VGAMextra (version 0.0-6)

invgammaDist: The Inverse Gamma Distribution

Description

Density, distribution function, quantile function and random numbers generator for the Inverse Gamma Distribution.

Usage

dinvgamma(x, scale = 1/rate, shape, rate = 1, log = FALSE)
pinvgamma(q, scale = 1/rate, shape, rate = 1, lower.tail = TRUE, log.p = FALSE)
qinvgamma(p, scale = 1/rate, shape, rate = 1, lower.tail = TRUE, log.p = FALSE)
rinvgamma(n, scale = 1/rate, shape, rate = 1)

Value

dinvgamma() returns the density, pinvgamma() gives the distribution function, qinvgamma() gives the quantiles, and

rinvgamma() generates random deviates.

Arguments

x, q, p, n

Same as GammaDist.

scale, shape

Scale and shape parameters, same as GammaDist. Both must be positive.

rate

Same as GammaDist.

log, log.p, lower.tail

Same as GammaDist.

Author

V. Miranda and T. W. Yee.

Warning

The order of the arguments scale and shape does not match GammaDist.

Details

The Inverse Gamma density with parameters scale = \(b\) and shape = \(s\) is given by $$f(y) = \frac{b^{s}}{\Gamma(s)} y^{-s-1} e^{-b/y},$$ for \(y > 0\), \(b > 0\), and \(s > 0\). Here, \(\Gamma(\cdot)\) is the gamma function as in gamma

The relation between the Gamma Distribution and the Inverse Gamma Distribution is as follows:

Let \(X\) be a random variable distributed as Gamma (\(b, s\)), then \(Y = 1 / X\) is distributed as Inverse Gamma (\(1/b, s\)). It is worth noting that the math relation between the scale paramaters of both, the Inverse Gamma and Gamma distributions, is inverse.

Thus, algorithms of dinvgamma(), pinvgamma(), qinvgamma() and rinvgamma() underlie on the algorithms GammaDist.

Let \(Y\) distributed as Inverse Gamma (\(b, s\)). Then the \(k^{th}\) moment of \(Y\) exists for \(-\infty < k < s\) and is given by

$$E[Y^k] = \frac{\Gamma(s - k)}{\Gamma(s)} b^k.$$

The mean (if \(s > 1\)) and variance (if \(s > 2\)) are $$E[Y] = \frac{b}{(s - 1)}; \ \ \ Var[Y] = \frac{b^2}{(s - 1)^2 \times (s - 2)}.$$

References

Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. Wiley Series in Probability and Statistics. Hoboken, New Jersey, USA.

See Also

GammaDist, gamma.

Examples

  # Example 1.______________________
  n        <- 20
  scale    <- exp(2)
  shape    <- exp(1)
  data.1   <- runif(n, 0, 1)
  data.q   <- qinvgamma(-data.1, scale = scale, shape = shape, log.p = TRUE)  
  data.p   <- -log(pinvgamma(data.q, scale = scale, shape = shape)) 
  arg.max  <- max(abs(data.p - data.1))     # Should be zero
  
  # \donttest{
  # Example 2.______________________
  scale <- exp(1.0)
  shape <- exp(1.2)
  xx    <- seq(0, 3.0, len = 201)
  yy    <- dinvgamma(xx, scale = scale, shape = shape)
  qtl   <- seq(0.1, 0.9, by = 0.1)
  d.qtl <- qinvgamma(qtl, scale = scale, shape = shape)
  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, pinvgamma(xx, scale = scale, shape = shape), col = "blue")
  lines(d.qtl, dinvgamma(d.qtl, scale = scale, shape = shape), 
        type ="h", col = "brown", lty = 3)
        # }