Showing papers by "Pranab Kumar Sen published in 1974"

Weak Convergence of Multidimensional Empirical Processes for Stationary $\phi$-Mixing Processes

Abstract: For a stationary $\phi$-mixing sequence of stochastic $p(\geqq 1)$-vectors, weak convergence of the empirical process (in the $J_1$-topology on $D^p\lbrack 0, 1 \rbrack)$ to an appropriate Gaussian process is established under a simple condition on the mixing constants $\{\phi_n\}$. Weak convergence for random number of stochastic vectors is also studied. Tail probability inequalities for Kolmogorov Smirnov statistics are provided.

Weak Convergence of Generalized $U$-Statistics

Abstract: Wichura (1969) studied an invariance principle for partial sums of a multi-dimensional array of independent random variables. It is shown that a similar invariance principle holds for a broad class of generalized $U$-statistics for which the different terms in the partial sums are not independent. Weak convergence of generalized $U$-statistics for random sample sizes is also studied. The case of (generalized) von Mises' functional is treated briefly.

Sequential rank tests for location

Abstract: In this paper, for the one and two sample location problems, a class of sequential tests based on robust rank order statistics is developed. The proposed tests terminate with probability one for square integrable score functions. Under more stringent regularity conditions and for local alternatives, the OC and ASN of the proposed tests are obtained, and the allied asymptotic relative efficiency results with respect to the sequential probability ratio and likelihood ratio tests are studied.

Almost Sure Behaviour of $U$-Statistics and Von Mises' Differentiable Statistical Functions

Abstract: For $U$-Statistics and von Mises' differentiable statistical functions, when the regular functional is stationary of order zero, almost sure convergence to appropriate Wiener processes is studied. A second almost sure invariance principle, particularly useful in the context of the law of iterated logarithm and the probability of moderate deviations, is also established.

On L p-convergence of U-statistics

Abstract: For a kernel belonging toL p-space,p>1, the rate of convergence of Hoeffding's [5]U-statistic to its expectation is studied; this includes as a special case a similar result on the sample mean previously studied by Chung [3]. Also, anL p-convergence result of Pyke and Root [8] on the sample partial sum is extended toU-statistics.

Sequential rank tests for regression

Abstract: Asymptotic theory of sequential rank order tests (srot) for the regression coefficient in a simple linear regression model is considered here. The proposed srot terminates with probability one. For local alternatives, the oc function of the proposed srot is the same as that of the Wald (sequential probability ratio test). Finally, the asymptotic ASN of the proposed srot is compared with that of the sprt.

On Unbiased Estimation for Randomized Response Models

Abstract: For quantitative randomized response models, optimal unbiased estimation of regular functionals of distribution functions is considered. In this context, distribution theory of Hoeffding's U-statistics and von Mises' differentiable statistical functions is extended to randomized response models. Estimation of the basic distributions is also considered.

The Invariance Principle for One-Sample Rank-Order Statistics

Abstract: Analogous to the Donsker theorem on partial cumulative sums of independent random variables, for a broad class of one-sample rank order statistics, weak convergence to Brownian motion processes is studied here. A simple proof of the asymptotic normality of these statistics for random sample sizes is also presented. Some asymptotic results on renewal theory for one-sample rank order statistics are derived.

Asymptotic theory of sequential tests based on linear functions of order statistics

Abstract: Based on an al.ost sure Wiener process approxi.ation for linear functions of order statistics along with the strong consistency of their variance esti.at.s, a class of sequential tests is considered. These test ter.inate with probability'. Asyaptotic OC and ASN functions are also studied. AHS 1970 Classification No. 62 L 10, 62 G 10.

On a class of simultaneous rank order tests in MANOCOVA

Abstract: For the one-criterion multivariate analysis of covariance (MANOCOVA) model, the rank order tests for the overall hypothesis of no treatment effect considered by Quade [9], Puri and Sen [7] and Sen and Puri [11] are extended here to some simultaneous tests for various component hypotheses. The theory is based on an extension of rank order estimates of contrasts in multivariate analysis of variance (MANOVA) developed by Puri and Sen [6] to the MANOCOVA problem, and is formulated in the set up of Gabriel and Sen [1] and Krishnaiah [3], [4].

On The Estimation of Basic Distributions In Randomized Response Models

Abstract: In randomized response models the estimates of the basic distributions based on the two empirical distributions, though unbiased do not necessarily preserve the characteristics of a distribution function. For this reason, an adjusted estimator is proposed and its properties studied. Confidence bands for the basic distributions are also considered.

Asymptotic Comparison Of A Class Of Multivariate Multiparameter Tests

Abstract: In the context of growth curve models and elsewhere, one is faced with the problem of studying the relative performances of two (or more) tests of significance for some multiparameter hypotheses where the associated test statistics have (limiting) chi-square distributions (central under the null and noncentral under alternative hypotheses) , with possible different degrees of freedom. Since the standard Pitman-efficiency does not work ¦ out well in such a case, three alternative criteria of asymptotic relative efficiency are developed here, and their properties studied. It is noted that the efficiencies for each of the three criteria are products of a scalar adjustment function and a term involving the noncentrality parameters. Values of the scalar adjustment factor are included for selected values of the degrees of freedom and the size of the tests.