Resumen
The problem of testing the equality of two normal means when variances are not known is called the Behrens–Fisher Problem. This problem has three known exact solutions, due, respectively, to Chapman, to Prokof’yev and Shishkin, and to Dudewicz and Ahmed. Each procedure has level alpha and power beta when the means differ by a given amount delta, both set by the experimenter. No single-sample statistical procedures can make this guarantee. The most recent of the three procedures, that of Dudewicz and Ahmed, is asymptotically optimal. We review the procedures, and then compare them with respect to both asymptotic efficiency and also (using simulation) in finite samples. Of these exact procedures, based on finite-sample comparisons the Dudewicz–Ahmed procedure is recommended for practical use.

Resumen
In this paper, we investigate geometric properties of local polynomial regression and show that the class of global polynomial models is nested within the class of functions generated by fitting local polynomials. The geometric properties are then utilized to construct nonparametric F-tests for testing whether a regression relationship is a polynomial function. The proposed F-tests can be seen as a “calculus” extension of the classical F-tests with analysis of variance interpretations. With the normality assumption, the test statistic is shown to have asymptotic F-distributions under the null hypothesis and fixed alternatives. Simulation results illustrate that the asymptotic null F-distribution approximates well in finite sample cases and the proposed tests enjoy robustness against heteroscedasticity and non-normality as do the classical F-tests.