In 1929 Behrens introduced the concept of testing equality of two population means without assuming the homogeneity of the two population variances. Since that time there have been extensive discussions of the validity of the test and of the interpretation of the results. Fisher (1939) published a detailed paper defending Behrens' work and thus the work started by Behrens became known as the Behrens-Fisher problem. Fisher presented the test statistic (unequal variance $t$-test; Fryer, 1966) he felt was best for handling this situation. However, Fisher could only approximate the distribution of his test statistic. This approximation was tabled in Fisher and Yates (1948). Fisher's test statistic can also be approximated by the $t$-distribution, but the $t$-distribution approximation is not very good for small sample sizes (Cochran and Cox, 1957). Box (1954, a and b) wrote two extensive papers concerned with the violation of the assumptions in the analysis of variance. Box states that the one-way analysis of variance with equal sample sizes (the $t$-test is the same as a two-sample one-way analysis of variance) is robust when the variances are heterogeneous. However, little is said about the power of the test when the variances are heterogeneous. F. N. David and N. L. Johnson (1951, a and b) and Bozivich, Bancroft, and Hartley (1956) have written papers on the approximate theoretical power of the analysis of variance when the assumptions are violated. B. L. Welch (1937) and D. G. C. Gronow (1951) have written papers examining both the robustness and the power of the unequal variance $t$-test when the variances are heterogeneous. This paper examines by simulation techniques both the power and the robustness of the $t$-test and several other tests when the variances are heterogeneous, and presents a new test statistic designed for the situation where the coefficients of variation are homogeneous. This new test is more powerful than the $t$-test for certain ranges of the coefficient of variation. The assumption of homogeneous coefficients of variation is a valid assumption in many types of agricultural, biological, and psychological experimentation, because many times the treatment that yields a larger mean also has a larger standard deviation.
R. K. Kohrding. "A Test of Equality of Two Normal Population Means Assuming Homogeneous Coefficients of Variation." Ann. Math. Statist. 40 (4) 1374 - 1385, August, 1969. https://doi.org/10.1214/aoms/1177697509