpareto generates random deviates.

Examples

library(LaplacesDemon)
x <- dpareto(1,1)
x <- ppareto(0.5,1)
x <- qpareto(0.5,1)
x <- rpareto(10,1)

#Plot Probability Functions
x <- seq(from=1, to=5, by=0.01)
plot(x, dpareto(x,0.1), ylim=c(0,1), type="l", main="Probability Function",
     ylab="density", col="red")
lines(x, dpareto(x,0.5), type="l", col="green")
lines(x, dpareto(x,1), type="l", col="blue")
legend(2, 0.9, expression(alpha==0.1, alpha==0.5, alpha==1),
     lty=c(1,1,1), col=c("red","green","blue"))

LaplacesDemon

Complete Environment for Bayesian Inference

v16.1.4

MIT + file LICENSE

Authors

Byron Hall [aut], Martina Hall [aut], Statisticat, LLC [aut], Eric Brown [ctb], Richard Hermanson [ctb], Emmanuel Charpentier [ctb], Daniel Heck [ctb], Stephane Laurent [ctb], Quentin F. Gronau [ctb], Henrik Singmann [cre]

Initial release