Statistical inference for geometric processes with gamma distributions

Abstract

A stochastic process { X i } is a geometric process if there exists a positive real number a such that { a i−1 X i } generates a renewal process. Under the assumption that X 1 follows a Gamma distribution, the statistical inference problem for the geometric process is studied. The parameters a, μ and σ 2, where μ and σ 2, are respectively, the mean and variance of X 1, are estimated by parametric methods including maximum likelihood method along with some nonparametric methods previously proposed by Y. Lam such as the modified moment method. Limiting distributions for the maximum likelihood estimators are derived and this enables us to construct confidence intervals and perform hypothesis testing on parameters. Then some suggestions on the choice of methods are made based on simulation experiments and real data analysis.