dist.Inverse.ChiSquare: (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

This is a vector of quantiles.

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

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.

Value

dinvchisq gives the density and rinvchisq generates random deviates.

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.

Examples

Run this code

# NOT RUN {
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"))
# }