Summary
In this paper we consider a class of rank order tests for the identity of two multiple regression surfaces
(0.1) Here X i = (X (1) i ,X (2) i , i=1,..., N are the observable random variables, c i (1) i ,..., c (p) i , i=1, ...,N are the vectors of known constants, Β's are the regression parameters, and Z i =(Z) (1) i , Z z (2) i , i=1, ..., N are independent and identically distributed random variables. It is assumed that (Z (1)i , Z (2)i ) are either interchangeable random variables or their joint distribution is diagonally symmetric about (0, 0). We wish to test the hypothesis H 0: Β (1)k =Β (2)k , 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 (1)i -X (2)i Zi=Z (1)i -Z (2)i , Βk=Β (1)k -Β (2)k , 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.
Article PDF
Similar content being viewed by others
References
Andrews, F.C.: Asymptotic behavior of some rank tests for analysis of variance. Ann. math. Statistics 25, 724–735 (1954).
Chernoff, H., and I.R. Savage: Asymptotic normality and efficiency of certain non-parametric test statistics. Ann. math. Statistics 29, 972–994 (1958).
Fraser, D.A.S.: Most powerful rank-type tests. Ann. math. Statistics 28, 1040–1043 (1957).
Graybill, F.A.: An introduction to linear statistical models. New York: McGraw-Hill Book Co. 1961.
Hájek, J.: Asymptotically most powerful rank order tests. Ann. math. Statistics 33, 1124–1147 (1962).
Puri, Madan Lal: Asymptotic efficiency of a class of c-sample tests. Ann. math. Statistics 35, 102–121 (1964).
—, and Pranab Kumar Sen: On the asymptotic normality of one sample rank order test statistics. Teor. Verojatn. Primen. 14, 172–177 (1969).
Author information
Authors and Affiliations
Additional information
This work was partially supported by a Sloan Foundation Grant for Statistics, and the National Institutes of Health, Public Health Service, Grant GM-12868.
Rights and permissions
About this article
Cite this article
Puri, M.L., Sen, P.K. On a class of rank order tests for the identity of two multiple regression surfaces. Z. Wahrscheinlichkeitstheorie verw Gebiete 12, 1–8 (1969). https://doi.org/10.1007/BF00538519
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00538519