The joint distribution of the maximum and duration of stochastic events driven by Pareto II observations
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Authors
Donkor, Foster
Issue Date
2025
Type
Thesis
Language
en_US
Keywords
GMOL Model , Heavy Tail , Lomax/Pareto Type II Distribution , Maximum Likelihood Estimation , Power Law , Precipitation Data
Alternative Title
Abstract
We study the joint distribution of X and N, where N has the geometric distribution and X is the maximum of the N independent and identically distributed Pareto II (Lomax) observations. The bivariate distribution is referred to as the geometric Marshall-Olkin Lomax distribution (GMOL). A related model for a geometric maximum of IID exponential observations was introduced by Kozubowski and Panorska (2008) and has proven useful in areas such as finance, hydrology and climate. However, the existence of heavy tails in environmental variables motivated this model. Our results for this research include derivations of the joint probability density function, cumulative distribution function, conditional and marginal distributions, conditional survival function, moment-generating function, Laplace transforms, and covariance matrix. We also address the problem of parameter estimation using the method of maximum likelihood. Estimation is empirically verified using a simulation study. We also present results of modeling precipitation and temperature data sets.