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Aminzadeh, M.S., and Min Deng. 2022. “Bayesian Estimation of Renewal Function Based on Pareto-Distributed Inter-Arrival Times via an MCMC Algorithm.” Variance 15 (2).
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  • Table 1. Accuracy of ML and Bayes estimators for \(m\), \(\alpha\), and \(M(t)\)
  • Table 2. P-values for goodness of fit for SPP, gamma, exponential, and IG
  • Table 3. Accuracy of ML and Bayes estimators for \(m\), \(\alpha\), and \(M(t)\)
  • Table 4. P-values for GOF for SPP, Gamma, Exponential, IG
  • Table 5. Accuracy of Bayes estimators for \(m\), \(\alpha\), and \(M(t)\) when MLEs are used to choose hyper-parameters
  • Figure 1. MCMC Chain PLot for Bayes Estimate of \(\alpha\)
  • Figure 2. Histogram of \(\alpha\)(Bayes)
  • Figure 3. MCMC Chain Plot for Bayes Estimate of m
  • Figure 4. Histogram of m(Bayes)
  • Table 6. P-values for GOF of Flood Data
  • Table 7. P-values for GOF of Tornado Data
  • Table 8. P-values for GOF of a generated sample from SPP
  • Table 9. Hurricane data for New Orleans 9/29/2021-1/18/2022
  • Table 10. P-values for goodness of fit of hurricane data for New Orleans
  • Table 11. Date and Insured Damage Losses of Natural Event Flood in USA from 2000 to 2021
  • Table 12. Date and Insured Damage Losses of Natural Event Tornado in USA from 2000 to 2021


The purpose of this article is to provide a computational tool via Maximum Likelihood (ML) and Markov Chain Mont Carlo (MCMC) methods for estimating the renewal function when the inter-arrival distribution of a renewal process is single-parameter Pareto (SPP). The proposed method has applications in a variety of applied fields such as insurance modeling and modeling self-similar network traffic, to name a few. It is shown that inter-arrivals of insured damages for floods and tornados during 2000-2020 in the USA have SPP distribution. It is also shown that inter-arrivals of recent hurricanes hitting New Orleans fit SPP distribution. For the Bayesian estimation of SPP parameters via the MCMC method, based on the Metropolis algorithm, gamma and shifted exponential distributions are used. Simulations confirm that the MCMC estimator of the renewal function outperforms maximum likelihood estimator (MLE) with regards to its accuracy when the sample size is relatively small. However, for large samples, the accuracies of ML and Bayes estimators for the renewal function are comparable.

Accepted: May 22, 2022 EDT