A New Type of Discrete Self-Decomposability and Its Application to Continuous-Time Markov Processes for Modeling Count Data Time Series

Abstract

We propose a family of extended thinning operators, indexed by a parameter γ in [0, 1), with the boundary case of γ=0 corresponding to the well-known binomial thinning operator. The extended thinning operators can be used to construct a class of continuous-time Markov processes for modeling count time series data. The class of stationary distributions of these processes is called generalized discrete self-decomposable, denoted by DSD (γ). We obtain characterization results for the DSD (γ) class and investigate relationships among the classes for different γ's.

Keywords:

• Binomial thinning
• Extended thinning
• Generalized discrete self-decomposability
• Continuous-time Markov processes
• Count data time series

Acknowledgments

This research has been supported by an NSERC Canada Grant. It is part of the first author's Ph.D. thesis. We are grateful to the referee and editors, especially the chief-editor Dr. Peter Taylor, for their valuable comments leading to an improved presentation.