Inverse Gamma Distribution — invgamma • flexCountReg

These functions provide the density function, distribution function, quantile function, and random number generation for the Inverse-Gamma (IG) Distribution

Usage

dinvgamma(x, shape = 2.5, scale = 1, log = FALSE)

pinvgamma(q, shape = 2.5, scale = 1, lower.tail = TRUE, log.p = FALSE)

qinvgamma(p, shape = 2.5, scale = 1, lower.tail = TRUE, log.p = FALSE)

rinvgamma(n, shape = 2.5, scale = 1)

Arguments

x: numeric value or a vector of values.
shape: numeric value or vector of shape values for the distribution (the values have to be greater than 0).
scale: single value or vector of values for the scale parameter of the distribution (the values have to be greater than 0).
log: logical; if TRUE, probabilities p are given as log(p).
q: quantile or a vector of quantiles.
lower.tail: logical; if TRUE, probabilities p are $P[X\leq x]$ otherwise, $P[X>x]$.
log.p: logical; if TRUE, probabilities p are given as log(p).
p: probability or a vector of probabilities.
n: the number of random numbers to generate.

Details

dinvgamma computes the density (PDF) of the Inverse-Gamma Distribution.

pinvgamma computes the CDF of the Inverse-Gamma Distribution.

qinvgamma computes the quantile function of the Inverse-Gamma Distribution.

rinvgamma generates random numbers from the Inverse-Gamma Distribution.

The compound Probability Mass Function (PMF) for the Inverse-Gamma distribution: $$f(x | \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} \left(\frac{1}{x}\right)^{\alpha+1} e^{-\frac{\beta}{x}}$$

Where $\alpha$ is the shape parameter and $\beta$ is a scale parameter with the restrictions that $\alpha > 0$ and $\eta > 0$, and $x > 0$.

The CDF of the Inverse-Gamma distribution is: $$F(x | \alpha, \beta) = \frac{\alpha. \Gamma \left(\frac{\beta}{x}\right)}{\Gamma(\alpha)} = Q\left(\alpha, \frac{\beta}{x} \right)$$

Where the numerator is the incomplete gamma function and $Q(\cdot)$ is the regularized gamma function.

The mean of the distribution is (provided $\alpha>1$): $$\mu=\frac{\beta}{\alpha-1}$$

The variance of the distribution is (for $\alpha>2$): $$\sigma^2=\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}$$

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

The length of the result is determined by n for rinvgamma, and is the maximum of the lengths of the numerical arguments for the other functions.

Examples

dinvgamma(1, shape = 3, scale = 2)
#> [1] 0.5413411
pinvgamma(c(0.1, 0.5, 1, 3, 5, 10, 30), shape = 3, scale = 2)
#> [1] 4.555150e-07 2.381033e-01 6.766764e-01 9.697879e-01 9.920737e-01
#> [6] 9.988515e-01 9.999530e-01
qinvgamma(c(0.1, 0.3, 0.5, 0.9, 0.95), shape = 3, scale = 2)
#> [1] 0.3757760 0.5531635 0.7479263 1.8147745 2.4459104
rinvgamma(30, shape = 3, scale = 2)
#>  [1] 0.2937822 0.4909691 0.5750880 0.9867140 0.6353773 1.9819641 0.8870580
#>  [8] 1.0057749 0.6611418 0.4277593 2.5997910 0.6948481 0.3511642 0.4924415
#> [15] 0.9255774 6.1444962 1.3036911 1.4501172 0.6066221 0.9371221 0.4320380
#> [22] 1.6987338 2.1125857 0.3442973 0.3036261 1.3134867 0.4868805 1.2093836
#> [29] 1.2170003 0.5214717