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Quantile mechanics


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.

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[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), 474480.
[2]Acklam, P. J. An algorithm for computing the inverse normal cumulative distribution function,
[3]Aletti, G., Naldi, G. & Toscani, G. (2007) First-order continuous models of opinion formation, SIAM J. Appl. Math. 67 (3), 827853.
[4]Bliss, C. I. (1934) The method of probits. Science 39, 3839.
[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.
[6]Cherubini, U., Luciano, E. & Vecchiato, W. (2004) Copula Methods in Finance, Wiley, New York.
[7]Csörgő, M. (1983) Quantile Processes with Statistical Applications, SIAM, Philadelphia.
[8]Dassios, A. (2005) On the quantiles of Brownian motion and their hitting times. Bernoulli 11 (1), 2936.
[9]Devroye, L.Non-uniform random variate generation, Springer 1986. Out of print – now available on-line from the author's web site at
[10]Etheridge, A. (2002) A Course in Financial Calculus, Cambridge University Press, Cambridge, UK.
[11]Fergusson, K. & Platen, E. (2006) On the distributional characterization of daily Log-returns of a World Stock Index, Applied Mathematical Finance, 13 (1), 1938.
[12]Gilchrist, W. (2000) Statistical Modelling with Quantile Functions, CRC Press, London.
[13]Hill, G. W. & Davis, A. W. (1968) Generalized asymptotic expansions of Cornish–Fisher Type, Ann. Math. Stat. 39 (4), 12641273.
[17]Jäckel, P. (2002) Monte–Carlo Methods in Finance, Wiley, New York.
[18]Johnson, N. L. & Kotz, S. (1970) Distributions in Statistics. Continuous Univariate Distributions, Wiley, New York.
[19]Miura, R. (1992) A note on a look-back option based on order statistics, Hitosubashi J. Com. Manage. 27, 1528.
[20]Pearson, K. (1916) Second supplement to a memoir on skew variation. Phil. Trans. A 216, 429457.
[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.
[22]Shaw, W. T. (2006) Sampling Student's T distribution – use of the inverse cumulative distribution function. J. Comput. Fin., 9 (4).
[23]Shaw, W. T. Refinement of the Normal quantile, King's College working paper,, accessed 20 Feb 2007.
[24]Steinbrecher, G. (2002) Taylor expansion for inverse error function around origin, University of Craiova working paper,
[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), 125003125006.
[26]Steinbrecher, G. & Weyssow, B. (2007) Extreme anomalous particle transport at the plasma edge, University of Craiova/Univ Libre de Bruxells working paper,
[27]Toscani, G. & Li, H. (2004) Long-time asymptotics of kinetic models of granular flows, Archiv. Ration. Mech. Anal. 172, 407428.
[28]Wichura, M. J. (1988) Algorithm AS 241: The Percentage Points of the Normal Distribution. Appl. Stat. 37, 477484.
[29] On line discussion of Pearson's system of distributions defined via differential equations. !
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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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