R: (Scaled) Inverse Chi-Squared Distribution

dist.Inverse.ChiSquare {LaplacesDemon} | R Documentation

## (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) = ((nu/2)^(nu/2))/(Γ(nu/2)) lambda^nu (1/theta)^((nu/2)+1) exp(-(nu lambda^2)/(2*theta)), theta >= 0

- Inventor: Derived from the chi-squared distribution

- Notation 1: theta ~ 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"))

[Package LaplacesDemon version 16.1.1 Index]