# The Inverse Gamma Distribution Density, distribution function, quantile function and random numbers generator for the Inverse Gamma Distribution. ```r 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) ``` ## 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`. ## Details The Inverse Gamma density with parameters scale = $b$ and shape = $s$ is given by $$ f(y) = \frac{b^{s}}{\Gamma(s)} y^{-s-1} e^{-b/y},%f(y) = (b^s / \Gamma(s)) * y^(-s-1) * e^(-b/y), $$ for $y > 0$, $b > 0$, and $ s > 0$. Here, $gamma()$ 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 $$ E[Y^k] = \frac{\Gamma(s - k)}{\Gamma(s)} b^k.%E[Y^k] = (b^k) * (\Gamma(s - k)/\Gamma(s)). $$ The mean (if $s > 1$) and variance (if $s >2$) are $$ E[Y] = \frac{b}{(s - 1)}; \ \ \ Var[Y] = \frac{b^2}{(s - 1)^2\times (s - 2)}.%E[Y] = b/(s - 1); Var[Y] = b^2 / [((s - 1)^2) * (s - 2)]. $$ ## Returns `dinvgamma()` returns the density, `pinvgamma()` gives the distribution function, `qinvgamma()` gives the quantiles, and `rinvgamma()` generates random deviates. ## 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. ## Author(s) V. Miranda and T. W. Yee. ## Note Unlike the `GammaDist`, small values of $shape$ (plus modest $mu$) or very large values of $mu$ (plus moderate $shape > 2$), generate Inverse Gamma values so near to zero. Thus, `rinvgamma` in `invgammaDist` may return either values too close to zero or values represented as zero in computer arithmetic. In addition, function `dinvgamma` will return zero for $x = 0$, which is the limit of the Inverse Gamma density when $'x'$ tends to zero. ## Warning The order of the arguments **scale** and **shape** does not match `GammaDist`. ## See Also `GammaDist`, `gamma`. ## Examples ```r # 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 # 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) ```

invgammaUC function