Recent Development on Estimation of the Mean Function of a Compound Cyclic Poisson Process

Abstract

Compound cyclic Poisson process is a special form of compound inhomogeneous Poisson process, which has many applications in applied sciences. The objective of this paper is to survey some recent development on estimation of the mean function of a compound cyclic Poisson process. The presented results will include formulation of the estimator, consistency, asymptotic approximations to its bias and variance, asymptotic normality, and a confidence interval for the mean function. We will also show that, in order to have asymptotic bias and asymptotic variance of the estimator, it is needed to modify the estimator. Furthermore, in order to have asymptotic normality of the estimator, it is needed to rewrite the estimator as a sum of independent components. Some simulation results on distribution of the estimator and some applications of this process will also be presented.