On Fitting Probability Distribution to Univariate Grouped Actuarial Data with Both Group Mean and Relative Frequencies

Abstract

Many publicly available datasets relevant to actuarial work contain data grouped in various ways. For example, operational loss data are often reported in a grouped format that includes group boundaries, loss frequency, and average or total amount of loss for each group. The process of fitting a parametric distribution to grouped data becomes more complex but potentially more accurate when additional information, such as group means, is incorporated in the estimation process. This article compares the relative performance of three methods of inference using distributions suitable for actuarial applications, particularly those that are right-skewed, heavy-tailed, and left-truncated. We compare the traditional maximum likelihood method, which only considers the group limits and frequency of observations in each group, to two research innovations: a modified maximum likelihood method and a modified generalized method of moments approach, both of which incorporate additional group mean information in the estimation process. We perform a simulation study where the proposed methods outperform the traditional maximum likelihood method and the maximum entropy when the true underlying distribution is both known and unknown. Further, we apply the methods to three actuarial datasets: operational loss data, pension fund data, and car insurance claims data. Here we compare the performance of the three methods along with the maximum entropy distribution (under the traditional maximum likelihood and the modified maximum likelihood methods) and find that for all three datasets the proposed methods outperform the traditional maximum likelihood method. We conclude that there is merit in considering the proposed methods while fitting a parametric distribution to grouped data.

ACKNOWLEDGMENTS

The authors thank Prof. Edward Frees for his valuable comments. The authors appreciate the operational loss data suggestion from Prof. Pavel Shevchenko. The authors express their thanks for the comments received from the 23rd International Congress on Insurance: Mathematics and Economics in 2019, a research seminar at the Research School of Finance, Actuarial Studies & Statistics, College of Business and Economics, The Australian National University and a research seminar at the School of Risk and Actuarial Studies, University of New South Wales in 2021.

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Notes

1 Previous results of Brockett et al. (1995) are discussed in Appendix B.

2 We also develop our simulation study when the simulated data come from an equally weighted mixture distribution of generalized Gamma, Gamma, Lognormal, and Weibull distributions. The results were broadly similar to the simulation study presented here and hence are not reported in this article. These results are available from the authors upon request.

3 We also performed the analysis for data left-truncated at 10,000 and found the same conclusions.

4 Australian term for retirement savings.

5 Over the last few years, the government-mandated minimum retirement age has undergone significant changes (based on the year of birth). Consequently, the data on individuals in the 55–65 age category are sparse. Hence, we have not included this age band in our analysis. The three age groups analyzed in this research are sufficient for the purposes of illustration of the methodology.

6 We also performed computational run time comparison. These are provided in Appendix D.