Gram-Charlier-like expansions of the convoluted hyperbolic-secant density. (English) Zbl 1437.62338
Summary: Since financial series are usually heavy tailed and skewed, research has formerly considered well-known leptokurtic distributions to model these series and, recently, has focused on the technique of adjusting the moments of a probability law by using its orthogonal polynomials. This paper combines these approaches by modifying the moments of the convoluted hyperbolic secant. The resulting density is a Gram-Charlier-like (GC-like) expansion capable to account for skewness and excess kurtosis. Multivariate extensions of these expansions are obtained on an argument using spherical distributions. Both the univariate and multivariate (GC-like) expansions prove to be effective in modeling heavy-tailed series and computing risk measures.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 91B84 | Economic time series analysis |
| 62H15 | Hypothesis testing in multivariate analysis |
| 62G32 | Statistics of extreme values; tail inference |
Keywords:
convoluted hyperbolic-secant distribution; orthogonal polynomials; kurtosis; skewness; Gram-Charlier-like expansionReferences:
| [1] | Adcock, Cj; Shutes, K., On the multivariate extended skew-normal, normal-exponential, and normal-gamma distributions, J Stat Theory Pract, 6-4, 636-664 (2012) · Zbl 1425.62027 · doi:10.1080/15598608.2012.719799 |
| [2] | Barakat, Mh, A new method for adding two parameters to a family of distributions with applications to the normal and exponential families, Stat Methods Appl, 24, 3, 359-372 (2015) · Zbl 1327.62076 · doi:10.1007/s10260-014-0265-8 |
| [3] | Barndorff-Nielsen, O.; Blæsild, P.; Johnson, Nl; Kotz, S.; Read, Cb, Hyperbolic distributions, Encyclopedia of statistical sciences (1983), New York: Wiley, New York |
| [4] | Baten, Wd, The probability law for the sum of n independent variables, each subject to the law ((1/2h)sech(πx/(2h))), Bull Am Math Soc, 40, 284-290 (1934) · Zbl 0009.21903 · doi:10.1090/S0002-9904-1934-05852-X |
| [5] | Bingham, Nh; Kiesel, R., Semi-parametric modelling in finance: theoretical foundations, Quant Finance, 2, 4, 241-250 (2002) · Zbl 1408.62171 · doi:10.1088/1469-7688/2/4/201 |
| [6] | Bracewell, Rn, The Fourier transform and its application (1986), New York: Mc Graw Hill, New York |
| [7] | Bronshtein, In; Semendyayev, Ka; Musiol, G.; Muhlig, H., Handbook of mathematics (1998), Berlin: Springer, Berlin |
| [8] | Carreau, J.; Bengio, Y., A hybrid Pareto model for asymmetric fat-tailed data: the univariate case, Extremes, 12, 53-76 (2009) · Zbl 1223.62030 · doi:10.1007/s10687-008-0068-0 |
| [9] | Carreau, Jp; Naveau, P.; Sauquet, E., A statistical rainfall-runoff mixture model with heavy-tailed components, Water Resour Res, 45, W10437 (2009) · doi:10.1029/2009WR007880 |
| [10] | Carreau, J.; Vrac, M., Stochastic down scaling of precipitation with neural network conditional mixture models, Water Resour Res, 47, W10502 (2011) · doi:10.1029/2010WR010128 |
| [11] | Cheah, Pk; Fraser, Das; Reid, N., Some alternatives to Edgeworth, Can J Stat, 21, II, 131-138 (1993) · Zbl 0779.62014 · doi:10.2307/3315806 |
| [12] | Chihara, Ts, An introduction to orthogonal polynomials (1978), New York: Gordon & Breach, New York · Zbl 0389.33008 |
| [13] | Cooley, D.; Nychka, D.; Nevada, P., Bayesian spatial modeling of extreme precipitation return levels, J Am Stat Assoc, 102, 479, 824-840 (2007) · Zbl 1469.62389 · doi:10.1198/016214506000000780 |
| [14] | Faliva, M.; Poti’, V.; Zoia, Mg, Orthogonal polynomials for tailoring density functions to excess kurtosis, asymmetry and dependence, Commun Stat Theory Methods, 45, 49-62 (2016) · Zbl 1381.62034 · doi:10.1080/03610926.2013.818698 |
| [15] | Fang, Kt; Kotz, S.; Ng, Kw, Symmetric multivariate and related distributions (1990), London: Chapman and Hall, London · Zbl 0699.62048 |
| [16] | Fernandez, C.; Steel, Mf, On Bayesian modeling of fat tails and skewness, J Am Stat Assoc, 93, 359-371 (1998) · Zbl 0910.62024 |
| [17] | Fisher, Mj, Generalized hyperbolic secant distribution with application to finance (2013), Berlin: Springer, Berlin |
| [18] | Frigessi, A.; Haug, O.; Rue, H., A dynamic mixture for unsupervised tail estimation without threshold selection, Extremes, 5, 219-235 (2002) · Zbl 1039.62042 · doi:10.1023/A:1024072610684 |
