expgamma

Expgamma, short for the exponentiated gamma distribution, is a flexible three-parameter family of continuous distributions defined on the positive real axis. It is constructed by taking a gamma distribution and raising its CDF to a positive power, thereby generalizing the gamma family.

Definition. Let X_G ~ Gamma(α, β) with shape α > 0 and scale β > 0. Denote its CDF by F_G(x) and its PDF by f_G(x). For λ > 0, define a random variable X with distribution ExpGamma(λ, α, β) by the CDF F_X(x) = [F_G(x; α, β)]^λ for x ≥ 0. The corresponding PDF is f_X(x) = λ f_G(x; α, β) [F_G(x; α, β)]^{λ−1}.

Special cases. When λ = 1, ExpGamma reduces to the base Gamma(α, β). If α = 1, the family specializes to the exponentiated exponential distribution. The parameter λ controls the concentration and tail behavior, providing greater flexibility than the standard gamma family.

Properties. The density is nonnegative and integrates to 1. The shape of the density and the hazard function can be adjusted through α, β, and λ, allowing a range of skewness and tail behaviors. Moments exist for finite orders, but closed-form expressions are generally not available; moments are typically computed numerically or via series representations.

Applications. The expgamma family is used in reliability analysis, lifetime data modeling, and other applied fields where gamma-like shapes are desirable but additional flexibility is needed to capture heavier or lighter tails and varying skewness.

See also. Gamma distribution; Exponentiated distributions; Exponential distribution.

References. Foundational work on exponentiated gamma families is attributed to researchers studying generalized gamma families and their extensions; practical parameter estimation follows standard maximum likelihood approaches for exponential-family models.