ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary In this paper we consider a class of rank order tests for the identity of two multiple regression surfaces 1 $$X_i^{\left( j \right)} = \beta _0^{\left( j \right)} + \sum\limits_{k = 1}^p {\beta _k^{\left( j \right)} C_i^{\left( k \right)} + Z_i^{\left( j \right)} ,{\text{ }}j = 1,2,....}$$ (0.1) Here X i = (X i (1) ,X i (2) , i=1,..., N are the observable random variables, c i i (1) ,..., c i (p) , i=1, ...,N are the vectors of known constants, Β's are the regression parameters, and Z i =(Z) i (1) , Z z i (2) , i=1, ..., N are independent and identically distributed random variables. It is assumed that (Z i (1) , Z i (2) ) are either interchangeable random variables or their joint distribution is diagonally symmetric about (0, 0). We wish to test the hypothesis H 0: Β k (1) =Β k (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 X i=X i (1) -X i (2) Zi=Z i (1) -Z i (2) , Βk=Β k (1) -Β k (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 Β k0 for at least one k. A class of permutationally distribution free rank order tests is proposed for this problem. Using the methods of Hájek (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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00538519
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