Quantile Functions

(aka Inverse Cumulative Distribution Functions)

I have some research projects running on the determination of quantile functions of distributions relevant to mathematical finance applications. Quantile functions are formally the inverse (when it exists) of the cumulative distribution function, but when people talk about quantiles they are usually talking in one of two (overlapping) contexts:

A. Simply-determined key percentage points of distributions for statistical applications;

B. The (fast) application of quantile functions to random samples from the unit interval to produce samples from non-uniform distributions.

My work is on topics related to item B, and applications to Monte Carlo sampling, low-discrepancy and copula methods. If you have come here looking for (A) statistical methodology, you should go here for a very quick introduction and see the book by Gilchrist for an extensive discussion.

Topics:

Recursive quantiles

With Gyorgy Steinbrecher I am investigating recursive algorithms for key distributions. The work looks at the creation of very-high-precision benchmark algorithms (with applications to calibrating new and fast rational and polynomial approximations), and faster methods for real-time use. The first version of our paper on this is downloadable here, and prototype F95 and C++ code is available now for the normal case and a first version in F95 for the Student is also available covering the entire unit interval. The quad precision F95 code is typically returning errors better than 1.0 E-28! A C++ code for the Student is under development.

C++ code for the normal quantile on [e, 1-e], e=0.0005

F95 code for the normal quantile on [e,1-e], e=0.0007, error < 1.0E-29

F95 code for the Student distribution on (0,1), (prototype, error < 1.0E-24)

Benchmarking Rational Approximations

I took a look at the existing methods for the normal case (Beasley-Springer-Moro, Acklam, AS241) and my results are in the working paper and related material below (Refinement of the Normal Quantile).

Adding skewness and kurtosis to an existing distribution

With Ian Buckley I have been looking at methods for modulating the skewness and kurtosis of an existing distribution in a way that retains tractable forms for the CDF and quantile functions. This has resulted in new definitions of a skew-normal and kurtotic-normal distribution, and indeed a skew-kurtotic-normal distribution. See Shaw and Buckley, 2007.

Analytical Methods

My work on the quantile function for the Student distribution appeared in the Journal of Computational Finance (Shaw, 2006). The power series developed there can be extended massively, and without the use of symbolic computation, using the recursive methods developed with Steinbrecher.

Publications and working papers

Steinbrecher, G. and Shaw, W.T., 2007, Differential Equations for Quantile Functions [PDF of May 1 07 working paper]. [Key Words: Inverse CDF, quantile function, Normal, Student, Beta, T-distribution, Simulation, Monte Carlo, Inverse Cumulative Distribution Function, non linear ordinary differential equations, recurrence relations]

Shaw, W.T., 2007, Refinement of the Normal Quantile, Simple improvements to the Beasley-Springer-Moro method of simulating the Normal Distribution, and a comparison with Acklam's method and Wichura’s AS241, [PDF], [Mathematica Notebook] [Key Words, Rational Approximation, Beasley Springer, Moro, Acklam, AS241, Wichura, Inverse Error function, Normal Quantile, Inverse Cumulative Distribution Function] (Working paper, updated 20 Feb 07). Also available – (a) AS241 (Wichura’s method alone) Notebook and PDF; (b) Acklam’s method (Notebook and PDF); (c) first version of C++ code for non-linear recursion.

Shaw, W.T., and Buckley, I.R.C. 2007, The Alchemy of Probability Distributions: Beyond Gram-Charlier & Cornish-Fisher Expansions, and Skew-Normal or Kurtotic-Normal Distributions. (Submitted, Feb 07) [PDF] [Key Words: Distributional Alchemy, Gram Charlier, Cornish Fisher, Skew Uniform, Skew Normal, Skew T, Skew Student, Student Distribution, T Distribution, Kurtotic Uniform, Kurtotic Normal, Skew Kurtotic Normal, Skew Exponential]

Sydney QMF 2006 presentation on distributional alchemy and related applications of computer algebra to Monte Carlo methods (Sydney, Australia, Dec 2006) [PDF of presentation] [Mathematica Notebook of presentation] [Key Words: Student Distribution, Distributional alchemy, Cornish Fisher expansion,  Gram Charlier expansion, Skew Normal Distribution]

Shaw, W.T., 2006, Sampling Student’s T distribution – use of the inverse cumulative distribution function. Journal of Computational Finance, Vol 9 Issue 4, pp 37-73, Summer 2006  Journal Link. On-line supplements to this article are available here. [Key words: Student, Student’s T Distribution, T-Distribution, Inverse CDF, Inverse Cumulative Distribution Function, Quantile, T-Quantile, Simulation, Monte Carlo, Copula]

Other Refs

W. Gilchrist, 2000, Statistical Modelling with Quantile Functions, CRC Press

P. Jaeckel, 2002, Monte Carlo Methods in Finance, Wiley