New Exact and Asymptotically Optimal Heteroscedastic Statistical Procedures and Tables, II

SYNOPTIC ABSTRACT

Heteroscedasticity has been a problem in statistics from the start of the field. For example, the Behrens-Fisher Problem (of testing the equality of two normal means when the variances are not known and cannot be assumed equal) has received the attention of Linnik, H. Scheffé, Welch, Chernoff, Chapman, Prokof'yev, Shishkin, B.K. Taneja, and others, and papers on it continue to appear (see Dudewicz and Ahmed (1998) for recent references). A new type of two-stage procedure was put forth by Aoshima, Hyakutake, and Dudewicz (1996) which allows an exact solution, and that solution can be (in some cases) asymptotically optimal. Dudewicz and Ahmed (1998) developed this solution, which is exact, for the Behrens-Fisher Problem, and gave tables needed to achieve level α. In the present paper we: review the previous work (Section 1), state the exact solution (Section 2), give tables needed for level α and power β (Sections 3 & 4), prove asymptotic optimality (Section 5), and give notes and programs on computational aspects (Section 6).

Key Words and Phrases:

• Behrens-Fisher Problem
• Tests of Hypotheses
• Heteroscedasticity
• Testing Equality of Means
• Exact Level Tests
• Tests With Specified Power
• Asymptotically Optimal Tests
• Statistical Tables