Skip to main content Accessibility help

Comparing the riskiness of dependent portfolios via nested L-statistics

  • Ranadeera G.M. Samanthi (a1), Wei Wei (a2) and Vytaras Brazauskas (a3)


A non-parametric test based on nested L-statistics and designed to compare the riskiness of portfolios was introduced by Brazauskas et al. (2007). Its asymptotic and small-sample properties were primarily explored for independent portfolios, though independence is not a required condition for the test to work. In this paper, we investigate how performance of the test changes when insurance portfolios are dependent. To achieve that goal, we perform a simulation study where we consider three different risk measures: conditional tail expectation, proportional hazards transform, and mean. Further, three portfolios are generated from exponential, Pareto, and lognormal distributions, and their interdependence is modelled with the three-dimensional t and Gaussian copulas. It is found that the presence of strong positive dependence (comonotonicity) makes the test very liberal for all the risk measures under consideration. For types of dependence that are more common in an insurance environment, the effect of dependence is less dramatic but the results are mixed, i.e., they depend on the chosen risk measure, sample size, and even on the test’s significance level. Finally, we illustrate how to incorporate such findings into sensitivity analysis of the decisions. The risks we analyse represent tornado damages in different regions of the United States from 1890 to 1999.


Corresponding author

*Correspondence to: Vytaras Brazauskas, Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA. Tel: 1-414-229-5656; Fax: 1-414-229-4907; E-mail:


Hide All
Albrecht, P. (2004). Risk measures. In B. Sundt & J. Teugels (Eds.), Encyclopedia of Actuarial Science, vol. 3 (pp. 14931501). Wiley, London.
Brazauskas, V., Jones, B.L., Puri, M.L. & Zitikis, R. (2007). Nested L-statistics and their use in comparing the riskiness of portfolios. Scandinavian Actuarial Journal, 2007(3), 162179.
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), 35903604.
Brazauskas, V. & Kaiser, T. (2004). Discussion of “Empirical estimation of risk measures and related quantities” by Jones and Zitikis. North American Actuarial Journal, 8(3), 114117.
Brooks, H.E. & Doswell, C.A. III (2001). Normalized damage from major tornadoes in the United States: 1890–1999. Weather and Forecasting, 16, 168176.
Chernoff, H., Gastwirth, J.L. & Jones, M.V. (1967). Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation. The Annals of Mathematical Statistics, 38(1), 5272.
Darolles, S., Gourieroux, C. & Jasiak, J. (2009). L-performance with an application to hedge funds. Journal of Empirical Finance, 16(4), 671685.
David, H.A. & Nagaraja, H.N. (2003). Order Statistics, 3rd edition. Wiley, Hoboken, NJ.
Embrechts, P., Lindskog, F. & McNeil, A. (2003). Modelling dependence with copulas and applications to risk management. In S. Rachev (Ed.), Handbook of Heavy Tailed Distributions in Finance (pp. 329384). Elsevier, Amsterdam.
Frees, E.W. & Valdez, E. (1998). Understanding relationships using copulas. North American Actuarial Journal, 2(1), 125.
Hosking, J.R.M. (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society: Series B, 52, 105124.
Joe, H. (2014). Dependence Modeling with Copulas. Chapman and Hall, Boca Raton, FL.
Jones, B.L., Puri, M.L. & Zitikis, R. (2006). Testing hypotheses about the equality of several risk measure values with applications in insurance. Insurance: Mathematics and Economics, 38, 253270.
Jones, B.L. & Zitikis, R. (2003). Empirical estimation of risk measures and related quantities. North American Actuarial Journal, 7(4), 4454.
Jones, B.L. & Zitikis, R. (2005). Testing for the order of risk measures: an application of L-statistics in actuarial science. Metron, LXIII(2), 193211.
Jones, B.L. & Zitikis, R. (2007). Risk measures, distortion parameters, and their empirical estimation. Insurance: Mathematics and Economics, 41, 279297.
Kaiser, T. & Brazauskas, V. (2006). Interval estimation of actuarial risk measures. North American Actuarial Journal, 10(4), 249268.
McNeil, A.J., Frey, R. & Embrechts, P. (2005). Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton, NJ.
Necir, A. & Meraghni, D. (2009). Empirical estimation of the proportional hazard premium for heavy-tailed claim amounts. Insurance: Mathematics and Economics, 45(1), 4958.
Necir, A. & Meraghni, D. (2010). Estimating L-functionals for heavy-tailed distributions and application. Journal of Probability and Statistics, 2010, article id 707146, 134.
Necir, A., Meraghni, D. & Meddi, F. (2007). Statistical estimate of the proportional hazard premium of loss. Scandinavian Actuarial Journal, 2007(3), 147161.
Nelson, R.B. (2006). An Introduction to Copulas, 2nd edition. Springer, New York, NY.
Samanthi, R.G.M., Wei, W. & Brazauskas, V. (2016). Ordering Gini indexes of multivariate elliptical risks. Insurance: Mathematics and Economics, 68(3), 8491.
Tapiero, C.S. (2004). Risk management: an interdisciplinary framework. In B. Sundt & J. Teugels (Eds.), Encyclopedia of Actuarial Science, vol. 3 (pp. 14831493). Wiley, London.
Young, V.R. (2004). Premium principles. In B. Sundt & J. Teugels (Eds.), Encyclopedia of Actuarial Science, vol. 3 (pp. 13221331). Wiley, London.


Comparing the riskiness of dependent portfolios via nested L-statistics

  • Ranadeera G.M. Samanthi (a1), Wei Wei (a2) and Vytaras Brazauskas (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed