A Distribution-Free Test for Regression Parameters

Abstract

Brown and Mood [1], [4] have recently given convenient distribution-free methods of testing and setting up confidence regions for the parameters of a linear regression model. Their technique, which is based on the use of medians, allows the parameters to be considered singly or simultaneously as required. Theil [5] gives two methods of constructing confidence intervals for single parameters, a "complete" method using rank correlation which is valid under the conditions assumed by Brown and Mood, and an "incomplete" method valid under wider conditions but not making full use of the data. For several parameters simultaneously, he obtains confidence regions in the weak sense of covering the true parameter point with probability not less than an assigned value. In the present paper we give a new distribution-free test for the hypothesis that all regression parameters have specified values, assuming only that the residuals are independent and have probability $\frac{1}{2}$ of being positive or negative. It can be used to set up exact confidence regions for the true parameter point. The new test avoids a defect which is shown to appear in the corresponding Brown and Mood test when the sample is not large. The distribution of the test statistic is found explicitly only for the case of two parameters, though in principle the idea extends to any number of parameters. The presence of repeated values of the independent variable necessitates certain modifications in the test, and a method of computing the appropriate distributions in such cases is described.