Generalized confidence intervals for the ratio of means of two normal populations

Based on the generalized p-values and generalized confidence interval developed by Tsui and Weerahandi (J. Amer. Statist. Assoc. 84 (1989) 602), Weerahandi (J. Amer. Statist. Assoc. 88 (1993) 899), respectively, hypothesis testing and confidence intervals for the ratio of means of two normal populations are developed to solve Fieller's problems. We use two different procedures to find two potential generalized pivotal quantities. One procedure is to find the generalized pivotal quantity based directly on the ratio of means. The other is to treat the problem as a pseudo Behrens-Fisher problem through testing the two-sided hypothesis on 0, and then to construct the 1 - alpha generalized confidence interval as a counterpart of generalized p-values. Illustrative examples show that the two proposed methods arc numerically equivalent for large sample sizes. Furthermore, our simulation study shows that confidence intervals based on generalized p-values without the assumption of identical variance are more efficient than two other methods, especially in the situation in which the heteroscedasticity of the two populations is serious. (C) 2003 Elsevier B.V. All rights reserved.