A general approximation to quantiles.

Area of Science:

• Statistics
• Numerical Analysis
• Probability Theory

Background:

• Quantiles for many continuous probability distributions lack closed-form expressions.
• Existing numerical quantile approximations are often distribution-specific, requiring tailored methods.
• This necessitates a more generalized approach for quantile approximation.

Purpose of the Study:

• To develop a unified, general approximation method for quantiles of continuous distributions.
• To utilize Taylor expansion for a broadly applicable numerical quantile estimation technique.
• To demonstrate the efficacy of this method across various common distributions.

Main Methods:

• Employed Taylor expansion as the core mathematical tool for approximation.
• Required only a continuous probability density function and its derivatives up to a certain order (typically 3 or 4).
• Applied the method to approximate quantiles for normal, exponential, and chi-square distributions.

Main Results:

• Successfully developed a general approximation for quantiles using Taylor expansion.
• The unified approach demonstrated good performance for the tested distributions (normal, exponential, chi-square).
• The method's requirements are minimal, relying on the existence of derivatives of the probability density function.

Conclusions:

• A general Taylor expansion-based method provides an effective way to approximate quantiles for continuous distributions.
• This unified approach simplifies quantile approximation compared to distribution-specific methods.
• The technique is robust and applicable to a range of important statistical distributions.