Binomial ARMA count series from renewal processes

TL;DR

This paper introduces a novel method for generating stationary integer-valued time series with ARMA-like autocovariance functions using renewal processes, providing a practical approach for modeling count data.

Contribution

It establishes conditions under which renewal processes produce ARMA-type count series and offers an estimation method for AR models in this context.

Findings

Generated binomial ARMA series from renewal processes.

Proved conditions for causality and invertibility.

Developed an estimation method for AR cases.

Abstract

This paper describes a new method for generating stationary integer-valued time series from renewal processes. We prove that if the lifetime distribution of renewal processes is nonlattice and the probability generating function is rational, then the generated time series satisfy causal and invertible ARMA type stochastic difference equations. The result provides an easy method for generating integer-valued time series with ARMA type autocovariance functions. Examples of generating binomial ARMA(p,p-1) series from lifetime distributions with constant hazard rates after lag p are given as an illustration. An estimation method is developed for the AR(p) cases.