Skip to main content
SpringerLink
Log in

Empirical tail conditional allocation and its consistency under minimal assumptions

  • Published:
Annals of the Institute of Statistical Mathematics Aims and scope Submit manuscript

This article has been updated

Abstract

Under minimal assumptions, we prove that an empirical estimator of the tail conditional allocation (TCA), also known as the marginal expected shortfall, is consistent. Examples are provided to confirm the minimality of the assumptions. A simulation study illustrates the performance of the estimator in the context of developing confidence intervals for the TCA. The philosophy adopted in the present paper relies on three principles: easiness of practical use, mathematical rigor, and practical justifiability and verifiability of assumptions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Change history

  • 12 January 2022

    The original version of this article was revised to update Reference Citation corrections

References

  • Ahsanullah, M., Nevzorov, V. B., Shakil, M. (2013). An introduction to order statistics. Paris: Atlantis Press.

    Book  Google Scholar 

  • Arnold, B. C. (2015). Pareto distributions, 2nd ed. Boca Raton: Chapman and Hall/CRC.

    Book  Google Scholar 

  • Arnold, B. C., Balakrishnan, N., Nagaraja, H. N. (2008). A first course in order statistics. Philadelphia: Society for Industrial and Applied Mathematics.

    Book  Google Scholar 

  • Arnold, B. C., Castillo, E., Sarabia, J. M. (1999). Conditional specification of statistical models. New York: Springer.

    MATH  Google Scholar 

  • Asimit, A. V., Vernic, R., Zitikis, R. (2016). Background risk models and stepwise portfolio construction. Methodology and Computing in Applied Probability, 18, 805–827.

    Article  MathSciNet  Google Scholar 

  • Bauer, D., Zanjani, G. (2016). The marginal cost of risk, risk measures, and capital allocation. Management Science, 62, 1431–1457.

    Article  Google Scholar 

  • BCBS (2016). Minimum capital requirements for market risk. (2016). Basel committee on banking supervision. Bank for international settlements, Basel. https://www.bis.org/bcbs/publ/d352.htm.

  • BCBS (2019). Minimum capital requirements for market risk. (2019). Basel committee on banking supervision. Bank for international settlements, Basel. https://www.bis.org/bcbs/publ/d457.htm.

  • Beirlant, J., Goegebeur, Y., Teugels, J., Segers, J. (2004). Statistics of extremes: Theory and applications. Chichester: Wiley.

    Book  Google Scholar 

  • Bickel, P. J., Sakov, A. (2008). On the choice of \(m\) in the \(m\) out of \(n\) bootstrap and confidence bounds for extrema. Statistica Sinica, 18, 967–985.

    MathSciNet  MATH  Google Scholar 

  • Bickel, P. J., Götze, F., van Zwet, W. R. (1997). Resampling fewer than \(n\) observations: Gains, losses, and remedies for losses. Statistica Sinica, 7, 1–31.

    MathSciNet  MATH  Google Scholar 

  • Cai, J.-J., Musta, E. (2020). Estimation of the marginal expected shortfall under asymptotic independence. Scandinavian Journal of Statistics, 47, 56–83.

    Article  MathSciNet  Google Scholar 

  • Cai, J.-J., Einmahl, J. H. J., de Haan, L., Zhou, C. (2015). Estimation of the marginal expected shortfall: the mean when a related variable is extreme. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77, 417–442.

    Article  MathSciNet  Google Scholar 

  • Carcea, M., Serfling, R. (2015). A Gini autocovariance function for time series modelling. Journal of Time Series Analysis, 36, 817–838.

    Article  MathSciNet  Google Scholar 

  • Castillo, E., Hadi, A. S., Balakrishnan, N., Sarabia, J. M. (2004). Extreme value and related models with applications in engineering and science. Chichester: Wiley.

    MATH  Google Scholar 

  • DasGupta, A. (2008). Asymptotic theory of statistics and probability. New York: Springer.

