# Inverse-Gamma: The Inverse Gamma Distribution In nimble: MCMC, Particle Filtering, and Programmable Hierarchical Modeling

 Inverse-Gamma R Documentation

## 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 ≤ 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` = a and `scale` = s has density

f(x)= (s^a / Gamma(a)) x^-(a+1) e^-(s/x)

for x ≥ 0, a > 0 and s > 0. (Here Gamma(a) is the function implemented by R's `gamma()` and defined in its help.

The mean and variance are E(X) = s/(a-1) and Var(X) = s^2 / ((a-1)^2 * (a-2)), with the mean defined only for a > 1 and the variance only for a > 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.

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