Behrens–Fisher Problem

Introduction

The Behrens–Fisher problem is the problem in statistics of hypothesis testing and interval estimation regarding the difference between the means of two independent normal populations without assuming the variances are equal. The solution of this problem was first offered by Behrens (1929) and reformulated later by Fisher (1939) using

$${t}^{{\prime}} = \frac{(\bar{{x}}_{1} -\bar{ {x}}_{2}) - ({\mu }_{1} - {\mu }_{2})} {\sqrt{{s}_{1 }^{2 }/{n}_{1 } + {s}_{2 }^{2 }/{n}_{2}}} = {t}_{1}\sin \theta - {t}_{2}\cos \theta,$$

where the sample mean $\bar{{x}}_{1}$ and sample variance s ²₁ are obtained from the random sample of size n ₁ from the normal distribution with mean μ ₁ and variance σ ²₁ , ${t}_{1} = (\bar{{x}}_{1} - {\mu }_{1})/\sqrt{{s}_{1 }^{2 }/{n}_{1}}$ has a t distribution with ν₁ = n ₁ − 1 degrees of freedom, the respective quantities with subscript 2 are defined similarly, and $\tan \theta = ({s}_{1}/\sqrt{{n}_{1}})/({s}_{2}/\sqrt{{n}_{2}})$ or \(\theta {=\tan...