Saibal Chattopadhyay

Asymmetric Penalized Prediction Using Adaptive Sampling Procedures

Abstract Prediction using a multiple-regression model is addressed when the penalties for overpredicting and underpredicting the true future value are not equal. Such asymmetric penalty functions are appropriate in many practical situations. If one imposes some preassigned precision on the prediction procedure, it is shown that in the presence of nuisance parameters in the model, the sample size needed to achieve the fixed precision is unknown. Some adaptive multistage sampling techniques are discussed that offer solutions to this problem. A prediction procedure based on a purely sequential sampling scheme is introduced, followed by a batch sequential scheme. Finally, a real-life example is provided to illustrate the use of these procedures, and computational evidence is supplied to demonstrate the efficiency of the latter procedure compared to the former one.

Three-stage and accelerated sequential point estimation of the normal mean using LINEX loss function

Consider a normal population with unknown mean μ and unknown variance σ2. We estimate μ under an asymmetric LINEX loss function such that the associated risk is bounded from above by a known quantity w. This necessitates the use of a random number (N) of observations. Under a fairly broad set of assumptions on N, we derive the asymptotic second-order expansion of the associated risk function. Some examples have been included involving accelerated sequential and three-stage sampling techniques. Performance comparisons of these procedures are considered using a Monte-Carlo study.

Sequential methodologies for comparing exponential mean survival times

Sequential methodologies are derived for comparing arbitrary number of exponential mean survival times when the loss function is squared error plus the cost of sampling. From the theoretical point of view, we have given derivations of the asymptotic second-order expansi6n of the. risk function and bias associated with the proposed sequential estimator. We have noted, via computer simulations, that these expansions lead to useful guidelines even for small as well as moderate sample sizes.

Further developments in estimation of the largest mean ofK normal populations

We revisit the bounded maximal risk point estimation problem as well as the fixed-width confidence interval estimation problem for the largest mean amongk(≥2) independent normal populations having unknown means and unknown but equal variance. In the point estimation setup, we devise appropriate two-stage and modified two-stage methodologies so that the associatedmaximal risk can bebounded from aboveexactly by a preassigned positive number. Kuo and Mukhopadhyay (1990), however, emphasized only the asymptotics in this context. We have also introduced, in both point and interval estimation problems,accelerated sequential methodologies thereby saving sampling operations tremendously over the purely sequential schemes considered in Kuo and Mukhopadhyay (1990), but enjoying at the same time asymptotic second-order characteristics, fairly similar to those of the purely sequential ones.

Efficient sequential sampling strategies for environmental monitoring

Assessments of resources at risk to anthropogenic pollution require extensive environmental monitoring. In addition, such assessments are required to have a long-term monitoring component in order to evaluate not only the status but also the trend of the resources at risk to ecological stresses. There is a need to identify statistical methodologies that would provide effective and cost-saving environmental monitoring designs, since such monitoring surveys are very expensive. In this paper the purely sequential, accelerated sequential, and three-stage procedures are evaluated as effective fixed-precision sampling procedures for environmental monitoring. Current monitoring designs utilize a sampling methodology where each resource is assigned a population inclusion probability, with the intent of describing the distribution of the whole population of resources at risk to anthropogenic environmental stresses. This study assumes that existing designs accurately describe the population distribution. A simultaneous fixed-precision estimation procedure is developed as an efficient method of estimating practically relevant percentiles of the cumulative distribution function, using water quality data from the Eastern Lake Survey as a lake population distribution. Accelerated sequential and three-stage procedures are shown to be better alternatives to the purely sequential procedure, requiring fewer sampling operations without any substantial loss of efficiency. Depending upon the precision required, all procedures showed potential reductions in sample size by as much as 60%. These types of designs for environmental monitoring are expected to be advantageous in national monitoring efforts directed toward the assessment of the status and trends of various ecological indicators.

Estimation of a Linear Function of Multinormal Mean Vectors by Combining Available Independent Sequential Studies

Suppose we have k independent p-variate normal populations having unknown mean vectors µi and dispersion matrices of the form σ   i 2 H i ,   w h e r e   σ   i 2 is unknown and Hi is a known p x p positive definite matrix, i = 1, 2, ... k. The investigators in these k populations may possibly have implemented different sequential or multistage sampling methodologies to estimate the mean vectors individually, having certain notions of “fixed-precision” tied with their individual problems of statistical inference. In this paper, we examine how such already available sample resources obtained from independent, but comparable studies can be fruitfully combined in order to provide a “fixed-size” confidence region for a linear combination of the k mean vectors. We discuss appropriate asymptotic second-order characteristics for the coverage probability under a fairly broad set of assumptions on the stopping variables. An example has been included involving purely sequential, accelerated sequential and three-stage sampling techniques.

Simultaneous Estimation of Proportions in a Finite Population

Sequential and multistage sampling strategies via simple random sampling without replacement, are proposed for simultaneously estimating several proportions in a finite population. Various asymptotic first-order properties are addressed, while some limited moderate sample performance have also been included. AMS (1980) Subject Classification: Primary 62L99; Secondary 62L12

Sequential Point Estimation Procedures for the Generalized Life Distributions

SYNOPTIC ABSTRACT Sequential procedures are developed for the point estimation of the parameter of the generalized life distributions under (i) squared-error loss function plus linear cost of sampling and (ii) squared-error loss function plus log-cost function. Second-order approximations are obtained for the risk and theoretical justifications to some of the numerical findings of Starr and Woodroofe (1972) is given. ‘Improved’ estimators are proposed and their dominance over the usual estimators is established.