Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 23, 2020

Estimating the distribution of a stochastic sum of IID random variables

Viktor Witkovský EMAIL logo , Gejza Wimmer and Tomas Duby
From the journal Mathematica Slovaca

Abstract

Suggested is a non-parametric method and algorithm for estimating the probability distribution of a stochastic sum of independent identically distributed continuous random variables, based on combining and numerically inverting the associated empirical characteristic function (CF) derived from the observed data. This is motivated by classical problems in financial risk management, actuarial science, and hydrological modelling. This approach can be naturally generalized to more complex semi-parametric modelling and estimating approaches, e.g., by incorporating the generalized Pareto distribution fit for modelling heavy tails of the considered continuous random variables, or by considering the weighted mixture of the parametric CFs (used to incorporate the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the observed or historical data). The suggested numerical approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the associated CF and the trapezoidal quadrature rule used for the required numerical integration. The presented non-parametric estimation method is related to the bootstrap estimation approach, and thus, it shares similar properties. Applicability of the proposed estimation procedure is illustrated by estimating the aggregate loss distribution from the well-known Danish fire losses data.


The work was supported by the Slovak Research and Development Agency, project APVV-15-0295, and by the Scientific Grant Agency VEGA of the Ministry of Education of the Slovak Republic and the Slovak Academy of Sciences, by the projects VEGA 2/0054/18 and VEGA 2/0081/19.


  1. Communicated by Anatolij Dvurečenskij

References

[1] Abate, J.—Whitt, W.: The Fourier-series method for inverting transforms of probability distributions, Queueing Syst. 10 (1992), 5–87.10.1007/BF01158520Search in Google Scholar

[2] Ambagaspitiya, R. S.: On the distribution of a sum of correlated aggregate claims, Insurance Math. Econom. 23 (1998), 15–19.10.1016/S0167-6687(98)00018-3Search in Google Scholar

[3] Asheim, A.—Huybrechs, D.: Complex Gaussian quadrature for oscillatory integral transforms, IMA J. Numer. Anal. 33 (2013), 1322–1341.10.1093/imanum/drs060Search in Google Scholar

[4] Bailey, D. H.—Swarztrauber, P. N.: The fractional Fourier transform and applications, SIAM Review 33 (1991), 389–404.10.1137/1033097Search in Google Scholar

[5] Carr, P.—Madan, D.: Option valuation using the fast Fourier transform, J. Comput. Finance 2 (1999), 61–73.10.21314/JCF.1999.043Search in Google Scholar

[6] Chourdakis, K.: Option pricing using the fractional FFT, J. Comput. Finance 8 (2004), 1–18.10.21314/JCF.2005.137Search in Google Scholar

[7] Cohen, H.—Villegas, F. R.—Zagier, D.: Convergence acceleration of alternating series, Exp. Math. 9 (2000), 3–12.10.1080/10586458.2000.10504632Search in Google Scholar

[8] Davies, R. B.: Algorithm AS 155: The distribution of a linear combinations of χ2random variables, Appl. Stat. 29 (1980), 232–333.10.2307/2346911Search in Google Scholar

[9] Davison, A. C.—Hinkley, D. V.: Bootstrap Methods and Their Application, Cambridge University Press, 1997.10.1017/CBO9780511802843Search in Google Scholar

[10] Deaño, A.—Huybrechs, D.—Iserles, A.: Computing Highly Oscillatory Integrals, SIAM (2017).10.1137/1.9781611975123Search in Google Scholar

[11] Efron, B.: Bootstrap methods: Another look at the jackknife, Ann. Statist. 7 (1979), 1–26.10.1007/978-1-4612-4380-9_41Search in Google Scholar

[12] Efron, B.—Tibshirani, R. J.: An Introduction to the Bootstrap, Boca Raton, Chapman & Hall/CRC, 1993.10.1007/978-1-4899-4541-9Search in Google Scholar

[13] Eling, M.: Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models?, Insurance Math. Econom. 51 (2012), 239–248.10.1016/j.insmatheco.2012.04.001Search in Google Scholar

