# Weighted allocations, their concomitant-based estimators, and asymptotics

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## 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

- 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 - 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 - Beirlant, J., Goegebeur, Y., Segers, J., Teugels, J. L. (2004).
*Statistics of extremes: Theory and applications*. Chichester: Wiley.Google Scholar - Bhattacharya, P. K. (1974). Convergence of sample paths of normalized sum of induced order statistics.
*Annals of Statistics*,*2*, 1034–1039.MathSciNetCrossRefzbMATHGoogle Scholar - 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 - 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 - Brazauskas, V. (2009). Robust and efficient fitting of loss models: diagnostic tools and insights.
*North American Actuarial Journal*,*13*, 356–369.MathSciNetCrossRefGoogle Scholar - 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 - Brazauskas, V., Kleefeld, A. (2016). Modeling severity and measuring tail risk of Norwegian fire claims.
*North American Actuarial Journal*,*20*, 1–16.Google Scholar - 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 - 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 - 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 - 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 - de Haan, L., Ferreira, A. (2006).
*Extreme value theory: An introduction*. New York: Springer.Google Scholar - Embrechts, P., Klüppelberg, C., Mikosch, T. (1997).
*Modelling extremal events: For insurance and finance*. New York: Springer.Google Scholar - Föllmer, H., Schied, A. (2016).
*Stochastic finance: An introduction in discrete time*(4th ed.). Berlin: Walter de Gruyter.Google Scholar - Furman, E., Landsman, Z. (2005). Risk capital decomposition for a multivariate dependent gamma portfolio.
*Insurance: Mathematics and Economics*,*37*, 635–649.Google Scholar - Furman, E., Landsman, Z. (2010). Multivariate Tweedie distributions and some related capital-at-risk analyses.
*Insurance: Mathematics and Economics*,*46*, 351–361.Google Scholar - Furman, E., Zitikis, R. (2008a). Weighted premium calculation principles.
*Insurance: Mathematics and Economics*,*42*, 459–465.Google Scholar - Furman, E., Zitikis, R. (2008b). Weighted risk capital allocations.
*Insurance: Mathematics and Economics*,*43*, 263–269.Google Scholar - 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 - Gonzalez, R., Wu, G. (1999). On the shape of the probability weighting function.
*Cognitive Psychology*,*38*, 129–166.Google Scholar - 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 - Gribkova, N. V. (2017). Cramér type large deviations for trimmed $L$-statistics.
*Probability and Mathematical Statistics*,*37*, 101–118.MathSciNetzbMATHGoogle Scholar - 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 - Helmers, R. (1982).
*Edgeworth expansions for linear combinations of order statistics*. Amsterdam: Mathematisch Centrum.zbMATHGoogle Scholar - Jones, B. L., Zitikis, R. (2003). Empirical estimation of risk measures and related quantities.
*North American Actuarial Journal*,*7*, 44–54.Google Scholar - Jones, B. L., Zitikis, R. (2007). Risk measures, distortion parameters, and their empirical estimation.
*Insurance: Mathematics and Economics*,*41*, 279–297.Google Scholar - Kamnitui, N., Santiwipanont, T., Sumetkijakan, S. (2015). Dependence measuring from conditional variances.
*Dependence Modeling*,*3*, 98–112.Google Scholar - Maesono, Y. (2005). Asymptotic representation of ratio statistics and their mean squared errors.
*Journal of the Japan Statistical Society*,*35*, 73–97.MathSciNetCrossRefzbMATHGoogle Scholar - Maesono, Y. (2010). Edgeworth expansion and normalizing transformation of ratio statistics and their application.
*Communications in Statistics: Theory and Methods*,*39*, 1344–1358.MathSciNetCrossRefzbMATHGoogle Scholar - Maesono, Y., Penev, S. (2013). Improved confidence intervals for quantiles.
*Annals of the Institute of Statistical Mathematics*,*65*, 167–189.Google Scholar - McNeil, A. J., Frey, R., Embrechts, P. (2015).
