Showing papers by "Pranab Kumar Sen published in 1969"

A Class of Rank Order Tests for a General Linear Hypothesis

Abstract: For a general multivariate linear hypothesis testing problem, a class of permutationally (conditionally) distribution-free tests is proposed and studied. The asymptotic distribution theory of the proposed class of test statistics is studied along with a generalization of the elegant results of Hajek (1968) to the multistatistics and multivariate situations. Asymptotic power and optimality of the proposed tests are established and a characterization of the multivariate multisample location problem [cf. Puri and Sen (1966)] in terms of the proposed linear hypothesis is also considered.

Analysis of Covariance Based on General Rank Scores

Abstract: The purpose of the present investigation is to develop a class of rank order tests for the equality of treatment-effects in the presence of a set of concomitant variates for the analysis of covariance (ANOCA) model relating to completely randomized layouts. The proposed procedures are shown to be conditionally distribution-free and to have some desirable large sample properties.

On a Class of Rank Order Tests for the Parallelism of Several Regression Lines

Abstract: For the regression model $Y_{ u i} = \alpha + \beta C_{ u i} + \epsilon_{ u i}, i = 1, \cdots, N_ u$, where the $\epsilon_{ u i}$ are independent and identically distributed random variables (iidrv), optimum rank order tests for the hypothesis that $\beta = 0$ are due to Hoeffding (1950), Terry (1952) and Hajek (1962), among others. In the present paper, the theory is extended to the problem of testing the homogeneity of the regression coefficients from $k(\geqq 2)$ independent samples. Allied efficiency results are also presented.

On nonparametric T-method of multiple comparisons for randomized blocks

Abstract: Some nonparametric generalizations of Tukey’s [9]T-method of multiple comparisons are considered for randomized blocks and the allied efficiency results are studied. For this, the distribution theory of aligned rank order statistics developed in [6], [7] is extended for multiple comparisons along the lines of [5] which deals with one-way layouts.

On the Asymptotic Normality of One Sample Rank Order Test Statistics

Abstract: The asymptotic normality of a class of one sample rank order test statistics is established This class includes among other test statistics the well-known normal scores test of symmetry developed by Fraser [2] and the Wilcoxon paired comparison test [8]

On the asymptotic theory of rank order tests for experiments involving paired comparisons

Abstract: The problem of testing the hypothesis of no difference among several treatments for the case when the comparisons between the treatments is possible only in pairs has been considered by Durbin [3], Bradley and Terry [1], Elteren and Noether [4], and Mehra and Puri [6], among others. Following the lines of Sen and Puri [9], a new approach to the asymptotic theory of rank order tests for this problem is developed. This avoids the unnecessarily complicated and lengthy conditional approach of Mehra and Puri [6] and also simplifies the proofs considerably.

A Generalization of the T-Method of Multiple Comparisons for Interactions

Abstract: Tukey's [4] T-(maximum modulus) method of multiple comparisons is applicable to homoscedastic and equally correlated normally distributed random variables. For multiple comparisons on interaction effects (in factorial experiments), the allied random variables are not equally correlated. Nevertheless, it is shown that the T-method holds in this situation. The theory is illustrated by an example on a partially balanced incomplete block (PBIB) design.

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.

On a Robustness Property of a Class of Nonparametric Tests Based on U­Statistics*

Abstract: A CLASS of nonparametric tests based on Hoeffding's (1948) U-statisticsis shown to be robust for heterogeneity of the distributions of the sample observations ; the theory is illustrated by means of some known tests. Let X10 • •• , Xn be independent (real or vector valued) random variables having cumulative distribution functions (cdf's) F1(x), ... Fn(x) respectively. Let n be the set of all cdf's closed under the following (countable) set operation: if FidJ. for i = 1, 2, ... , then (1/n)Lt~tFtt:!2 for all n;>l. It is assumed that Fn = (F1, ••. , Fn) is an element of the product set Qn, and the null hypothesis states that