Multivariate Behrens–Fisher problem

In statistics, the multivariate Behrens–Fisher problem is the problem of testing for the equality of means from two multivariate normal distributions when the covariance matrices are unknown and possibly not equal.

Notation and Problem Formulation

Let $X_{ij}\sim {\mathcal {N}}_{p}(\mu _{i},\,\Sigma _{i})\ \ (j=1,\dots ,n_{i};\ \ i=1,2)\$ be independent random samples from two $p$-variate normal distributions with unknown mean vectors $\mu _{i}$ and unknown dispersion matrices $\Sigma _{i}$. The index $i$ refers to the first or second population, and the $j$th observation from the $i$th population is $X_{ij}$.

The multivariate Behrens-Fisher problem is to to test the null hypothesis $H_{0}$ that the means are equal versus the alternative $H_{1}$ of non-equality:

$H_{0}:\mu _{1}=\mu _{2}\ \ {\text{vs}}\ \ H_{1}:\mu _{1}\neq \mu _{2}.$

Define some statistics, which are used in the various attempts to solve the multivariate Behrens-Fisher problem, by

${\begin{aligned}{\bar {X_{i}}}&={\frac {1}{n_{i}}}\sum _{j=1}^{n_{i}}X_{ij},\\A_{i}&=\sum _{j=1}^{n_{i}}(X_{ij}-{\bar {X_{i}}})(X_{ij}-{\bar {X_{i}}})',\\S_{i}&={\frac {1}{n_{i}-1}}A_{i},\\{\tilde {S_{i}}}&={\frac {1}{n_{i}}}S_{i},\\{\tilde {S}}&={\tilde {S_{1}}}+{\tilde {S_{2}}},\quad {\text{and}}\\T^{2}&=({\bar {X_{1}}}-{\bar {X_{2}}})'{\tilde {S}}^{-1}({\bar {X_{1}}}-{\bar {X_{2}}}).\end{aligned}}$

The sample means ${\bar {X_{i}}}$ and sum-of-squares matrices $A_{i}$ are sufficient for the multivariate normal parameters $\mu _{i},\Sigma _{i},\ (i=1,2)$, so it suffices to perform inference be based on just these statistics. The distributions of ${\bar {X_{i}}}$ and $A_{i}$ are independent and are, respectively, multivariate normal and Wishart :

${\begin{aligned}{\bar {X_{i}}}&\sim {\mathcal {N}}_{p}\left(\mu _{i},\Sigma _{i}/n_{i}\right),\\A_{i}&\sim W_{p}(\Sigma _{i},n_{i}-1).\end{aligned}}$

Since this is a generalization of the univariate Behrens-Fisher problem, it inherits all of the difficulties that arise in the univariate problem.

References

• W. F. Christensen and A. C. Rencher, "A comparison of type I error rates and power levels for seven solutions to the multivariate Beherns-Fisher problem", Communications in Statist. Simulation and Computation, 26 (1997), 1251-1273.
• Junyong Park and Bimal Sinha, "Some aspects of multivariate Behrens-Fisher problem", Tech-Report (2007), http://www.math.umbc.edu/~kogan/technical_papers/2007/Park_Sinha.pdf