Behrens–Fisher t' Statistic

The Behrens–Fisher t′ statistic can be employed when one seeks to make inferences about the means of two normal populations without assuming the variances are equal. The statistic was offered first by W. U. Behrens in 1929 and reformulated by Ronald A. Fisher in 1939:

$$t^{\prime }=\frac{\left({\overline{x}}_1-{\overline{x}}_2\right)-\left({\mu }_1-{\mu }_2\right)}{\sqrt{s_1^2/n_1+s_2^2/n_2}}=t_1\text{}\sin \theta -t_2\text{}\cos \theta ,$$

where sample mean ${\overline{x}}_1$ and sample variance $s_1^2$ are obtained from the random sample of size n1 from the normal distribution with mean ${\mu }_1$ and variance ${\sigma }_1^2$, $t_1=\left({\overline{x}}_1-{\mu }_1\right)/\sqrt{s_1^2{n}_1}$ has a t distribution with ${\nu }_1=n_1-1$ degrees of freedom, the respective quantities with subscript 2 are defined similarly, and $\tan \theta =\text{}\left(s_1/\sqrt{n_1}\right)\left(s_2/\sqrt{n_2}\right)\text{or}\theta =\text{}{\tan }^{-1}\text{}\left[\left(s_1/\sqrt{n_1}\right)/\left(s_2/\sqrt{n_2}\right)\right]\text{}.$ The distribution of t′ is the Behrens–Fisher distribution. It is, hence, a mixture of the two distributions. The problem arising when one tries to test the normal population means without making any ...