# invchisquaredUC: The Inverse Chi-squared Distribution In VGAMextra: Additions and Extensions of the 'VGAM' Package

 inv.chisqDist R Documentation

## The Inverse Chi–squared Distribution

### Description

Density, CDF, quantile function and random number generator for the Inverse Chi–squared distribution.

### Usage

    dinv.chisq(x, df, log = FALSE)
pinv.chisq(q, df, lower.tail = TRUE, log.p = FALSE)
qinv.chisq(p, df, lower.tail = TRUE, log.p = FALSE)
rinv.chisq(n, df)



### Arguments

 x, q, p, n Same as Chisquare. df, lower.tail, log, log.p Same as Chisquare.

### Details

The inverse chi–squared distribution with non–negative df = \nu degrees of freedom implemented here has density

f(x; \nu) = \frac{ 2^{-\nu / 2} x^{-\nu/2 - 1} e^{-1 / (2x)} }{ \Gamma(\nu / 2) }, 

where x > 0, and \Gamma is the gamma function.

The mean is 1 / (\nu - 2), for \nu > 2, and the variance is given by 2 / [(\nu - 2)^2 (\nu - 4)], for \nu > 4.

Also, as with Chisquare, the degrees of freedom can be non–integer.

### Value

dinv.chisq returns the density, pinv.chisq returns the distribution function, qinv.chisq gives the quantiles, and rinv.chisq generates random numbers from this distribution.

### Source

Specifically, it is the probability distribution of a random variable whose reciprocal follows a chi–squared distribution, i.e.,

If Y \sim chi–squared(\nu), then 1/Y \sim Inverse chi–squared(\nu).

As a result, dinv.chisq, pinv.chisq, qinv.chisq and rinv.chisq use the functions Chisquare as a basis. Hence, invalid arguments will lead these functions to return NA or NaN accordingly.

### Note

Yet to do: A non–central parameter as an argument, if amenable.

Two similar versions of the Inverse chi–squared distribution with \nu degrees of freedom may be found in the literature, as follows:

Let Y \sim chi–squared(\nu), then

I. 1 / Y \sim Inverse chi–squared(\nu), and II.  \nu / Y \sim Inverse chi–squared(\nu).

Here, the former, which is the popular version, has been implemented.

V. Miranda

### References

Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions. Chapters 18 (volume 1) and 29 (volume 2). Wiley, New York.

Chisquare, gamma.

### Examples


##  Example 1  ##
nn <- 50; df <- 1.4
data.1   <- ppoints(nn)
data.q   <- qinv.chisq(-data.1, df = df, log.p = TRUE)
data.p   <- -log(pinv.chisq(data.q, df = df))
max(abs(data.p - data.1))     # Should be zero

##  Example 2  ##

xx    <- seq(0, 3.0, len = 301)
yy    <- dinv.chisq(xx, df = df)
qtl   <- seq(0.1, 0.9, by = 0.1)
d.qtl <- qinv.chisq(qtl, df = df)
plot(xx, yy, type = "l", col = "orange",
main = "Orange is density, blue is cumulative distribution function",
sub  = "Brown dashed lines represent the 10th, ... 90th percentiles",
las = 1, xlab = "x", ylab = "", ylim = c(0, 1))
abline(h = 0, col= "navy", lty = 2)
lines(xx, pinv.chisq(xx, df = df), col = "blue")
lines(d.qtl, dinv.chisq(d.qtl, df = df), type ="h", col = "brown", lty = 3)



VGAMextra documentation built on Nov. 2, 2023, 5:59 p.m.