Weighted allocations, their concomitant-based estimators, and asymptotics

Article
  • 22 Downloads

Abstract

Various members of the class of weighted insurance premiums and risk capital allocation rules have been researched from a number of perspectives. Corresponding formulas in the case of parametric families of distributions have been derived, and they have played a pivotal role when establishing parametric statistical inference in the area. Nonparametric inference results have also been derived in special cases such as the tail conditional expectation, distortion risk measure, and several members of the class of weighted premiums. For weighted allocation rules, however, nonparametric inference results have not yet been adequately developed. In the present paper, therefore, we put forward empirical estimators for the weighted allocation rules and establish their consistency and asymptotic normality under practically sound conditions. Intricate statistical considerations rely on the theory of induced order statistics, known as concomitants.

Keywords

Weighted allocation Insurance premium Concomitant Consistency Asymptotic normality 

Notes

Acknowledgements

We are indebted to two anonymous reviewers for suggestions, insightful comments, and constructive criticism that guided our work on the revision.

References

  1. Asimit, A. V., Vernic, R., Zitikis, R. (2013). Evaluating risk measures and capital allocations based on multi-losses driven by a heavy-tailed background risk: The multivariate Pareto-II model. Risks, 1, 14–33.Google Scholar
  2. Asimit, A. V., Furman, E., Vernic, R. (2016). Statistical inference for a new class of multivariate Pareto distributions. Communications in Statistics: Simulation and Computation, 45, 456–471.Google Scholar
  3. Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J. L. (2004). Statistics of extremes: Theory and applications. Chichester: Wiley.Google Scholar
  4. Bhattacharya, P. K. (1974). Convergence of sample paths of normalized sum of induced order statistics. Annals of Statistics, 2, 1034–1039.MathSciNetCrossRefMATHGoogle Scholar
  5. Brahimi, B., Meddi, F., Necir, A. (2012). Bias-corrected estimation in distortion risk premiums for heavy-tailed losses. Afrika Statistika, 7, 474–490.Google Scholar
  6. Brahimi, B., Meraghni, D., Necir, A., Zitikis, R. (2011). Estimating the distortion parameter of the proportional-hazard premium for heavy-tailed losses. Insurance: Mathematics and Economics, 49, 325–334.Google Scholar
  7. Brazauskas, V. (2009). Robust and efficient fitting of loss models: diagnostic tools and insights. North American Actuarial Journal, 13, 356–369.MathSciNetCrossRefGoogle Scholar
  8. Brazauskas, V., Kleefeld, A. (2009). Robust and efficient fitting of the generalized Pareto distribution with actuarial applications in view. Insurance: Mathematics and Economics, 45, 424–435.Google Scholar
  9. Brazauskas, V., Kleefeld, A. (2016). Modeling severity and measuring tail risk of Norwegian fire claims. North American Actuarial Journal, 20, 1–16.Google Scholar
  10. Brazauskas, V., Jones, B. L., Puri, M. L., Zitikis, R. (2008). Estimating conditional tail expectation with actuarial applications in view. Journal of Statistical Planning and Inference, 138(11), 3590–3604 (special issue in Honor of Junjiro Ogawa: Design of experiments, multivariate analysis and statistical inference).Google Scholar
  11. Brazauskas, V., Serfling, R. (2003). Favourable estimators for fitting Pareto models: A study using goodness-of-fit measures with actual data. ASTIN Bulletin, 33, 365–381.Google Scholar
  12. Castillo, E., Hadi, A. S., Balakrishnan, N., Sarabia, J. M. (2005). Extreme value and related models with applications in engineering and science. Hoboken: Wiley.Google Scholar
  13. Chernoff, H., Gastwirth, J. L., Johns, M. V. (1967). Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation. Annals of Mathematical Statistics, 38, 52–72.Google Scholar
