The determination of a test for the equality of means for two normal distributions with different variances given samples from each. There exists an exact test which, however, does not give a unique answer because it does not use all the data. There also exist approximate tests which do not use all the data.
Fisher-Behrens Problem
See also
Normal DistributionExplore with Wolfram|Alpha
References
Aspin, A. A. "An Examination and Further Development of a Formula Arising in the Problem of Comparing Two Mean Values." Biometrika 35, 88-96, 1948.Chernoff, H. "Asymptotic Studentization in Testing of Hypothesis." Ann. Math. Stat. 20, 268-278, 1949.Fisher, R. A. "The Fiducial Argument in Statistical Inference." Ann. Eugenics 6, 391-398, 1935.Kenney, J. F. and Keeping, E. S. "The Behrens-Fisher Test." §9.8 in Mathematics of Statistics, Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, pp. 257-260 and 261-264, 1951.Sukhatme, P. V. "On Fisher and Behrens' Test of Significance of the Difference in Means of Two Normal Samples." Sankhya 4, 39, 1938.Trickett, W. H. and Welch, B. L. "On the Comparison of Two Means: Further Discussion of Iterative Methods for Calculating Tables." Biometrika 41, 361-374, 1954.Trickett, W. H.; Welch, B. L.; and James, G. S. "Further Critical Values for the Two-Means Problems." Biometrika 43, 203-205, 1956.Wallace, D. L. "Asymptotic Approximations to Distributions." Ann. Math. Stat. 29, 635-654, 1958.Wald, A. "Testing the Difference Between the Means of Two Normal Populations with Unknown Standard Deviations." In Selected Papers in Statistics and Probability by Abraham Wald. New York: McGraw-Hill, pp. 669-695, 1955.Welch, B. L. "The Generalization of 'Student's' Problem when Several Different Populations are Involved." Biometrika 34, 28-35, 1947.Referenced on Wolfram|Alpha
Fisher-Behrens ProblemCite this as:
Weisstein, Eric W. "Fisher-Behrens Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Fisher-BehrensProblem.html