*Quantitative risk management: Concepts, techniques and tools*(Revised ed.). Princeton, NJ: Princeton University Press.Google Scholar - 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 - Necir, A., Meraghni, D., Meddi, F. (2007). Statistical estimate of the proportional hazard premium of loss.
*Scandinavian Actuarial Journal*,*2007*, 147–161.Google Scholar - 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 - 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 - Pflug, G. C., Römisch, W. (2007).
*Modeling, measuring and managing risk*. Singapore: World Scientific.Google Scholar - Quiggin, J. (1993).
*Generalized expected utility theory*. Dordrecht: Kluwer.CrossRefzbMATHGoogle Scholar - Rao, C. R., Zhao, L. C. (1995). Convergence theorems for empirical cumulative quantile regression function.
*Mathematical Methods of Statistics*,*4*, 81–91.Google Scholar - Rassoul, A. (2013). Kernel-type estimator of the conditional tail expectation for a heavy-tailed distribution.
*Insurance: Mathematics and Economics*,*53*, 698–703.MathSciNetzbMATHGoogle Scholar - 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 - Rüschendorf, L. (2013).
*Mathematical risk analysis: Dependence, risk bounds, optimal allocations and portfolios*. New York: Springer.CrossRefzbMATHGoogle Scholar - Serfling, R. J. (1980).
*Approximation theorems of mathematical statistics*. New York: Wiley.CrossRefzbMATHGoogle Scholar - Shorack, G. R. (1972). Functions of order statistics.
*Annals of Mathematical Statistics*,*43*, 412–427.MathSciNetCrossRefzbMATHGoogle Scholar - Shorack, G. R. (2017).
*Probability for statisticians*(2nd ed.). New York: Springer.CrossRefzbMATHGoogle Scholar - Stigler, S. M. (1974). Linear functions of order statistics with smooth weight functions.
*Annals of Statistics*,*2*, 676–693.MathSciNetCrossRefzbMATHGoogle Scholar - Su, J. (2016).
*Multiple risk factors dependence structures with applications to actuarial risk management*. Ph.D. thesis, York University, Ontario, Canada.Google Scholar - Su, J., Furman, E. (2017). A form of multivariate Pareto distribution with applications to financial risk measurement.
*ASTIN Bulletin*,*47*, 331–357.Google Scholar - Tse, S. M. (2009). On the cumulative quantile regression process.
*Mathematical Methods of Statistics*,*18*, 270–279.MathSciNetCrossRefzbMATHGoogle Scholar - Tse, S. M. (2015). The cumulative quantile regression function with censored and truncated response.
*Mathematical Methods of Statistics*,*24*, 147–155.MathSciNetCrossRefzbMATHGoogle Scholar - Tversky, A., Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty.
*Journal of Risk and Uncertainty*,*5*, 297–323.Google Scholar - van Zwet, W. R. (1980). A strong law for linear functions of order statistics.
*Annals of Probability*,*8*, 986–990.MathSciNetCrossRefzbMATHGoogle Scholar - Vernic, R. (2017). Capital allocation for Sarmanov’s class of distributions.
*Methodology and Computing in Applied Probability*,*19*, 311–330.MathSciNetCrossRefzbMATHGoogle Scholar - von Neumann, J., Morgenstern, O. (1944).
*Theory of games and economic behavior*. Princeton, NJ: Princeton University Press.Google Scholar - Wakker, P. P. (2010).
*Prospect theory: For risk and ambiguity*. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar - Wang, S. (1995). Insurance pricing and increased limits ratemaking by proportional hazards transforms.
*Insurance: Mathematics and Economics*,*17*, 43–54.MathSciNetzbMATHGoogle Scholar - Wang, S. S. (1996). Premium calculation by transforming the layer premium density.
*ASTIN Bulletin*,*26*, 71–92.CrossRefGoogle Scholar - Yaari, M. E. (1987). The dual theory of choice under risk.
*Econometrica*,*55*, 95–115.MathSciNetCrossRefzbMATHGoogle Scholar - 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.MathSciNetCrossRefzbMATHGoogle Scholar - 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