Aging Renewal Point Processes and Exchangeability of Event Times

Abstract

We investigate the impact of aging on exchangeable inter-arrival times in mixed renewal processes, exploring its implications for reliability and survival analysis. In this study, first, we revisit the definition of renewal point processes, where inter-event time intervals are considered as exchangeable non-negative random variables. Then, we define the concept of statistical aging as latency in the observational process of event counting. Latency affects event detection but preserves exchangeability. However, it may alter the statistical properties of inter-event time intervals. Our analytical and numerical assessments highlight the significance of aging in exchangeable lifetimes, offering insights into key metrics such as the failure survival function, renewal function, and hazard rate function. Through archetypal examples and empirical findings, we illustrate the implications of aging on renewal processes. In particular, employing a Bayesian perspective, we analyze high-frequency currency exchange rate data to assess the impact of aging on volatility risk evaluation. This study contributes novel insights to the literature of renewal theory and survival analysis, emphasizing the role of latency aging in stochastic point processes under exchangeability assumption.