  14. de Haan, L., Ferreira, A. (2006). Extreme value theory: An introduction. New York: Springer.Google Scholar
  15. Embrechts, P., Klüppelberg, C., Mikosch, T. (1997). Modelling extremal events: For insurance and finance. New York: Springer.Google Scholar
  16. Föllmer, H., Schied, A. (2016). Stochastic finance: An introduction in discrete time (4th ed.). Berlin: Walter de Gruyter.Google Scholar
  17. Furman, E., Landsman, Z. (2005). Risk capital decomposition for a multivariate dependent gamma portfolio. Insurance: Mathematics and Economics, 37, 635–649.Google Scholar
  18. Furman, E., Landsman, Z. (2010). Multivariate Tweedie distributions and some related capital-at-risk analyses. Insurance: Mathematics and Economics, 46, 351–361.Google Scholar
  19. Furman, E., Zitikis, R. (2008a). Weighted premium calculation principles. Insurance: Mathematics and Economics, 42, 459–465.Google Scholar
  20. Furman, E., Zitikis, R. (2008b). Weighted risk capital allocations. Insurance: Mathematics and Economics, 43, 263–269.Google Scholar
  21. Furman, E., Zitikis, R. (2017). Beyond the Pearson correlation: Heavy-tailed risks, weighted Gini correlations, and a Gini-type weighted insurance pricing model. ASTIN Bulletin, 47, 919–942.Google Scholar
  22. Gonzalez, R., Wu, G. (1999). On the shape of the probability weighting function. Cognitive Psychology, 38, 129–166.Google Scholar
  23. Greselin, F., Zitikis, R. (2018). From the classical Gini index of income inequality to a new Zenga-type relative measure of risk: A modeller’s perspective. Econometrics, 6, 1–20 (special issue on econometrics and income inequality, with Guest Editors Martin Biewen and Emmanuel Flachaire).Google Scholar
  24. Gribkova, N. V. (2017). Cramér type large deviations for trimmed $L$-statistics. Probability and Mathematical Statistics, 37, 101–118.MathSciNetMATHGoogle Scholar
  25. Gribkova, N. V., Zitikis, R. (2017). Statistical foundations for assessing the difference between the classical and weighted-Gini betas. Mathematical Methods of Statistics, 26, 267–281.Google Scholar
  26. Helmers, R. (1982). Edgeworth expansions for linear combinations of order statistics. Amsterdam: Mathematisch Centrum.MATHGoogle Scholar
  27. Jones, B. L., Zitikis, R. (2003). Empirical estimation of risk measures and related quantities. North American Actuarial Journal, 7, 44–54.Google Scholar
  28. Jones, B. L., Zitikis, R. (2007). Risk measures, distortion parameters, and their empirical estimation. Insurance: Mathematics and Economics, 41, 279–297.Google Scholar
  29. Kamnitui, N., Santiwipanont, T., Sumetkijakan, S. (2015). Dependence measuring from conditional variances. Dependence Modeling, 3, 98–112.Google Scholar
  30. Maesono, Y. (2005). Asymptotic representation of ratio statistics and their mean squared errors. Journal of the Japan Statistical Society, 35, 73–97.MathSciNetCrossRefMATHGoogle Scholar
  31. Maesono, Y. (2010). Edgeworth expansion and normalizing transformation of ratio statistics and their application. Communications in Statistics: Theory and Methods, 39, 1344–1358.MathSciNetCrossRefMATHGoogle Scholar
  32. Maesono, Y., Penev, S. (2013). Improved confidence intervals for quantiles. Annals of the Institute of Statistical Mathematics, 65, 167–189.Google Scholar
  33. McNeil, A. J., Frey, R., Embrechts, P. (2015). Quantitative risk management: Concepts, techniques and tools (Revised ed.). Princeton, NJ: Princeton University Press.Google Scholar
  34. Necir, A., Meraghni, D. (2009). Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts. Insurance: Mathematics and Economics, 45, 49–58.Google Scholar
  35. Necir, A., Meraghni, D., Meddi, F. (2007). Statistical estimate of the proportional hazard premium of loss. Scandinavian Actuarial Journal, 2007, 147–161.Google Scholar
  36. Necir, A., Rassoul, A., Zitikis, R. (2010). Estimating the conditional tail expectation in the case of heady-tailed losses. Journal of Probability and Statistics, 2010, 1–17.Google Scholar