    MATH  Google Scholar 

  • David, H. A., Nagaraja, H. N. (2003). Order statistics, 3rd ed. Hoboken: Wiley.

    Book  Google Scholar 

  • Davison, A. C., Hinkley, D. V. (1997). Bootstrap methods and their application. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • de Haan, L., Ferreira, A. (2006). Extreme value theory: An introduction. New York: Springer.

    Book  Google Scholar 

  • Dudkina, O. I., Gribkova, N. V. (2020). On the strong law of large numbers for linear combinations of concomitants. Vestnik St. Petersburg University, Mathematics, 53, 282–286.

    Article  MathSciNet  Google Scholar 

  • Efron, B., Tibshirani, R. J. (1993). An introduction to the bootstrap. Boca Raton: Chapman and Hall/CRC.

    Book  Google Scholar 

  • Embrechts, P., Mao, T., Wang, Q., Wang, R. (2021). Bayes risk, elicitability, and the expected shortfall. Mathematical Finance (in press). https://onlinelibrary.wiley.com/doi/abs/10.1111/mafi.12313.

  • Frees, E. W., Valdez, E. A. (1998). Understanding relationships using copulas. North American Actuarial Journal, 2, 1–25.

    Article  MathSciNet  Google Scholar 

  • Furman, E., Kye, Y., Su, J. (2021a). A reconciliation of the top-down and bottom-up approaches to risk capital allocations: Proportional allocations revisited. North American Actuarial Journal, 25, 395–416.

    Article  MathSciNet  Google Scholar 

  • Furman, E., Kye, Y., Su, J. (2021b). Multiplicative background risk models: Setting a course for the idiosyncratic risk factors distributed phase-type. Insurance: Mathematics and Economics, 96, 153–167.

  • Furman, E., Zitikis, R. (2008). Weighted risk capital allocations. Insurance: Mathematics and Economics, 43, 263–269.

  • Gribkova, N. V., Helmers, R. (2007). On the Edgeworth expansion and the \(M\) out of \(N\) bootstrap accuracy for a Studentized trimmed mean. Mathematical Methods of Statistics, 16, 142–176.

    Article  MathSciNet  Google Scholar 

  • Gribkova, N. V., Helmers, R. (2011). On the consistency of the \(M \ll N\) bootstrap approximation for a trimmed mean. Theory of Probability and Its Applications, 55, 42–53.

    Article  MathSciNet  Google Scholar 

  • Gribkova, N., Zitikis, R. (2017). Statistical foundations for assessing the difference between the classical and weighted-Gini betas. Mathematical Methods of Statistics, 26, 267–281.

    Article  MathSciNet  Google Scholar 

  • Gribkova, N., Zitikis, R. (2019). Weighted allocations, their concomitant-based estimators, and asymptotics. Annals of the Institute of Statistical Mathematics, 71, 811–835.

    Article  MathSciNet  Google Scholar 

  • Guo, Q., Bauer, D., Zanjani, G. (2021). Capital allocation techniques: Review and comparison. Variance (in press).

  • Hall, P. (1992). The bootstrap and Edgeworth expansion. New York: Springer.

    Book  Google Scholar 

  • Insurance Information Institute (2021). What Is Covered by Standard Homeowners Insurance? Accessed online on September 17, 2021, at the address https://www.iii.org/article/what-covered-standard-homeowners-policy.

  • Kattumannil, S. K., Sreelakshmi, N., Balakrishnan, N. (2020). Non-parametric inference for Gini covariance and its variants. Sankhyā A . https://doi.org/10.1007/s13171-020-00218-z.

  • Kulik, R., Tong, Z. (2019). Estimation of the expected shortfall given an extreme component under conditional extreme value model. Extremes, 22, 29–70.

    Article  MathSciNet  Google Scholar 

  • Laidi, M., Rassoul, A., Ould Rouis, H. (2020). Improved estimator of the conditional tail expectation in the case of heavy-tailed losses. Statistics, Optimization and Information Computing, 8, 98–109.

