Showing papers by "Pranab Kumar Sen published in 1970"

A Note on Order Statistics for Heterogeneous Distributions

Abstract: For non-identically distributed random variables, some inequalities on the median and the tails of the distribution of a sample order statistic are derived, and a simple condition for the existence of its moments is studied.

Asymptotic theory of likelihood ratio and rank order tests in some multivariate linear models

Abstract: The purpose of this paper is two-fold: (i) to develop the asymptotic distribution theory of the normal theory likelihood ratio test statistic for the (multivariate) general linear hypothesis problem when the parent distribution is not necessarily normal and (ii) to develop the theory of the multivariate analysis of covariance based on general rank scores. The problem (i) extends the distribution theory of the likelihood ratio statistic developed by the authors in [9] for the multivariate general linear hypothesis problem (for a class of simple alternatives) to the more general case where one has also to deal with a set of concomittant variables, and the problem (ii) extends the results of the authors' earlier paper [8] on the rank order theory of the univariate analysis of covariance to the corresponding multivariate case. Let $\mathbf{Z}_{k\alpha} = (\mathbf{Y}_{k\alpha}, \mathbf{X}_{k\alpha})$; \lbrack where $\mathbf{Y}_{k\alpha} = (Y^{(1)}_{k\alpha}, \cdots, Y^{(p)}_{k\alpha}) \text{and} \mathbf{X}_{k\alpha} = (X^{(1)}_{k\alpha}, \cdots, X^{(q)}_{k\alpha}), p, q \geqq 1 \rbrack, \alpha = 1, \cdots, n_k$ be $n_k$ independent and identically distributed random vectors (i.i.d.r.v.) having a $(p + q)$-variate continuous cumulative distribution function (cdf) $G_k(\mathbf{z}), \mathbf{z} \in R^{p+q}$, for $k = 1, \cdots, c$. It is assumed that $\mathbf{Z}_{11}, \cdots, \mathbf{Z}_{cn_c}$ are mutually independent. Let us denote by $F_k^{(1)}(\mathbf{x})$ the (marginal) joint cdf of $\mathbf{X}_{k\alpha}$, and let $F_k^{(2)}(\mathbf{y}\mid\mathbf{x})$ be the conditional cdf of $\mathbf{Y}_{k\alpha}$, given $\mathbf{X}_{k\alpha} = \mathbf{x}, k = 1, \cdots, c$. As in the univariate theory (cf. [8, 10]), we assume that \begin{equation*} \tag{1.1} F_1^{(1)}(\mathbf{x}) = \cdots = F_c^{(1)}(\mathbf{x}),\quad\mathbf{x} \in R^q\end{equation*} and frame the null hypothesis as \begin{equation*} \tag{1.2} H_0:F_1^{(2)}(\mathbf{y}|\mathbf{x}) = \cdots = F_c^{(2)}(\mathbf{y}|\mathbf{x}).\end{equation*} We may note that under the usual additive model, viz., \begin{equation*} \tag{1.3} F_k^{(2)}(\mathbf{y}|\mathbf{x}) = F^{(2)}(\mathbf{y} - \mathbf{\tau}\_k|\mathbf{x}), \tau_k = (\tau_k^{(1)}, \cdots, \tau_k^{(p)}),\end{equation*} $k = 1, \cdots, c$, the null hypothesis $H_0$ in (1.2) implies that $\tau_1 = \cdots = \tau_c$. We are interested in the set of alternatives that (1.2) does not hold, which under the model (1.3) implies that not all $\tau_k, k = 1, \cdots, c$ are identical. The problem of multivariate analysis of covariance (MANOCA) can be viewed as a special case of the general linear hypothesis problem, considered in Anderson (1958, chapter 8). Two problems arise in this context: (i) how the likelihood ratio (1.r.) test behaves when the parent distribution is not necessarily normal, and (ii) how the multivariate generalizations of the tests considered in [8, 10] compare with the normal theory l.r. test. The purpose of the present investigation is to study these problems thoroughly.

On Some Convergence Properties of One-Sample Rank Order Statistics

Abstract: For a broad class of one-sample rank order statistics, almost sure (a.s.) convergence and exponential bounds for the probability of large deviations, when the basic random variables are not necessarily identically distributed, are established here. In this context, extending a result of Brillinger (1962) to the case of non-iidrv (independent and identically distributed random variables), a result on the a.s. convergence of sample means for a double sequence of random variables is derived. These results are of importance for the study of the properties of sequential tests and estimates based on rank order statistics.

Asymptotic Distribution of a Class of Multivariate Rank Order Statistics

Abstract: A VARIETY of rank tests for the multivariate multi-sample location and scale problems are now available in the literature. Chatterjee and Sen (1964, 1966) considered the median and the rank-sum tests for location. Puri and Sen (1966) extended the rank permutation idea of the previous authors to general Chernoff-Savage (1958) statistics and obtained the multivariate generalizations of the latter theorems (see also Tamura (1966)). Finally, Sen (1965) and Sugiura (1965) considered some other tests based on appropriate U-statistics. The rank statistics to be considered here are neither U-statistics nor statistics that satisfy the regularity conditions of the ChernoffSavage theorems. These of course include the median test statistics as a special case, and more generally, are useful for the censored data problem, where the censoring is made by the pooled sample quantiles. In this respcet, the theory developed here generalizes the results of Gastwirth (1965, 1966) and Sen (1967) to the multivariate case, though by an entirely different approach _based on the asymptotic behaviour of the empirical distribution functions. Multivariate generalization of a different type of tests for censored data (cf. Basu (1967)) follows more easily and is just appended as a remark.

Nonparametric inference in n replicated 2 m factorial expriments

Abstract: For experiments involvingm factors (A 1,…,A m), each at 2 levels (1, 2), and replicated inn(≧2) blocks, a class of nonparametric procedures for estimating and testing the various main effects and interactions are considered. The procedures are based on a simple alignment process and involve the use of some well known rank statistics. Their performance characteristics are compared with those of the standard (normal-theory) parametric procedures. Extensions to confounded or partially confounded designs are also considered.