  37. Nešlehová, J., Embrechts, P., Chavez-Demoulin, V. (2006). Infinite-mean models and the LDA for operational risk. Journal of Operational Risk, 1, 3–25.Google Scholar
  38. Pflug, G. C., Römisch, W. (2007). Modeling, measuring and managing risk. Singapore: World Scientific.Google Scholar
  39. Quiggin, J. (1993). Generalized expected utility theory. Dordrecht: Kluwer.CrossRefMATHGoogle Scholar
  40. Rao, C. R., Zhao, L. C. (1995). Convergence theorems for empirical cumulative quantile regression function. Mathematical Methods of Statistics, 4, 81–91.Google Scholar
  41. Rassoul, A. (2013). Kernel-type estimator of the conditional tail expectation for a heavy-tailed distribution. Insurance: Mathematics and Economics, 53, 698–703.MathSciNetMATHGoogle Scholar
  42. Ratovomirija, G., Tamraz, M., Vernic, R. (2017). On some multivariate Sarmanov mixed Erlang reinsurance risks: Aggregation and capital allocation. Insurance: Mathematics and Economics, 74, 197–209.Google Scholar
  43. Rüschendorf, L. (2013). Mathematical risk analysis: Dependence, risk bounds, optimal allocations and portfolios. New York: Springer.CrossRefMATHGoogle Scholar
  44. Serfling, R. J. (1980). Approximation theorems of mathematical statistics. New York: Wiley.CrossRefMATHGoogle Scholar
  45. Shorack, G. R. (1972). Functions of order statistics. Annals of Mathematical Statistics, 43, 412–427.MathSciNetCrossRefMATHGoogle Scholar
  46. Shorack, G. R. (2017). Probability for statisticians (2nd ed.). New York: Springer.CrossRefMATHGoogle Scholar
  47. Stigler, S. M. (1974). Linear functions of order statistics with smooth weight functions. Annals of Statistics, 2, 676–693.MathSciNetCrossRefMATHGoogle Scholar
  48. Su, J. (2016). Multiple risk factors dependence structures with applications to actuarial risk management. Ph.D. thesis, York University, Ontario, Canada.Google Scholar
  49. Su, J., Furman, E. (2017). A form of multivariate Pareto distribution with applications to financial risk measurement. ASTIN Bulletin, 47, 331–357.Google Scholar
  50. Tse, S. M. (2009). On the cumulative quantile regression process. Mathematical Methods of Statistics, 18, 270–279.MathSciNetCrossRefMATHGoogle Scholar
  51. Tse, S. M. (2015). The cumulative quantile regression function with censored and truncated response. Mathematical Methods of Statistics, 24, 147–155.MathSciNetCrossRefMATHGoogle Scholar
  52. Tversky, A., Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5, 297–323.Google Scholar
  53. van Zwet, W. R. (1980). A strong law for linear functions of order statistics. Annals of Probability, 8, 986–990.MathSciNetCrossRefMATHGoogle Scholar
  54. Vernic, R. (2017). Capital allocation for Sarmanov’s class of distributions. Methodology and Computing in Applied Probability, 19, 311–330.MathSciNetCrossRefMATHGoogle Scholar
  55. von Neumann, J., Morgenstern, O. (1944). Theory of games and economic behavior. Princeton, NJ: Princeton University Press.Google Scholar
  56. Wakker, P. P. (2010). Prospect theory: For risk and ambiguity. Cambridge: Cambridge University Press.CrossRefMATHGoogle Scholar
  57. Wang, S. (1995). Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics, 17, 43–54.MathSciNetMATHGoogle Scholar
  58. Wang, S. S. (1996). Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26, 71–92.CrossRefGoogle Scholar
  59. Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica, 55, 95–115.MathSciNetCrossRefMATHGoogle Scholar
  60. Yang, S. S. (1981). Linear combinations of concomitants of order statistics with application to testing and estimation. Annals of the Institute of Statistical Mathematics, 33, 463–470.MathSciNetCrossRefMATHGoogle Scholar
  61. Zitikis, R., Gastwirth, J. L. (2002). Asymptotic distribution of the S-Gini index. Australian and New Zealand Journal of Statistics, 44, 439–446.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2018

Authors and Affiliations

  1. 1.Faculty of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.School of Mathematical and Statistical SciencesWestern UniversityLondonCanada

Personalised recommendations