[14] Embrechts, P.—Klüppelberg, C.—Mikosch, T.: Modelling Extremal Events: For Insurance and Finance, Springer Science & Business Media, 2013.Search in Google Scholar

[15] Epps, T. W.: Characteristic functions and their empirical counterparts: Geometrical interpretations and applications to statistical inference, Amer. Statist. 47 (1993), 33–38.Search in Google Scholar

[16] Feng, L.—Lin, X.: Inverting analytic characteristic functions and financial applications, SIAM J. Financial Math. 4 (2013), 372–398.10.1137/110830319Search in Google Scholar

[17] Gil-Pelaez, J.: Note on the inversion theorem, Biometrika 38 (1951), 481–482.10.1093/biomet/38.3-4.481Search in Google Scholar

[18] Hogg, R. V.—Klugman, S. A.: Loss Distributions, John Wiley & Sons, 1984.10.1002/9780470316634Search in Google Scholar

[19] Hürlimann, W.: Improved FFT approximations of probability functions based on modified quadrature rules, Int. Math. Forum 8 (2013), 829–840.10.12988/imf.2013.13087Search in Google Scholar

[20] Imhof, J. P.:. Computing the distribution of quadratic forms in normal variables, Biometrika 48 (1961), 419–426.10.1093/biomet/48.3-4.419Search in Google Scholar

[21] Kaas, R.—Goovaerts, M.—Dhaene, J.—Denuit, M.: Modern Actuarial Risk Theory: Using R, Springer Science & Business Media, 2008.10.1007/978-3-540-70998-5Search in Google Scholar

[22] Kim, Y. S.—Rachev, S.—Bianchi, M. L.—Fabozzi, F. J.: Computing VaR and AVaR in infinitely divisible distributions, Probab. Math. Statist. 30 (2010), 223–245.10.2139/ssrn.1400965Search in Google Scholar

[23] Levin, D.: Fast integration of rapidly oscillatory functions, J. Comput. Appl. Math. 67 (1996), 95–101.10.1016/0377-0427(94)00118-9Search in Google Scholar

[24] Lukacs, E. Characteristics Functions, London: Griffin, 1970.Search in Google Scholar

[25] McNeil, A. J.: Estimating the tails of loss severity distributions using extreme value theory, ASTIN Bull. 27 (1997), 117–137.10.2143/AST.27.1.563210Search in Google Scholar

[26] McNeil, A. J.—Saladin, T.: The peaks over thresholds method for estimating high quantiles of loss distributions. In: Proceedings of 28th International ASTIN Colloquium, 1997, pp. 23–43.Search in Google Scholar

[27] Mikosch, T.—Nagaev, A. Rates in approximations to ruin probabilities for heavy-tailed distributions, Extremes 4 (2001), 67–78.10.1023/A:1012237524316Search in Google Scholar

[28] Milovanović, G. V.: Numerical calculation of integrals involving oscillatory and singular kernels and some applications of quadratures, Comput. Math. Appl. 36 (1998), 19–39.10.1016/S0898-1221(98)00180-1Search in Google Scholar

[29] Petrov, V. V. Sums of Independent Random Variables, Springer Science & Business Media, volume 82, 2012.Search in Google Scholar

[30] Potocký, R.—Waldl, H.—Stehlík, M. On sums of claims and their applications in analysis of pension funds and insurance products, Prague Economic Papers 23 (2014).10.18267/j.pep.488Search in Google Scholar

[31] Resnick, S. I.: Discussion of the Danish data on large fire insurance losses, Astin Bull. 27 (1997), 139–151.10.2143/AST.27.1.563211Search in Google Scholar

[32] Rolski, T.—Schmidli, H.—Schmidt, V.—Teugels, J.: Stochastic Processes for Insurance and Finance, John Wiley & Sons (2009).Search in Google Scholar

[33] Roncalli, T.: Lecture Notes on Risk Management & Financial Regulation, 2016.10.2139/ssrn.2776813Search in Google Scholar

[34] Schmidli, H.: Accumulated Claims. In: Encyclopedia of Quantitative Finance, Wiley and Sons, Chichester, 2010.10.1002/9780470061602.eqf21013Search in Google Scholar

