Approximation to probability density functions in sampling distributions based on Fourier cosine series

TL;DR

This paper introduces a Fourier cosine series-based method for accurately approximating probability density functions in sampling distributions, demonstrated through examples involving sums of uniform variables and sample skewness.

Contribution

It presents a novel, simple approximation technique for sampling distribution densities using Fourier cosine series, applicable even when explicit formulas are unavailable.

Findings

Accurately approximates distributions of sums of uniform variables.

Effectively estimates distribution of sample skewness from normal populations.

Provides a practical method for cases lacking explicit density expressions.

Abstract

We derive a simple and precise approximation to probability density functions in sampling distributions based on the Fourier cosine series. After clarifying the required conditions, we illustrate the approximation on two examples: the distribution of the sum of uniformly distributed random variables, and the distribution of sample skewness drawn from a normal population. The probability density function of the first example can be explicitly expressed, but that of the second example has no explicit expression.