Showing papers by "Pranab Kumar Sen published in 1968"

Estimates of the Regression Coefficient Based on Kendall's Tau

Abstract: The least squares estimator of a regression coefficient β is vulnerable to gross errors and the associated confidence interval is, in addition, sensitive to non-normality of the parent distribution. In this paper, a simple and robust (point as well as interval) estimator of β based on Kendall's [6] rank correlation tau is studied. The point estimator is the median of the set of slopes (Yj - Yi )/(tj-ti ) joining pairs of points with ti ≠ ti , and is unbiased. The confidence interval is also determined by two order statistics of this set of slopes. Various properties of these estimators are studied and compared with those of the least squares and some other nonparametric estimators.

On a Class of Aligned Rank Order Tests in Two-way Layouts

Abstract: 1. Summary and introduction. The present investigation is concerned with the formulation of a multivariate approach for the construction of a class of aligned rank order tests for the analysis of variance (ANOVA) problem relating to two-way layouts. The problems of simultaneous testing and testing for ordered alternatives based on aligned rank order statistics are also considered. Various efficiency results pertaining to the proposed tests are studied. Let us consider a two factor experiment comprising n blocks, each block containing p( > 2) plots receiving p different treatments. In accordance with the two-way ANOVA model, we express the yield Xij of the plot receiving the jth treatment in the ith block as

Asymptotic Normality of Sample Quantiles for $m$-Dependent Processes

Abstract: The usual technique of deriving the asymptotic normality of a quantile of a sample in which the random variables are all independent and identically distributed [cf. Cramer (1946), pp. 367-369] fails to provide the same result for an $m$-dependent (and possibly non-stationary) process, where the successive observations are not independent and the (marginal) distributions are not necessarily all identical. For this reason, the derivation of the asymptotic normality is approached here indirectly. It is shown that under certain mild restrictions, the asymptotic almost sure equivalence of the standardized forms of a sample quantile and the empirical distribution function at the corresponding population quantile, studied by Bahadur (1966) [see also Kiefer (1967)] for a stationary independent process, extends to an $m$-dependent process, not necessarily stationary. Conclusions about the asymptotic normality of sample quantiles then follow by utilizing this equivalence in conjunction with the asymptotic normality of the empirical distribution function under suitable restrictions. For this purpose, the results of Hoeffding (1963) and Hoeffding and Robbins (1948) are extensively used. Useful applications of the derived results are also indicated.

Robustness of Some Nonparametric Procedures in Linear Models

Abstract: For the random variables $X_{ij}(i = 1, \cdots, N; j = 1, \cdots, r)$ consider the linear model \begin{equation*}\tag{1.1} X_{ij} = \mu + \beta_i + \tau_j + Y_{ij} (\sum \beta_i = 0, \sum\tau_j = 0),\end{equation*} where the $\tau$'s are treatment effects, the $\beta$'s are nuisance parameters (block effects), and the $Y_{ij}$'s are error components. Nonparametric procedures for estimating and testing contrasts in the $\tau$'s, based on the Wilcoxon signed rank statistics, are due to Lehmann (1964), Hollander (1967) and Doksum (1967), among others. These rest on the assumption that the $Y_{ij}$'s are independent with a common continuous distribution. Since these procedures are actually based on the paired differences $X^\ast_{ijk}$, defined by (2.1), they are unaffected by the addition of a random variable $V_i$ to $\beta_i$ (or to $Y_{ij}$) for $i = 1, \cdots, N$. The object of the present investigation is to show that these procedures are valid even if $Y_{i1}, \cdots, Y_{ir}$ are interchangeable random variables, for each $i( = 1, \cdots, N)$. It may be noted that if in (1.1) the superimposed random variable $V_i$ is absorbed in $Y_{ij}$, then of course $Y_{i1}, \cdots, Y_{ir}$ are interchangeable, but the interchangeability of $Y_{i1},\cdots, Y_{ir}$ does not necessarily imply that $Y_{ij} = W_{ij} + V_i$, where $W_{ij}$'s are independent and identically distributed random variables (iidrv). In fact, in `mixed model' experiments, interchangeability of $Y_{i1}, \cdots, Y_{ir}$ (of quite arbitrary nature) may arise when there is no block versus treatment interaction [cf. Koch and Sen (1968) for details]. It is also shown that the procedures mentioned above are robust against possible heterogeneity of the distributions of the error vectors $\mathbf{Y}_i = (Y_{i1}, \cdots, Y_{ir}), i = 1, \cdots, N$. This situation may arise when the block effects are not additive or the errors are heteroscedastic. Thus, in this paper the independence of the errors is replaced by within block symmetric dependence, while the additivity of the block effects and homoscedasticity of the errors are relaxed.

Some aspects of the statistical analysis of the 'mixed model'.

Abstract: This paper deals with the statistical analysis (both parametric and non-parametric) of 'mixed model' experiments. The general structure of such experiments involves n randomly chosen subjects who respond once to each of p distinct treatments. Thus the subject or block effects are random and treatment effects are fixed. The hypothesis of no treatment effects is considered under several different combinations of assumptions concerning the joint distribution of the observations corresponding to each of the particular subjects. For each situation, an appropriate test procedure is discussed and its properties studied. The different methods considered in the paper are illustrated in detail in two numerical examples. These examples have been chosen to illustrate the relative performances of the different test criteria for a situation in which the null hypothesis is essentially true (Example 1) and for a situation in which the null hypothesis is essentially false (Example 2). The reader may wish to begin by studying these examples for a better understanding of the theory. Finally, the section on examples contains algorithms for the efficient computation of the various test criteria. A computer program based on these algorithms has been written and can be made available to any interested persons.

On Chernoff-Savage Tests for Ordered Alternatives in Randomized Blocks

Abstract: The object of the present paper is to generalize the results of Hollander [4] (concerning rank tests for randomized blocks for ordered alternatives) to Chernoff-Savage [1] class of tests which includes his test as a special case. Allied efficiency results are also studied.

Paired comparisons for paired characteristics

Abstract: The present investigation is concerned with the proposal and study of a class of nonparametric paired comparison tests for the hypothesis of no differences among several objects with respect to a pair of characteristics.