| [19] | Johnson, Nl; Kotz, S.; Balakrishnan, N., Continuous univariate distributions (1994), New York: Wiley, New York · Zbl 0811.62001 |
| [20] | García, Cb; Van García Perez, J.; Dorp, Jr, Modelling heavy-tailed, skewed and peaked uncertainty phenomena with bounded support, Stat Methods Appl, 20, 463-486 (2011) · Zbl 1238.62012 · doi:10.1007/s10260-011-0173-0 |
| [21] | Genton, Mg; Genton, Mg, Skew-symmetric and generalized skew-elliptical distributions, Skew-elliptical distributions and their applications, 101-120 (2004), London: Chapman and Hall, London · Zbl 1069.62045 |
| [22] | Gomez, E.; Gomez-Villegas, Ma; Marin, Jm, A survey on continuous elliptical vector distributions, Rev Mat Complut, 16, 1, 345-361 (2003) · Zbl 1041.60016 · doi:10.5209/rev_REMA.2003.v16.n1.16889 |
| [23] | Gradshteyn, Is; Ryzhik, Im, Table of integrals, series and products (1980), London: Academic Press, London · Zbl 0521.33001 |
| [24] | Handcock MS (2014) Relative distribution methods. Version 1.6-3. Project home page at http://www.stat.ucla.edu/ handcock/RelD ist http://CRAN.R-project.org/package=reldist |
| [25] | Hayfield, T.; Racine, Js, Nonparametric econometrics: the np package, J Stat Softw, 27, 5, 1-32 (2008) · doi:10.18637/jss.v027.i05 |
| [26] | Jondeau, E.; Rockinger, M., Gram-Charlier densities, J Econ Dyn Control, 25, 10, 1457-1483 (2001) · Zbl 1056.91512 · doi:10.1016/S0165-1889(99)00082-2 |
| [27] | Kandem, Js, Value at risk and expected shortfall for linear portfolios with elliptically distributed risk factors, J Theor Appl Finance, 8, 537-551 (2005) · Zbl 1138.91453 · doi:10.1142/S0219024905003104 |
| [28] | Katz, R.; Parlange, M.; Naveau, P., Extremes in hydrology, Adv Water Resour, 25, 1287-1304 (2002) · doi:10.1016/S0309-1708(02)00056-8 |
| [29] | Kelker, D., Distribution theory of spherical distributions and a location-scale parameter generalization. \(Sankhy\bar{a} \), Indian J Stat, Series A, 419-430 (1970) · Zbl 0223.60008 |
| [30] | Landsman, Z.; Valdez, Ae, Tail conditional expectations for elliptical distributions, N Am Actuar J, 8, 3, 118-122 (2004) · Zbl 1085.62501 · doi:10.1080/10920277.2004.10596157 |
| [31] | Li, C.; Singh, Vp; Mishra, Ak, Simulation of the entire range of daily precipitation using a hybrid probability distribution, Water Resour Res, 48, W03521 (2012) · doi:10.1029/2011WR011446 |
| [32] | Lutkepohl, H., Handbook of matrices (1996), Hoboken: Wiley, Hoboken · Zbl 0856.15001 |
| [33] | Macdonald, A.; Scarrot; Lee, D.; Darlow, B.; Reale, M.; Russel, G., A flexible extreme value mixture model, Comput Stat Data Anal, 55, 2137-2157 (2011) · Zbl 1328.62296 · doi:10.1016/j.csda.2011.01.005 |
| [34] | Maasoumi, E.; Racine, J., Entropy and predictability of stock market returns, J Econom, 107, 1, 291-312 (2002) · Zbl 1043.62090 · doi:10.1016/S0304-4076(01)00125-7 |
| [35] | Mills, Tc, The econometric modelling of financial time series (1999), Cambridge: Cambridge University Press, Cambridge · Zbl 1145.62402 |
| [36] | Naveau, P.; Toreti, I.; Xoplaki, E., A fast nonparametric spatio-temporal regression scheme for generalized Pareto distributed heavy precipitation, Water Resour Res, 50, 4011-4017 (2014) · doi:10.1002/2014WR015431 |
| [37] | Papastahopoulos, I.; Tawn, Aj, Extended generalized Pareto models for tails estimation, J Stat Plan Inference, 143, 131-143 (2013) · Zbl 1251.62020 · doi:10.1016/j.jspi.2012.07.001 |
| [38] | Philippe, J., Value at risk: the new benchmark for managing financial risk (2001), New York: McGraw-Hill Professional, New York |
| [39] | Rachev, St; Hoechstoetter, M.; Fabozzi, Fj, Probability and statistics for finance (2010), New York: Wiley, New York |
| [40] | Ruppert, D.; Wand, Mp, Correcting for kurtosis in density estimation, Aust N Z J Stat, 34, I, 19-29 (1992) · Zbl 0751.62019 · doi:10.1111/j.1467-842X.1992.tb01039.x |
| [41] | Szego G (1967) Orthogonal polynomials. American Mathematical Society Colloquium Publications, vol. 23, American Mathematical Society, New York, 1939, 3rd edn · Zbl 0023.21505 |
| [42] | Szego, G., Risk measures for the 21 century (2004), New York: Wiley, New York |
| [43] | Zoia, M., Tailoring the Gaussian law for excess kurtosis and asymmetry by Hermite polynomials, Commun Stat Theory Methods, 39, 52-64 (2010) · Zbl 1183.33016 · doi:10.1080/03610920802696596 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
