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. Since this is a generalization of the univariate Behrens-Fisher problem, it inherits all of the difficulties that arise in the univariate problem. Notation and problem formulation. Let formula_1 be independent random samples from two formula_2-variate normal distributions with unknown mean vectors formula_3 and unknown dispersion matrices formula_4. The index formula_5 refers to the first or second population, and the formula_6th observation from  the formula_5th population is formula_8. The multivariate Behrens–Fisher problem is to test the null hypothesis formula_9 that the means are equal versus the alternative formula_10 of non-equality: Define some statistics, which are used in the various attempts to solve the multivariate Behrens–Fisher problem, by The sample means formula_13 and sum-of-squares matrices formula_14 are sufficient for the multivariate normal parameters formula_15, so it suffices to perform inference be based on just these statistics. The distributions of formula_13 and formula_14 are independent and are, respectively, multivariate normal and Wishart: Background. In the case where the dispersion matrices  are equal, the distribution of the formula_19 statistic is known to be an F distribution under the null and a noncentral F-distribution under the alternative. The main problem is that when the true values of the dispersion matrix are unknown, then under the null hypothesis the probability of rejecting formula_9 via a formula_19 test depends on the unknown dispersion matrices. In practice, this dependency harms inference when the dispersion matrices are far from each other or when the sample size is not large enough to estimate them accurately. Now, the mean vectors are independently and normally distributed, but the  sum formula_23 does not follow the Wishart distribution, which makes inference more difficult. Proposed solutions. Proposed solutions are based on a few main strategies: Approaches using the "T" with approximate degrees of freedom. Below, formula_26 indicates the trace operator. Yao (1965). where Johansen (1980). where and Nel and Van der Merwe's (1986). where Comments on performance. Kim (1992) proposed a solution that is based on a variant of formula_19. Although its power is high, the fact that it is not invariant makes it less attractive. Simulation studies by Subramaniam and Subramaniam (1973) show that the size of Yao's  test is closer to the nominal level than that of James's. Christensen and Rencher (1997) performed numerical studies comparing several of these testing procedures and concluded that Kim and Nel and Van der Merwe's tests had the highest power. However, these two procedures are not invariant. Krishnamoorthy and Yu (2004). Krishnamoorthy and Yu (2004) proposed a procedure which adjusts in Nel and Var der Merwe (1986)'s approximate df for the denominator of formula_19 under the null distribution to make it invariant. They show that the approximate degrees of freedom lies in the interval formula_36 to ensure that  the degrees of freedom is not negative. They report numerical studies that indicate that their procedure is as powerful as Nel and Van der Merwe's test for smaller dimension, and more powerful for larger dimension. Overall, they claim that their procedure is the better than the invariant procedures of Yao (1965) and Johansen (1980). Therefore, Krishnamoorthy and Yu's (2004) procedure has the best known size and power as of 2004. The test statistic formula_19 in Krishnmoorthy and Yu's procedure follows the distribution formula_38 where