Inverse-Gamma: The Inverse Gamma Distribution

In nimble: MCMC, Particle Filtering, and Programmable Hierarchical Modeling

The Inverse Gamma Distribution

Description

Density, distribution function, quantile function and random generation for the inverse gamma distribution with rate or scale (mean = scale / (shape - 1)) parameterizations.

Usage

dinvgamma(x, shape, scale = 1, rate = 1/scale, log = FALSE)

rinvgamma(n = 1, shape, scale = 1, rate = 1/scale)

pinvgamma(
  q,
  shape,
  scale = 1,
  rate = 1/scale,
  lower.tail = TRUE,
  log.p = FALSE
)

qinvgamma(
  p,
  shape,
  scale = 1,
  rate = 1/scale,
  lower.tail = TRUE,
  log.p = FALSE
)

Arguments

x	vector of values.
shape	vector of shape values, must be positive.
scale	vector of scale values, must be positive.
rate	vector of rate values, must be positive.
log	logical; if TRUE, probability density is returned on the log scale.
n	number of observations.
q	vector of quantiles.
lower.tail	logical; if TRUE (default) probabilities are P[X \le x]; otherwise, P[X > x].
log.p	logical; if TRUE, probabilities p are given by user as log(p).
p	vector of probabilities.

Details

The inverse gamma distribution with parameters shape =\alpha and scale =\sigma has density

f(x)= \frac{s^a}{\Gamma(\alpha)} {x}^{-(\alpha+1)} e^{-\sigma/x}%

for x \ge 0, \alpha > 0 and \sigma > 0. (Here \Gamma(\alpha) is the function implemented by R's gamma() and defined in its help.

The mean and variance are E(X) = \frac{\sigma}{\alpha}-1 and Var(X) = \frac{\sigma^2}{(\alpha-1)^2 (\alpha-2)}, with the mean defined only for \alpha > 1 and the variance only for \alpha > 2.

See Gelman et al., Appendix A or the BUGS manual for mathematical details.

Value

dinvgamma gives the density, pinvgamma gives the distribution function, qinvgamma gives the quantile function, and rinvgamma generates random deviates.

Author(s)

Christopher Paciorek

References

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (2004) Bayesian Data Analysis, 2nd ed. Chapman and Hall/CRC.

See Also

Distributions for other standard distributions

Examples

x <- rinvgamma(50, shape = 1, scale = 3)
dinvgamma(x, shape = 1, scale = 3)

nimble documentation built on Sept. 11, 2024, 7:10 p.m.