Behrens–Fisher distribution

The Behrens–Fisher distribution is the distribution of a random variable of the form

${\displaystyle T_{2}\cos \theta -T_{1}\sin \theta \,}$

where T₁ and T₂ are independent random variables each with a Student's t-distribution, with respective degrees of freedom ν₁ = n₁ − 1 and ν₂ = n₂ − 1, and θ is a constant. Thus the family of Behrens–Fisher distributions is parametrized by ν₁, ν₂, and θ.

Suppose it were known that the two population variances are equal, and samples of sizes n₁ and n₂ are taken from the two populations:

${\displaystyle {\begin{aligned}X_{1,1},\ldots ,X_{1,n_{1}}&\sim \operatorname {i.i.d.} N(\mu _{1},\sigma ^{2}),\\[6pt]X_{2,1},\ldots ,X_{2,n_{2}}&\sim \operatorname {i.i.d.} N(\mu _{2},\sigma ^{2}).\end{aligned}}}$

where "i.i.d" are independent and identically distributed random variables and N denotes the normal distribution. The two sample means are

${\displaystyle {\begin{aligned}{\bar {X}}_{1}&=(X_{1,1}+\cdots +X_{1,n_{1}})/n_{1}\\[6pt]{\bar {X}}_{2}&=(X_{2,1}+\cdots +X_{2,n_{2}})/n_{2}\end{aligned}}}$

The usual "pooled" unbiased estimate of the common variance σ² is then

${\displaystyle S_{\mathrm {pooled} }^{2}={\frac {\sum _{k=1}^{n_{1}}(X_{1,k}-{\bar {X}}_{1})^{2}+\sum _{k=1}^{n_{2}}(X_{2,k}-{\bar {X}}_{2})^{2}}{n_{1}+n_{2}-2}}={\frac {(n_{1}-1)S_{1}^{2}+(n_{2}-1)S_{2}^{2}}{n_{1}+n_{2}-2}}}$

where S₁² and S₂² are the usual unbiased (Bessel-corrected) estimates of the two population variances.

Under these assumptions, the pivotal quantity

${\displaystyle {\frac {(\mu _{2}-\mu _{1})-({\bar {X}}_{2}-{\bar {X}}_{1})}{\displaystyle {\sqrt {{\frac {S_{\mathrm {pooled} }^{2}}{n_{1}}}+{\frac {S_{\mathrm {pooled} }^{2}}{n_{2}}}}}}}}$

has a t-distribution with n₁ + n₂ − 2 degrees of freedom. Accordingly, one can find a confidence interval for μ₂ − μ₁ whose endpoints are

${\displaystyle {\bar {X}}_{2}-{\bar {X_{1}}}\pm A\cdot S_{\mathrm {pooled} }{\sqrt {{\frac {1}{n_{1}}}+{\frac {1}{n_{2}}}}},}$

where A is an appropriate quantile of the t-distribution.

However, in the Behrens–Fisher problem, the two population variances are not known to be equal, nor is their ratio known. Fisher considered the pivotal quantity

${\displaystyle {\frac {(\mu _{2}-\mu _{1})-({\bar {X}}_{2}-{\bar {X}}_{1})}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}.}$

This can be written as

${\displaystyle T_{2}\cos \theta -T_{1}\sin \theta ,\,}$

where

${\displaystyle T_{i}={\frac {\mu _{i}-{\bar {X}}_{i}}{S_{i}/{\sqrt {n_{i}}}}}{\text{ for }}i=1,2\,}$

are the usual one-sample t-statistics and

${\displaystyle \tan \theta ={\frac {S_{1}/{\sqrt {n_{1}}}}{S_{2}/{\sqrt {n_{2}}}}}}$

and one takes θ to be in the first quadrant. The algebraic details are as follows:

${\displaystyle {\begin{aligned}{\frac {(\mu _{2}-\mu _{1})-({\bar {X}}_{2}-{\bar {X}}_{1})}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}&={\frac {\mu _{2}-{\bar {X}}_{2}}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}-{\frac {\mu _{1}-{\bar {X}}_{1}}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}\\[10pt]&=\underbrace {\frac {\mu _{2}-{\bar {X}}_{2}}{S_{2}/{\sqrt {n_{2}}}}} _{{\text{This is }}T_{2}}\cdot \underbrace {\left({\frac {S_{2}/{\sqrt {n_{2}}}}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}\right)} _{{\text{This is }}\cos \theta }-\underbrace {\frac {\mu _{1}-{\bar {X}}_{1}}{S_{1}/{\sqrt {n_{1}}}}} _{{\text{This is }}T_{1}}\cdot \underbrace {\left({\frac {S_{1}/{\sqrt {n_{1}}}}{\displaystyle {\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}}\right)} _{{\text{This is }}\sin \theta }.\qquad \qquad \qquad (1)\end{aligned}}}$

The fact that the sum of the squares of the expressions in parentheses above is 1 implies that they are the squared cosine and squared sine of some angle.

The Behren–Fisher distribution is actually the conditional distribution of the quantity (1) above, given the values of the quantities labeled cos θ and sin θ. In effect, Fisher conditions on ancillary information.

Fisher then found the "fiducial interval" whose endpoints are

${\displaystyle {\bar {X}}_{2}-{\bar {X}}_{1}\pm A{\sqrt {{\frac {S_{1}^{2}}{n_{1}}}+{\frac {S_{2}^{2}}{n_{2}}}}}}$

where A is the appropriate percentage point of the Behrens–Fisher distribution. Fisher claimed that the probability that μ₂ − μ₁ is in this interval, given the data (ultimately the Xs) is the probability that a Behrens–Fisher-distributed random variable is between −A and A.

• Kendall, Maurice G., Stuart, Alan (1973) The Advanced Theory of Statistics, Volume 2: Inference and Relationship, 3rd Edition, Griffin. ISBN 0-85264-215-6 (Chapter 21)