Quantile mechanics: Issues arising from critical review

Abstract

Approximations are the alternative way of obtaining the Quantile function when the inversion method cannot be applied to distributions whose cumulative distribution functions do not have close form expressions. Approximations come in form of functional approximation, numerical algorithm, closed form expressed in terms of others and series expansions. Several quantile approximations are available which have been proven to be precise, but some issues like the presence of shape parameters, inapplicability of existing methods to complex distributions and low computational speed and accuracy place undue limitations to their effective use. Quantile mechanics (QM) is a series expansion method that addressed these issues as evidenced in the paper. Quantile mechanics is a generalization of the use of ordinary differential equations (ODE) in quantile approximation. The paper is a review that critically examined with evidences; the formulation, applications and advantages of QM over other surveyed methods. Some issues bothering on the use of QM were also discussed. The review concluded with areas of further studies which are open for scientific investigation and exploration.