[35] Shephard, N. G.: From characteristic function to distribution function: A simple framework for the theory, Econometric Theory 7 (1991), 519–529.10.1017/S0266466600004746Search in Google Scholar

[36] Shevchenko, P. V. Calculation of aggregate loss distributions, J. Operational Risk 5 (2010), 3–40.10.21314/JOP.2010.077Search in Google Scholar

[37] Sidi, A.: The numerical evaluation of very oscillatory infinite integrals by extrapolation, Math. Comp. 38 (1982), 517–529.10.1090/S0025-5718-1982-0645667-5Search in Google Scholar

[38] Sidi, A.: A user-friendly extrapolation method for oscillatory infinite integrals, Math. Comp. 51 (1988), 249–266.10.1090/S0025-5718-1988-0942153-5Search in Google Scholar

[39] Sidi, A.: A user-friendly extrapolation method for computing infinite range integrals of products of oscillatory functions, IMA J. Numer. Anal. 32 (2012), 602–631.10.1093/imanum/drr022Search in Google Scholar

[40] Strawderman, R. L.: Computing tail probabilities by numerical Fourier inversion: The absolutely continuous case, Statist. Sinica 14 (2004), 175–201.Search in Google Scholar

[41] Suhaila, J.—Ching-Yee, K.—Fadhilah, Y.—Hui-Mean, F.: Introducing the mixed distribution in fitting rainfall data, Open Journal of Modern Hydrology 1 (2011), 11.10.4236/ojmh.2011.12002Search in Google Scholar

[42] Thompson, V.—Dunstone, N. J.—Scaife, A. A.—Smith, D. M.—Slingo, J. M.—Brown, S.—Belcher, S. E.: High risk of unprecedented UK rainfall in the current climate, Nature Communications 8 (2017), 107.10.1038/s41467-017-00275-3Search in Google Scholar

[43] Waller, L. A.—Turnbull, B. W.—Hardin, J. M.: Obtaining distribution functions by numerical inversion of characteristic functions with applications, Amer. Statist. 49 (1995), 346–350.Search in Google Scholar

[44] Witkovský, V.: Computing the distribution of a linear combination of inverted gamma variables, Kybernetika 37 (2001), 79–90.Search in Google Scholar

[45] Witkovský, V.: On the exact computation of the density and of the quantiles of linear combinations of t and F random variables, J. Statist. Plann. Inference 94 (2001), 1–13.10.1016/S0378-3758(00)00208-1Search in Google Scholar

[46] Witkovský, V.: CharFunTool: The Characteristic Functions Toolbox (MATLAB), 2019, https://github.com/witkovsky/CharFunTool.Search in Google Scholar

[47] Witkovský, V.: Numerical inversion of a characteristic function: An alternative tool to form the probability distribution of output quantity in linear measurement models, Acta IMEKO 5 (2016), 1–13.10.21014/acta_imeko.v5i3.382Search in Google Scholar

[48] Witkovský, V.—Wimmer, G.—Duby, T.: Logarithmic Lambert W × 𝓕 random variables for the family of chi-squared distributions and their applications, Statist. Probab. Lett. 96 (2015), 223–231.10.1016/j.spl.2014.09.028Search in Google Scholar

[49] Witkovský, V.: Computing the exact distribution of the Bartlett’s test statistic by numerical inversion of its characteristic function, J. Appl. Stat. (2019); https://doi.org/10.1080/02664763.2019.1675608.10.1080/02664763.2019.1675608Search in Google Scholar

[50] Zieliński, R.: High-accuracy evaluation of the cumulative distribution function of α-stable symmetric distributions, J. Math. Sci. 105 (2001), 2630–2632.10.1023/A:1011327706906Search in Google Scholar

Received: 2019-06-06
Accepted: 2019-11-15
Published Online: 2020-05-23
Published in Print: 2020-06-25

© 2020 Mathematical Institute Slovak Academy of Sciences

Downloaded on 29.1.2023 from https://www.degruyter.com/document/doi/10.1515/ms-2017-0389/html
Scroll Up Arrow