Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-14T14:46:43.063Z Has data issue: false hasContentIssue false
  • English
  • Français

ROBUST ESTIMATION OF LOSS MODELS FOR LOGNORMAL INSURANCE PAYMENT SEVERITY DATA

Published online by Cambridge University Press:  05 March 2021

Chudamani Poudyal Open the ORCID record for Chudamani Poudyal [Opens in a new window]
Chudamani Poudyal*
Affiliation:
Department of Mathematics, Tennessee Technological University, Cookeville, TN38505, USA E-Mail: cpoudyal@tntech.edu
*
E-Mail: cpoudyal@tntech.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The primary objective of this scholarly work is to develop two estimation procedures – maximum likelihood estimator (MLE) and method of trimmed moments (MTM) – for the mean and variance of lognormal insurance payment severity data sets affected by different loss control mechanism, for example, truncation (due to deductibles), censoring (due to policy limits), and scaling (due to coinsurance proportions), in insurance and financial industries. Maximum likelihood estimating equations for both payment-per-payment and payment-per-loss data sets are derived which can be solved readily by any existing iterative numerical methods. The asymptotic distributions of those estimators are established via Fisher information matrices. Further, with a goal of balancing efficiency and robustness and to remove point masses at certain data points, we develop a dynamic MTM estimation procedures for lognormal claim severity models for the above-mentioned transformed data scenarios. The asymptotic distributional properties and the comparison with the corresponding MLEs of those MTM estimators are established along with extensive simulation studies. Purely for illustrative purpose, numerical examples for 1500 US indemnity losses are provided which illustrate the practical performance of the established results in this paper.

Keywords

Dynamic estimationlognormal insurance severitymaximum likelihood estimatorstrimmed momentsinsurance paymentsloss modelsrobust estimationtruncated and censored data
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 51 , Issue 2 , May 2021 , pp. 475 - 507
Copyright
© 2021 by Astin Bulletin. All rights reserved

