Amoroso distribution

Scalar Amoroso utilities used as flexible positive-support base kernels and mixture components. The package parameterization uses loc, scale, shape1, and shape2, allowing the family to represent a broad range of skewed bulk shapes.

Usage

dAmoroso(x, loc, scale, shape1, shape2, log = 0)

pAmoroso(q, loc, scale, shape1, shape2, lower.tail = 1, log.p = 0)

qAmoroso(p, loc, scale, shape1, shape2, lower.tail = TRUE, log.p = FALSE)

rAmoroso(n, loc, scale, shape1, shape2)

Arguments

x: Numeric scalar giving the point at which the density is evaluated.
loc: Numeric scalar location parameter.
scale: Numeric scalar scale parameter.
shape1: Numeric scalar first shape parameter.
shape2: Numeric scalar second shape parameter.
log: Logical; if TRUE, return the log-density (integer flag 0/1 in NIMBLE).
q: Numeric scalar giving the point at which the distribution function is evaluated.
lower.tail: Logical; if TRUE (default), probabilities are $P(X \le q)$.
log.p: Logical; if TRUE, probabilities are returned on the log scale.
p: Numeric scalar probability in $(0,1)$ for the quantile function.
n: Integer giving the number of draws. For portability inside NIMBLE, the RNG implementation supports n = 1.

Value

Density/CDF/RNG functions return numeric scalars. The quantile function returns a numeric scalar or vector matching the length of p.

Details

Writing loc = a, scale = theta, shape1 = alpha, and shape2 = beta, the transformed quantity $((X - a) / \theta)^\beta$ follows a gamma law with shape $\alpha$.

These uppercase NIMBLE-compatible functions are scalar (x/q and n = 1). For vectorized R usage, use base_lowercase().

The Amoroso family used in the package is defined by the density $$ f(x) = \left|\frac{\beta}{\theta}\right| \frac{z^{\alpha \beta - 1}\exp(-z^\beta)}{\Gamma(\alpha)}, \qquad z = \frac{x-a}{\theta}, $$ on the side of the location parameter determined by the sign of $\theta$. Equivalently, $Z = ((X-a)/\theta)^\beta$ follows a Gamma distribution with shape $\alpha$ and unit scale. That representation explains why the quantile function is computed from a gamma quantile and then mapped back through the inverse transformation.

The mean exists whenever $\alpha + 1/\beta$ lies in the domain of the gamma function used by the moment formula. In the package this family serves as a flexible positive-support bulk kernel capable of reproducing gamma-like, Weibull-like, and other skewed shapes with a single parameterization.

Functions

dAmoroso(): Density Function of Amoroso Distribution
pAmoroso(): Distribution Function of Amoroso Distribution
qAmoroso(): Quantile Function of Amoroso Distribution
rAmoroso(): Sample generating Function of Amoroso Distribution

Examples

loc <- 0
scale <- 1.5
shape1 <- 2
shape2 <- 1.2

dAmoroso(1.0, loc, scale, shape1, shape2, log = 0)
#> [1] 0.2452362
pAmoroso(1.0, loc, scale, shape1, shape2, lower.tail = 1, log.p = 0)
#> [1] 0.126778
qAmoroso(0.50, loc, scale, shape1, shape2)
#> [1] 2.309366
qAmoroso(0.95, loc, scale, shape1, shape2)
#> [1] 5.489526
replicate(10, rAmoroso(1, loc, scale, shape1, shape2))
#>  [1] 2.7006836 1.1196594 0.2684797 2.1882428 2.3012404 1.5863122 3.3273216
#>  [8] 3.5601145 4.3648873 1.1861036