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Quantile mechanics

Published online by Cambridge University Press:  01 April 2008

GYÖRGY STEINBRECHER  and
WILLIAM T. SHAW
GYÖRGY STEINBRECHER
Affiliation:
Association EURATOM-MEC Department of Theoretical Physics, Physics Faculty, University of Craiova, Str.A.I.Cuza 13, Craiova-200585, Romania email: [email protected]
WILLIAM T. SHAW
Affiliation:
Department of Mathematics King's College, The Strand, London WC2R 2LS, England, UK email: [email protected]
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Abstract

In both modern stochastic analysis and more traditional probability and statistics, one way of characterizing a static or dynamic probability distribution is through its quantile function. This paper is focused on obtaining a direct understanding of this function via the classical approach of establishing and then solving differential equations for the function. We establish ordinary differential equations and power series for the quantile functions of several common distributions. We then develop the partial differential equation for the evolution of the quantile function associated with the solution of a class of stochastic differential equations, by a transformation of the Fokker–Planck equation. We are able to utilize the static formulation to provide elementary time-dependent and equilibrium solutions.

Such a direct understanding is important because quantile functions find important uses in the simulation of physical and financial systems. The simplest way of simulating any non-uniform random variable is by applying its quantile function to uniform deviates. Modern methods of Monte–Carlo simulation, techniques based on low-discrepancy sequences and copula methods all call for the use of quantile functions of marginal distributions. We provide web resources for prototype implementations in computer code. These implementations may variously be used directly in live sampling models or in a high-precision benchmarking mode for developing fast rational approximations also for use in simulation.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 2 , April 2008 , pp. 87 - 112
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Abernathy, R. W. & Smith, R. P. (1993) Applying series expansion to the inverse beta distribution to find percentiles of the F-distribution. ACM Tran. Math. Soft. 19 (4), 474–480.CrossRefGoogle Scholar
[2]Acklam, P. J. An algorithm for computing the inverse normal cumulative distribution function, http://home.online.no/~pjacklam/notes/invnorm/Google Scholar
[3]Aletti, G., Naldi, G. & Toscani, G. (2007) First-order continuous models of opinion formation, SIAM J. Appl. Math. 67 (3), 827–853.CrossRefGoogle Scholar
[4]Bliss, C. I. (1934) The method of probits. Science 39, 38–39.CrossRefGoogle Scholar
[5]Carrillo, J. A. & Toscani, G. (2004) Wasserstein metric and large-time asymptotics of nonlinear diffusion equations. In New Trends in Mathematical Physics, World Sci. Publ., Hackensack, NJ, pp. 234–244.Google Scholar
[6]Cherubini, U., Luciano, E. & Vecchiato, W. (2004) Copula Methods in Finance, Wiley, New York.CrossRefGoogle Scholar
[7]Csörgő, M. (1983) Quantile Processes with Statistical Applications, SIAM, Philadelphia.CrossRefGoogle Scholar
[8]Dassios, A. (2005) On the quantiles of Brownian motion and their hitting times. Bernoulli 11 (1), 29–36.CrossRefGoogle Scholar
[9]Devroye, L.Non-uniform random variate generation, Springer 1986. Out of print – now available on-line from the author's web site at http://cg.scs.carleton.ca/~luc/rnbookindex.htmlCrossRefGoogle Scholar
[10]Etheridge, A. (2002) A Course in Financial Calculus, Cambridge University Press, Cambridge, UK.CrossRefGoogle Scholar
[11]Fergusson, K. & Platen, E. (2006) On the distributional characterization of daily Log-returns of a World Stock Index, Applied Mathematical Finance, 13 (1), 19–38.CrossRefGoogle Scholar
[12]Gilchrist, W. (2000) Statistical Modelling with Quantile Functions, CRC Press, London.CrossRefGoogle Scholar
[13]Hill, G. W. & Davis, A. W. (1968) Generalized asymptotic expansions of Cornish–Fisher Type, Ann. Math. Stat. 39 (4), 1264–1273.CrossRefGoogle Scholar
[14]http://functions.wolfram.com/GammaBetaErf/InverseErf/13/01/Google Scholar
[15]http:/functions.wolfram.com/GammaBetaErf/InverseBetaRegularized/13/01/Google Scholar
[16]http://functions.wolfram.com/GammaBetaErf/InverseBetaRegularized/06/01/Google Scholar
[17]Jäckel, P. (2002) Monte–Carlo Methods in Finance, Wiley, New York.Google Scholar
[18]Johnson, N. L. & Kotz, S. (1970) Distributions in Statistics. Continuous Univariate Distributions, Wiley, New York.Google Scholar
[19]Miura, R. (1992) A note on a look-back option based on order statistics, Hitosubashi J. Com. Manage. 27, 15–28.Google Scholar
[20]Pearson, K. (1916) Second supplement to a memoir on skew variation. Phil. Trans. A 216, 429–457.Google Scholar
[21]Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. (2007) Numerical Recipes. The Art of Scientific Computing, 3rd ed., Cambridge University Press, Cambridge, UK.Google Scholar
[22]Shaw, W. T. (2006) Sampling Student's T distribution – use of the inverse cumulative distribution function. J. Comput. Fin., 9 (4).Google Scholar
[23]Shaw, W. T. Refinement of the Normal quantile, King's College working paper, http://www.mth.kcl.ac.uk/~shaww/web_page/papers/NormalQuantile1.pdf, accessed 20 Feb 2007.Google Scholar
[24]Steinbrecher, G. (2002) Taylor expansion for inverse error function around origin, University of Craiova working paper, http://functions.wolfram.com/GammaBetaErf/InverseErf/06/01/0002/Google Scholar
[25]Steinbrecher, G. & Weyssow, B. (2004) Generalized randomly amplified linear system driven by Gaussian noises: Extreme heavy tail and algebraic correlation decay in plasma turbulence. Phys. Rev. Lett. 92 (12), 125003–125006.CrossRefGoogle ScholarPubMed
[26]Steinbrecher, G. & Weyssow, B. (2007) Extreme anomalous particle transport at the plasma edge, University of Craiova/Univ Libre de Bruxells working paper, http://www.fz-juelich.de/sfp/talks/2007/talks_index.htmlGoogle Scholar
[27]Toscani, G. & Li, H. (2004) Long-time asymptotics of kinetic models of granular flows, Archiv. Ration. Mech. Anal. 172, 407–428.Google Scholar
[28]Wichura, M. J. (1988) Algorithm AS 241: The Percentage Points of the Normal Distribution. Appl. Stat. 37, 477–484.CrossRefGoogle Scholar
[29] On line discussion of Pearson's system of distributions defined via differential equations. !http://mathworld.wolfram.com/PearsonSystem.htmlGoogle Scholar