Showing papers by "Pranab Kumar Sen published in 1997"

Introduction to Rao (1948) Large Sample Tests of Statistical Hypotheses Concerning Several Parameters with Applications to Problems of Estimation

Abstract: Professor Calyampudi Radhakrishna Rao has emerged as one of the most outstanding mathematical statisticians of our time, and his fundamental research contributions covering a wider spectrum of mathematical statistics, design of experiment, combinatorial mathematics, multivariate analysis, information theory, sample surveys, biometry, econometrics, and a variety of other fields, span over a period of little more than 50 years. While C.R. Rao, currently in his mid-seventies, has been embarking on new research projects, the classical work he accomplished in the 1940s is gaining new momentum through vigorous extensions, fruitful applications in diverse fields, and novel interpretations. In Breakthroughs in Statistics, Volume 1, Dr. P.K. Pathak has written an Introduction to another outstanding (1945) article of C.R. Rao, relating to the classical Cramer—Rao Inequality, wherein he has also made detailed comments on Rao’s significant contributions in related fields. In view of this, we will avoid the duplications, and mainly confine ourselves to the scientific aspects of the current article and its profound impact on up-to-date developments in statistical inference and its applications.

Statistical models and asymptotic results for multivariate failure time data with generalized competing risks

Abstract: SUMMARY. This research develops methods which bring together models for multivariate failure time data and competing risks under a unified framework. We refer to the situation where observation continues past the first failure so that the remaining failures can be observed as one of generalized competing risks. Under this more general setup, event-specific proportional hazards models for the first failure are formulated, and given the time and type of the first failure, conditional proportional hazards models for the remaining failures are similarly formulated. Estimation involves cross-classifying subjects into disjoint sets and the (conditional) stratum-specific partial likelihoods are pooled across strata to yield a total (conditional) partial likelihood for the first two failures. Maximum partial likelihood estimators (MPLEs) are obtained and their large-sample properties are examined. Weak convergence results for martingale stochastic processes are used to establish the weak consistency and asymptotic normality of the MPLEs and statistical tests are derived to assess the significance of the dependence of failure times on previous failures and on realized random covariates.

The asymptotic covariance matrix of estimates of factor loadings with normalized varimax rotation

Abstract: We derive the asymptotic covariance matrix of the estimates of normalized varimax-rotated factor loadings. We partition the process of obtaining the estimates of normalized varimax-rotated factor loadings into the three stages: (i) normalization, (ii) raw varimax rotation, and (iii) denormalization, and use the chain rule to combine the matrix of partial derivatives from each of the three stages. For the stage (ii) we make use of the existing formulas for the asymptotic covariance matrix of the estimates of raw varimax-rotated factor loadings.

17 Asymptotic representations and interrelations of robust estimators and their applications

Abstract: Publisher Summary This chapter highlights the asymptotic representations and interrelations of robust estimators and their applications. Various types of robust estimators of location, regression, and scale parameters have been proposed and studied. These include the L-estimators based on linear functions of order statistics, maximum likelihood type (or M-estimators), R-estimators based on appropriate rank statistics, statistical functionals including Hoeffding's U-statistics and von Mises' differentiable statistical functions, and others. Regression quantiles and regression rank scores estimators are the hybrid of robust estimators, such as minimum-distance estimators, Pitman-type (or P-) estimators, and Bayes-type (or B-) estimators. Typically, such robust estimators are nonlinear; they are often defined implicitly in terms of some estimating functions, whose solution may require an iterative procedure. Moreover such classes of robust estimators are not necessarily disjoint, though they have been mostly studied by using apparently different analysis tools. To appreciate fully the robustness picture, the general asymptotic properties and interrelations of various robust estimators in a systematic and unified manner should be looked into.

Stochastic approximation and selection of a conditional extermal function in continuum

Abstract: Stochastic approximation aims to estimate efficiently level crossing or extermal points of a nonparametric regression function in a parametric or nonparametric manner. For bundle strength of parallel filaments (with stochastic length, cross section etc.,) and in some other problems, one has an extended statistical functional where ψ is a suitable function and G(x|y) stands for a conditional distribution function of X, the dependent variate, given the independent variate Y=y. For a compact set C chosen from extraneous considerations, the problem of interest is to locate an optimal y0∊C, in the sense that . Nonparametric estimation of the conditional distribution G(.|y), nonlinear nature of the functional θ(·), and the fact that y is in continuum in C add complexity to possible statistical resolutions and create impasses for a direct application of the well known Bechhofer-Gupta-Sobel methodology. General asympotics for the selection procedures arising in this context are considered with due emphasis on a s...

A change point problem for some conditional functionals

Abstract: A change point problem for certain conditional sample functionals is considered. This type of functionals includes the strength of a bundle of parallel filaments as a special case. The consistency along with a first order representation of the proposed procedure is established under appropriate regularity conditions.