Abstract
We study a three-parameter model named the gamma generalized Pareto distribution. This distribution extends the generalized Pareto model, which has many applications in areas such as insurance, reliability, finance and many others. We derive some of its characterizations and mathematical properties including explicit expressions for the density and quantile functions, ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating function, R\'enyi entropy and order statistics. We discuss the estimation of the model parameters by maximum likelihood. A small Monte Carlo simulation study and two applications to real data are presented. We hope that this distribution may be useful for modeling survival and reliability data.
Highlights
The Pareto distribution is a very known statistical model, widely used for accommodate heavy-tailed distributions and many of its applications in areas such as economics, biology and physics can be found in the literature (Alzaatreh, A., et al, 2012)
We present the results of a Monte Carlo simulation study, which was carried out to evaluate the performance of the MLEs for the parameters a, ξ and σ of the gamma generalized Pareto (GGP) distribution in finite samples
It is noted that the GGP model has the lowest values of the Akaike information criterion (AIC), Bayesian information criterion (BIC) and consistent Akaike information criterion (CAIC) statistics, so it could be chosen as the best model to the these data
Summary
The Pareto distribution is a very known statistical model, widely used for accommodate heavy-tailed distributions and many of its applications in areas such as economics, biology and physics can be found in the literature (Alzaatreh, A., et al, 2012). It is considered that the scale parameter has a direct effect on the tails of this distribution, so that it has heavy-tailed when ξ > 0, medium-tailed when ξ = 0, and short-tailed when ξ < 0 & Balakrishnan, N., 2009) proposed the gamma-G family with an extra shape parameter a > 0 and cdf F(x) given by. ∫z γ(a, z) = ta−1 e−tdt denotes the lower incomplete gamma function and Γ(·) is the gamma function The pdf of this family is given by f (x; a). Some of its mathematical properties are derived in Section including a mixture representation for its pdf and explicit expressions for the quantile function (qf), ordinary and incomplete moments, mean deviations, Bonferroni and Lorenz curves, generating function, Renyi entropy and order statistics.
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