    Article  MathSciNet  Google Scholar 

  • Mardia, K. V. (1962). Multivariate Pareto distributions. Annals of Mathematical Statistics, 33, 1008–1015.

    Article  MathSciNet  Google Scholar 

  • Necir, A., Rassoul, A., Zitikis, R. (2010). Estimating the conditional tail expectation in the case of heavy-tailed losses. Journal of Probability and Statistics, 596839, 1–17 (Special issue on “Actuarial and Financial Risks: Models, Statistical Inference, and Case Studies”).

    MATH  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.

    Article  Google Scholar 

  • Pedersen, H., Campbell, M. P., Christiansen, S. L., Cox, S. H., Finn, D., Griffin, K., et al. (2016). Economic scenario generators: A practical guide. Schaumburg: The Society of Actuaries.

    Google Scholar 

  • Schechtman, E., Shelef, A., Yitzhaki, S., Zitikis, R. (2008). Testing hypotheses about absolute concentration curves and marginal conditional stochastic dominance. Econometric Theory, 24, 1044–1062.

    Article  Google Scholar 

  • Shalit, H., Yitzhaki, S. (1994). Marginal conditional stochastic dominance. Management Science, 40, 549–684.

    Article  Google Scholar 

  • Shao, J., Tu, D. (1995). The jackknife and bootstrap. New York: Springer.

    Book  Google Scholar 

  • Shelef, A. (2013). Statistical analyses based on Gini for time series data. PhD Dissertation, Ben-Gurion University of the Negev, Beer Sheva.

  • Shelef, A. (2016). A Gini-based unit root test. Computational Statistics and Data Analysis, 100, 763–772.

    Article  MathSciNet  Google Scholar 

  • Shelef, A., Schechtman, E. (2019). A Gini-based time series analysis and test for reversibility. Statistical Papers, 60, 687–716.

    Article  MathSciNet  Google Scholar 

  • Tasche, D. (2004). Allocating portfolio economic capital to sub-portfolios. In A. Dev (Ed.), Economic capital: A practitioner guide (pp. 275–302). London: Risk Books.

    Google Scholar 

  • Wang, R., Zitikis, R. (2020). An axiomatic foundation for the expected shortfall. Management Science, 67, 1413–1429.

    Article  Google Scholar 

  • Wang, Q., Wang, R. Zitikis, R. (2021). Risk Measures Induced by Efficient Insurance Contracts. Available at arXiv:org/abs/2109.00314.

  • Yitzhaki, S., Schechtman, E. (2013). The Gini methodology. New York: Springer.

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, Saint Petersburg State University, 7/9 Universitetskaya Embankment, Saint Petersburg, 199034, Russia

    N. V. Gribkova

  2. Department of Mathematics, Emperor Alexander I St. Petersburg State Transport University, 9 Moskovsky Prospekt, Saint Petersburg, 199034, Russia

    N. V. Gribkova

  3. Department of Statistics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907, USA

    J. Su

  4. Risk and Insurance Studies Centre, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada

    J. Su & R. Zitikis

  5. School of Mathematical and Statistical Sciences, Western University, 1151 Richmond Street North, London, ON, N6A 5B7, Canada

    R. Zitikis

Authors
  1. N. V. Gribkova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Su
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Zitikis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Su.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We are indebted to Chief Editor Hironori Fujisawa, an Associate Editor, and two referees for constructive criticism and generous suggestions. This research has been supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the national research organization Mathematics of Information Technology and Complex Systems (MITACS) of Canada.

The original version of this article was revised to update Reference Citation corrections.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gribkova, N.V., Su, J. & Zitikis, R. Empirical tail conditional allocation and its consistency under minimal assumptions. Ann Inst Stat Math 74, 713–735 (2022). https://doi.org/10.1007/s10463-021-00813-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10463-021-00813-3

Keywords

Search

Navigation