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abu Bakar, S.A., Hamzah, N.A., Maghsoudi, M. and Nadarajah, S. (2015) Modeling loss data using composite models. Insurance: Mathematics & Economics, 61, 146–154.Google Scholar
Barrow, D.F. and Cohen, Jr., A.C. (1954) On some functions involving Mill’s ratio. Annals of Mathematical Statistics, 25, 405–408.CrossRefGoogle Scholar
Blostein, M. and Miljkovic, T. (2019) On modeling left-truncated loss data using mixtures of distributions. Insurance: Mathematics & Economics, 85, 35–46.Google Scholar
Brazauskas, V., Jones, B.L. and Zitikis, R. (2009). Robust fitting of claim severity distributions and the method of trimmed moments. Journal of Statistical Planning and Inference, 139(6), 2028–2043.CrossRefGoogle Scholar
Brazauskas, V. and Kleefeld, A. (2016) Modeling severity and measuring tail risk of Norwegian fire claims. North American Actuarial Journal, 20(1), 1–16.CrossRefGoogle Scholar
Brazauskas, V. and Serfling, R. (2000) Robust and efficient estimation of the tail index of a single-parameter Pareto distribution. North American Actuarial Journal, 4(4), 12–27.CrossRefGoogle Scholar
Chan, J.S.K., Choy, S.T.B., Makov, U.E. and Landsman, Z. (2018). Modelling insurance losses using contaminated generalized beta type-II distribution. ASTIN Bulletin, 48(2), 871–904.CrossRefGoogle Scholar
Chernoff, H., Gastwirth, J.L. and Johns, Jr., M.V. (1967) Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation. Annals of Mathematical Statistics, 38(1), 52–72.CrossRefGoogle Scholar
Cohen, A.C. Jr. (1950) Estimating the mean and variance of normal populations from singly truncated and doubly truncated samples. Annals of Mathematical Statistics, 21(4), 557–569.CrossRefGoogle Scholar
Cohen, A.C. Jr. (1951) On estimating the mean and variance of singly truncated normal frequency distributions from the first three sample moments. Annals of the Institute of Statistical Mathematics, 3, 37–44.CrossRefGoogle Scholar
Cooray, K. and Ananda, M.M.A. (2005) Modeling actuarial data with a composite lognormal-Pareto model. Scandinavian Actuarial Journal, 2005(5), 321–334.CrossRefGoogle Scholar
Ergashev, B., Pavlikov, K., Uryasev, S. and Sekeris, E. (2016) Estimation of truncated data samples in operational risk modeling. The Journal of Risk and Insurance, 83(3), 613–640.CrossRefGoogle Scholar
Frees, E.W. and Valdez, E.A. (1998) Understanding relationships using copulas. North American Actuarial Journal, 2(1), 1–25.CrossRefGoogle Scholar
Grün, B. and Miljkovic, T. (2019) Extending composite loss models using a general framework of advanced computational tools. Scandinavian Actuarial Journal, 2019(8), 642–660.CrossRefGoogle Scholar
Gui, W., Huang, R. and Lin, X.S. (2021) Fitting multivariate Erlang mixtures to data: A roughness penalty approach. Journal of Computational and Applied Mathematics, 386, 113216, 17.CrossRefGoogle Scholar
Hampel, F.R. (1974) The influence curve and its role in robust estimation. Journal of the American Statistical Association, 69, 383–393.CrossRefGoogle Scholar
Hewitt, C.C., Jr. and Lefkowitz, B. (1979) Methods for fitting distributions to insurance loss data. In Proceedings of the Casualty Actuarial Society, vol. LXVI, pp. 139–160. VA: Casualty Actuarial Society.Google Scholar
Huber, P.J. (1964) Robust estimation of a location parameter. Annals of Mathematical Statistics, 35(1), 73–101.CrossRefGoogle Scholar
Huber, P.J. and Ronchetti, E.M. (2009) Robust Statistics. 2nd edn. Hoboken, NJ: John Wiley & Sons, Inc.CrossRefGoogle Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2019) Loss Models: From Data to Decisions. 5th edn. Hoboken, NJ: John Wiley & Sons.Google Scholar
Michael, S., Miljkovic, T. and Melnykov, V. (2020) Mixture modeling of insurance loss data with multiple level partial censoring. Advances in Data Analysis and Classification, 14, 355–378.CrossRefGoogle Scholar
Miljkovic, T. and Grün, B. (2016) Modeling loss data using mixtures of distributions. Insurance: Mathematics & Economics, 70, 387–396.Google Scholar
Poudyal, C. (2018) Robust estimation of parametric models for insurance loss data. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–The University of Wisconsin - Milwaukee.Google Scholar
Poudyal, C. (2021) Truncated, censored, and actuarial payment-type moments for robust fitting of a single-parameter Pareto distribution. Journal of Computational and Applied Mathematics, 388, 113310, 18.CrossRefGoogle Scholar
Punzo, A., Bagnato, L. and Maruotti, A. (2018) Compound unimodal distributions for insurance losses. Insurance: Mathematics & Economics, 81, 95–107.Google Scholar
Reynkens, T., Verbelen, R., Beirlant, J. and Antonio, K. (2017) Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions. Insurance: Mathematics & Economics, 77, 65–77.Google Scholar
Scollnik, D.P.M. (2007) On composite lognormal-Pareto models. Scandinavian Actuarial Journal, 2007(1), 20–33.CrossRefGoogle Scholar
Scollnik, D.P.M. and Sun, C. (2012) Modeling with Weibull-Pareto models. North American Actuarial Journal, 16(2), 260–272.CrossRefGoogle Scholar
Serfling, R. (2002) Efficient and robust fitting of lognormal distributions. North American Actuarial Journal, 6(4), 95–109.CrossRefGoogle Scholar
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. New York: John Wiley & Sons.CrossRefGoogle Scholar
Shah, S.M. and Jaiswal, M.C. (1966) Estimation of parameters of doubly truncated normal distribution from first four sample moments. Annals of the Institute of Statistical Mathematics, 18, 107–111.CrossRefGoogle Scholar
Tukey, J.W. (1960) A survey of sampling from contaminated distributions. In Contributions to Probability and Statistics, pp. 448–485. Stanford, CA: Stanford University Press.Google Scholar
van der Vaart, A.W. (1998) Asymptotic Statistics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Verbelen, R., Gong, L., Antonio, K., Badescu, A. and Lin, S. (2015) Fitting mixtures of Erlangs to censored and truncated data using the EM algorithm. ASTIN Bulletin, 45(3), 729–758.CrossRefGoogle Scholar
Zhao, Q., Brazauskas, V. and Ghorai, J. (2018) Robust and efficient fitting of severity models and the method of Winsorized moments. ASTIN Bulletin, 48(1), 275–309.CrossRefGoogle Scholar