A Class of ADF Tests for Subhypothesis in the Multiple Linear Regression

Abstract

In the regression model $Y = \beta_1x_1 + \beta_2x_2 + Z$ a class of asymptotically distribution free (ADF) tests for testing $H_0:\beta_1 = 0$ when $\beta_2$ is unknown is given. It turns out that if one uses Wilcoxon type tests then there is no reasonable distribution of $Z$ under which the test would be ADF unless $x_1$ and $x_2$ are orthogonal to each other when centered. On the other hand if one is sampling from double exponential, then the class of tests is reasonably large. The tests of the Freund-Ausari type, Mood-type, among others, are in the class. Section 1 consists of introduction, notation and assumptions. In Section 2, we prove a uniform continuity Theorem 2.2 for rank statistics. Theorem 2.1 and Lemma 2.3 are proved before Theorem 2.2. These latter two results are based on the work done in [4]. Theorem 2.4 gives the desired result. Finally generalization to the situation where one has multiple linear regression model and has more than one parameter under $H_0$ with more than one unknown is mentioned.