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).
• 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

## Abstract

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

# Appendix

Table 11.Date and Insured Damage Losses of Natural Event Flood in USA from 2000 to 2021
 Start Year StartMonth StartDay EndYear EndMonth EndDay InsuredDamages ('000US$) 2002 6 30 2002 7 23 200500 2004 7 12 2004 7 14 370000 2005 2 17 2005 2 23 200000 2005 1 7 2005 1 11 200000 2005 10 8 2005 10 16 200000 2006 6 25 2006 7 1 401000 2006 5 11 2006 5 22 62500 2007 8 16 2007 8 27 450000 2008 6 9 2008 6 30 400000 2009 9 20 2009 9 21 200000 2011 4 18 2011 5 23 500000 2011 8 6 2011 8 9 86000 2013 9 12 2013 9 19 200000 2014 12 2 2014 12 5 50000 2014 8 11 2014 8 13 540000 2015 12 15 2016 1 6 200000 2015 5 23 2015 5 30 1500000 2016 4 16 2016 4 19 1000000 2016 5 27 2016 6 2 200000 2016 8 9 2016 8 16 2500000 2016 3 8 2016 3 13 333000 2017 4 28 2017 5 1 800000 2019 3 14 2019 3 31 2500000 The following method is used to compute interarrival times. Let $$X_n,$$ the interarrival time between the $$(n-1)$$st and $$n$$th renewal, be defined as the difference between the end date of an occurrence and the start date of the subsequent occurrence. $$n=1,2,3,....$$ Without loss of generally, we assume that there are 30 days in a month and 360 days in a year. Therefore, interarrival times are found as a fraction of a year. $\frac{30\times \text{number of months + number of days}}{360}.$ For example, in Table 3 above, the first flood began at 6/30/2002 and ended at 7/23/2002, and the second flood began at 7/12/2004 and ended at 7/14/2004. Therefore, $X_1=\frac{30 \times 31 \ \ \text{months}}{360}=2.583 \ \ \text{years.}$ $$2.583\times 12=30.996$$ months, which is the last number in the flood data. $X_2=\frac{30\times 24 \ \ \text{months - 11 days}}{360}=1.969\ \ \text{years,}$ which is also converted to 23.628 months. Table 12.Date and Insured Damage Losses of Natural Event Tornado in USA from 2000 to 2021  Start Year StartMonth StartDay EndYear EndMonth EndDay InsuredDamages ('000US$) 2000 1 2 2000 1 4 200500 2000 9 20 2000 9 20 62500 2001 2 24 2001 2 24 200500 2002 4 27 2002 5 3 2000500 2002 11 5 2002 11 10 450500 2003 3 18 2003 3 20 450000 2003 5 4 2003 5 10 3200000 2004 3 4 2004 3 7 200000 2006 3 11 2006 3 13 800000 2006 11 15 2006 11 16 62500 2006 12 25 2006 12 25 56000 2007 2 2 2007 2 2 200000 2007 3 1 2007 3 2 450000 2007 5 4 2007 5 8 200000 2007 2 2 2007 2 2 200000 2008 5 22 2008 5 26 1325000 2008 5 10 2008 5 12 450000 2008 2 5 2008 2 6 955000 2008 3 14 2008 3 14 450000 2008 4 9 2008 4 11 800000 2008 4 28 2008 4 28 62500 2009 4 9 2009 4 10 1150000 2009 6 10 2009 6 18 1100000 2009 2 10 2009 2 13 1350000 2009 5 7 2009 5 9 600000 2010 5 12 2010 5 16 2000000 2011 5 29 2011 6 1 450000 2011 4 22 2011 4 29 8000000 2011 4 14 2011 4 16 1500000 2011 5 20 2011 5 25 6900000 2012 3 2 2012 3 4 2500000 2012 2 28 2012 2 29 450000 2012 4 2 2012 4 3 800000 2012 4 14 2012 4 15 910000 2013 5 18 2013 5 22 1800000 2013 5 15 2013 5 17 200000 2013 11 16 2013 11 18 800000 2013 1 29 2013 1 30 200000 2013 5 26 2013 6 6 1425000 2015 4 24 2015 4 28 800000 2015 5 3 2015 5 5 200000 2015 4 7 2015 4 10 990000 2015 3 25 2015 3 26 450000 2015 12 26 2015 12 30 800000 2016 5 8 2016 5 11 800000 2017 8 5 2017 8 8 200000 2017 3 25 2017 3 28 2000000 2017 2 27 2017 3 2 1400000 2018 7 19 2018 7 22 400000 2019 3 3 2019 3 4 140000 2020 4 6 2020 4 9 2200000 2020 4 10 2020 4 14 2600000