A method for calculating quantile function and its further use for data fitting

TL;DR

This paper presents a polynomial transformation model based on Weibull distribution to analytically derive quantile functions for various distributions, enabling accurate data fitting especially for complex distributions.

Contribution

Introduces a polynomial Weibull-based transformation model for deriving quantile functions and fitting data, improving handling of complex distributions.

Findings

Accurately approximates quantile functions within the probit range

Handles distributions close to binomial more effectively

Demonstrates the model's effectiveness through numerical experiments

Abstract

This paper introduces a polynomial transformation model based on Weibull distribution, whereby the analytical representation of the quantile function for many probability distributions can be obtained. Firstly, the target random variable $x$ with specified distribution is expressed as a polynomial of a Weibull random variable $z$, the coefficients are conveniently determined by the percentile matching method. Then, substituting $z$ with its quantile function $z=\lambda [-ln(1-u)]^{1/k}$ gives the analytical expression of the quantile function of $x$. Furthermore, using the probability weighted moments matching method, this polynomial transformation model can be used for data fitting. Through numerical experiment, it makes evident that the proposed model is capable of handling some distributions close to binomial which are difficult for the extant approaches, and the quantile functions of various distributions are accurately approximated within the probit range $[10^{-4},1-10^{-4}]$.