A unified approach to the Behrens-Fisher problem

TL;DR

This paper introduces a unified mathematical framework for the Behrens-Fisher problem, deriving explicit formulas for the null distribution of the test statistic, enabling precise tail analysis and critical value tabulation.

Contribution

It develops a Mellin--Barnes factorization and hypergeometric function representation, providing a new, stable, and exact approach to the two-sample Behrens-Fisher problem.

Findings

Derived a compact expression for the null distribution of the test statistic.

Provided exact critical value tables for various sample sizes and variance ratios.

Quantified the accuracy of Welch's approximation across different parameter regimes.

Abstract

A unified framework is presented to study the two-sample Behrens--Fisher problem -- testing equality of means when two normal populations have unequal, unknown variances -- and a compact expression is derived for the null distribution of the classical test statistic. Our new approach involves a Mellin--Barnes factorization that decouples the square root of a weighted sum of independent chi-square variates, thereby collapsing a challenging two-dimensional integral to a tractable single-contour integral. Closing the contour yields a residue series that terminates whenever either sample's degrees of freedom is odd. A complementary Euler--Beta reduction identifies the density as a Gauss hypergeometric function with explicit parameters, yielding a numerically stable form that recovers Student's $t$ under equal variances. Ramanujan's master theorem supplies exact inverse-power tail coefficients, which bound Lugannani--Rice saddle-point approximation errors and support reliable tail analyses. The proposed framework reveals why hypergeometric structure appears, why certain finite-sum cases arise, and how one can pass from the bulk of the distribution to its tails without altering the analytic framework. Finally, it lets us tabulate exact two-sided critical values over a broad grid of sample sizes and variance ratios that reveal the parameter surface on which the well-known Welch's approximation switches from conservative to liberal, quantifying its maximum size distortion.