dist.Inverse.ChiSquare {LaplacesDemon}

(Scaled) Inverse Chi-Squared Distribution

Description

This is the density function and random generation for the (scaled) inverse chi-squared distribution.

Usage

dinvchisq(x, df, scale, log=FALSE)
rinvchisq(n, df, scale=1/df)

Arguments

x

This is a vector of quantiles.

n

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

df

This is the degrees of freedom parameter, usually represented as \nu.

scale

This is the scale parameter, usually represented as \lambda.

log

Logical. If log=TRUE, then the logarithm of the density is returned.

Details

• Application: Continuous Univariate

• Density:

  p(\theta) = \frac{{\nu/2}^{\nu/2}}{\Gamma(\nu/2)} \lambda^\nu \frac{1}{\theta}^{\nu/2+1} \exp(-\frac{\nu \lambda^2}{2\theta}), \theta \ge 0

• Inventor: Derived from the chi-squared distribution

• Notation 1: \theta \sim \chi^{-2}(\nu, \lambda)

• Notation 2: p(\theta) = \chi^{-2}(\theta | \nu, \lambda)

• Parameter 1: degrees of freedom parameter \nu > 0

• Parameter 2: scale parameter \lambda

• Mean: E(\theta) = unknown

• Variance: var(\theta) = unknown

• Mode: mode(\theta) =

The inverse chi-squared distribution, also called the inverted chi-square distribution, is the multiplicate inverse of the chi-squared distribution. If x has the chi-squared distribution with \nu degrees of freedom, then 1 / x has the inverse chi-squared distribution with \nu degrees of freedom, and \nu / x has the inverse chi-squared distribution with \nu degrees of freedom.

These functions are similar to those in the GeoR package.

Value

dinvchisq gives the density and rinvchisq generates random deviates.

See Also

dchisq

Examples

library(LaplacesDemon)
x <- dinvchisq(1,1,1)
x <- rinvchisq(10,1)

#Plot Probability Functions
x <- seq(from=0.1, to=5, by=0.01)
plot(x, dinvchisq(x,0.5,1), ylim=c(0,1), type="l", main="Probability Function",
     ylab="density", col="red")
lines(x, dinvchisq(x,1,1), type="l", col="green")
lines(x, dinvchisq(x,5,1), type="l", col="blue")
legend(3, 0.9, expression(paste(nu==0.5, ", ", lambda==1),
     paste(nu==1, ", ", lambda==1), paste(nu==5, ", ", lambda==1)),
     lty=c(1,1,1), col=c("red","green","blue"))