Showing papers by "Pranab Kumar Sen published in 1976"

The Locally Optimal Combination of Independent Test Statistics

Abstract: A class of locally optimal tests based on linear functions of certain transformations of independent test statistics is proposed and studied. These overall tests are useful in combining independent tests for certain multivariate or multiparameter hypotheses where local optimality is interpreted in a specialized sense. For the two-sample case, along with the expressions for the null distributions of the test statistics, tabulations of percentile points are provided.

Weak Convergence of Progressively Censored Likelihood Ratio Statistics and its Role in Asymptotic Theory of Life Testing

Abstract: Progressive censoring schemes are often adopted in clinical trials and life testing problems with a view to monitoring the experiment from the start with the objective of a possible early termination of the experiment depending on the cumulative evidence at its various stages. Along with a basic martingale property, a Wiener process approximation for progressively censored likelihood ratio statistics is established here and the same is incorporated in the formulation of some asymptotic sequential tests for the life testing problem.

A Note on Invariance Principles for Induced Order Statistics

Abstract: Weak convergence of a sequence of two-dimensional time parameter stochastic processes constructed from partial sums of induced order statistics to a standard Brownian sheet process is established.

A Two-Dimensional Functional Permutational Central Limit Theorem for Linear Rank Statistics

Abstract: Some two-dimensional time-parameter stochastic processes are constructed from a sequence of linear rank statistics by considering their projections on the spaces generated by the (double) sequence of anti-ranks. Under appropriate regularity conditions, it is shown that these processes weakly converge to Brownian sheets in the Skorokhod $J_1$-topology on the $D^2\lbrack 0, 1 \rbrack$ space. This unifies and extends earlier one-dimensional invariance principles for linear rank statistics to the two-dimensional case. The case of contiguous alternatives is treated briefly.

Invariance Principles for the Coupon Collector's Problem: A Martingale Approach

Abstract: : For the coupon collector's problem, invariance principles for the partial sequence of bonus sums after n coupons as well as for the waiting times to obtain the bonus sum t are studied through a construction of a triangular array of martingales related to these sequences and verifying the invariance principles for these martingales. This approcah provides simpler and neater proofs than given in Rosen (1969, 1970) and strengthens his convergence of finite dimensional results to those of weak invariance principles. (Author)

Asymptotically Optimal Rank Order Tests For Progressive Censoring

Abstract: For progressive censoring schemes, Chatterjee and Sen (1973) have considered a general class of time-sequential tests based on suitable rank order statistics and have studied their asymptotic relative efficiency results in the light of the Bahadur-efficiency ,criterion. The object of the present note is to study the power properties of these tests for local (contiguous) alternatives, and, within this framework, to throw some light on their asymptotic optimality. Unlike the case of the Bahadur-efficiency, for our purpose, we confine ourselves to local altsrntatives and also need an additional regularity condition which localizes (to some extent) the scope of the comparisons. Along with the preliminary notions, the rank order tests are briefly presented in Section 2. In Section 3, a general class of contiguous location and scale alternatives is introduced and the asymptotic power properties of the rank order tests for such alternatives are studied. Asymptotic optimality results are then considered in Section 4.

An Almost Sure Invariance Principle for the Extrema of Certain Sample Functions

Abstract: For a general class of statistics expressible as extrema of certain sample functions, an almost sure invariance principle, particularly useful in the context of the law of iterated logarithm and the probabilities of moderate deviations, is established, and its applications are stressed. AMS 1970 Classification Numbers: 60FlO, 60F15

Asymptotically Optimal Estimators of General Parameters in Randomized Response Models

Abstract: Summary General estimable parameters in randomized response models have been characterized as functionals of several independent distributions for which the so-called generalized U-statistics are known to be optimal unbiased estimators. The object of the present paper is to determine the individual sample sizes in such a way that the efficiency of these generalized U-statistics is maximized in some sense. Specifically, along the general lines of Williams and Sen (1973, 1974), the problem of sequential estimation of general parameters in randomized response models is formulated and some applications to live data are made. selects one of the two questions at random without revealing the outcome to the interviewer. The probabilities of selecting the two questions are set beforehand. Thus, the randomization affords protection to the respondent and hence potential embarrassments and stigma for answering a sensitive question are avoided, and therefore, the primary reason for either a refusal or an evasive answer does not exist. In a recent paper (Sen, 1974), the present author has formulated the theory of estimation of general estimable parameters in a RRM. It turns out that such estimable parameters (of individual or composite distributions) are functionals of the two randomized response distributions. This characterization enables one to make use of the theory of generalized U-statistics and von Mises' differentiable statistical functions for studying the general properties of natural estimators in RRM. The current investigation attempts to provide sequential solutions to some related problems of estimation in RRM. Two basic problems are considered here. First, for the problem of point estimation of one or more estimable parameters in a RRM, one is naturally interested in choosing the individual sample sizes in such a manner that the precision of the estimators is maximized in some sense. Since the covariance matrix of the individual kernels is not generally known, no fixed sample size procedure sounds feasible. A sequential procedure based on sequentially updated versions of estimators of this covariance matrix is developed here, and this procedure is optimal in a certain sense. Second, for unknown covariance matrices, a fixed-width confidence interval for an estimable parameter in a RRM can also be achieved only in a sequential setup. In this context, the problem of determining the individual sample sizes so as to maximize the efficiency (or minimize the average sample number of the total sample) is considered.

Chi-square tests for general models under progressive censoring with batch arrivals

Abstract: For some batch-arrival models relating to categorical data under timesequential studies, suitably progressively censored tests based on chisquare statistics are proposed and studied. The necessary (asymptotic) distribution theory is considered for the null as well as local alternative hypotheses situations. To facilitate comparisons of the different proposed tests, a numerical illustration is presented at the end. AMS 1970 Classification No: 62E20, 62FOS, 62GIO, 62L99.