On a class of rank order tests for the identity of two multiple regression surfaces

Abstract

In this paper we consider a class of rank order tests for the identity of two multiple regression surfaces $$X_i^{\left( j \right)} = \beta _0^{\left( j \right)} + \sum\limits_{k = 1}^p {\beta _k^{\left( j \right)} C_i^{\left( k \right)} + Z_i^{\left( j \right)} ,{\text{ }}j = 1,2,....}$$ (1) (0.1) Here Xi= (Xi(1),Xi(2), i=1,..., N are the observable random variables, cii(1),..., ci(p), i=1, ...,N are the vectors of known constants, Β's are the regression parameters, and Zi=(Z)i(1), Zzi(2), i=1, ..., N are independent and identically distributed random variables. It is assumed that (Zi(1), Zi(2)) are either interchangeable random variables or their joint distribution is diagonally symmetric about (0, 0). We wish to test the hypothesis H0: Βk(1)=Βk(2), k=0, 1,...,p, p≧1 (0.2) against the alternative that at least one of the p+1 equalities above is not true. If we make the transformation Xi=Xi(1)-Xi(2)Zi=Zi(1)-Zi(2), Βk=Βk(1)-Βk(2), i=1, ...,N, k=0,1, ...,p then the above problem reduces to that of testing H′0: Βk=0, k=0,1,...,p (0.3) against the alternative that Βk0 for at least one k. A class of permutationally distribution free rank order tests is proposed for this problem. Using the methods of Hajek (1962), based on the concept of contiguity of probability distributions, the asymptotic properties of the proposed tests are studied. These results are used to derive general formulas for the asymptotic relative efficiencies of these tests with respect to one another and to the least squares procedure.