SOME PROBLEMS ON ESTIMATION AFTER SELECTION

Abstract

In the practical world of everyday life, full of complexities, we face baffling problem of making the myriad of judgments. Every situation needs formulation of the problem under an appropriate theoretical model followed by sifting and a careful analysis of the evidence leading to a conclusion whose validity is based on reason. In the progression of scientific reasoning, statistical inference provides the methodology developed to meet these requirements. In many practical situations it may be of interest to select the best (or worst) of k ( 2) available populations (or options), where the quality of a population is defined in terms of unknown parameters associated with it. In the statistical literature, these type of problems are classified as “Ranking and Selection Problem”. A problem of practical interest after selection of the best (or worst) population, using a given selection procedure, is estimation of the worth of the selected population. In the statistical literature, these problems are called “Estimation After Selection Problems”. In this thesis, we study this problem of estimating parameters of the selected population(s) for certain distributions. An application of this theory is shown in this thesis. Most of the previous works are studied under the squared error loss function. In this thesis, some problems are studied under some other loss functions. In this thesis, we study this problem of estimating parameters of the selected population(s) for certain distributions. A brief summary of the thesis is give below. In chapter 1, a review of available work on the problem of estimation after selection is given. A summary of the results in the thesis is also given. In Chapter 2, some basic definitions results and techniques are explained which are of use in this thesis. In Chapter 3, two normal populations with different unknown means and same known variance are considered. The population with the smaller sample mean is selected. Various estimators are constructed for the mean of the selected normal population. Finally, we are compared with respect to the bias and Mean Squared Error (MSE) risks by the method of Monte-Carlo simulation and their performances are analyzed with the help of graphs. In Chapter 4, we consider two competing pairs of random variables (X;Y1) and (X;Y2) satisfying linear regression models with equal intercepts. We describe the model which connects the i ii selection between two regression lines with the selection between two normal populations for estimating regression coefficients of the selected regression line. We apply this model to a problem in finance which involves selecting security with lower risk. We assume that an investor being risk averse always chooses the security with lower risk (or, volatility ) while choosing one of two securities available to him for investment and further is interested in estimating the risk of the chosen security. We construct several estimators and apply the theory to real data sets. Finally, graphical representation of the results is given. In Chapter 5, independent random samples are drawn from two normal populations with same unknown mean and different unknown variances. The population corresponding to the smallest sum of the squared deviations from the mean is selected as the best population. We consider estimation of quantiles of the selected population. Admissible class of estimators for the quantile of the selected population is found in certain subclasses of estimators. The biases and mean squared error risks of these estimators are compared numerically by Monte-Carlo simulation. Finally, the biases and risks of different estimators are represented by graphs. In Chapter 6, we consider independent random samples Xi1; : : : ;Xin drawn from k(k 2) population Pi; i = 1; :::;k. The observations from Pi follows Pareto distribution with an unknown scale (qi) and common known shape parameters. In this chapter, estimation of an unknown scale parameter of the selected population from the given k Pareto population are discussed. The uniformly minimum risk unbiased (UMRU) estimator of scale parameter of the population corresponding to the largest and smallest qi, are determined under the Generalized Stein loss function. Sufficient condition for minimaxity of an estimator of qL(scale parameter of the population corresponding to the largest qi) and qS (scale parameter of the population corresponding to the smallest qi) are given, and we determine that the generalized Bayes estimator of qS is minimax for k = 2. Also, found the class of linear admissible estimators of qL(qS). Further, we demonstrate that the UMRU estimator of qS is inadmissible. Finally some results and discussions are reported. In Chapter 7, we consider P1; : : : ;Pk, k ( 2) independent populations, where Pi follows the uniform distribution over the interval (0;qi) and qi > 0 (i = 1; : : : ;k) is an unknown scale parameter. The population associated with the largest scale parameter is called the best population. The problem of estimating the scale parameter qL of the selected uniform population when sample sizes are unequal and the loss is measured by the squared log error (SLE) loss function is considered. We derive the uniformly minimum risk unbiased (UMRU) estimator of qL under the SLE loss function and two natural estimators of qL are also studied. For k = 2, we derive a sufficient condition for inadmissibility of an estimator of qL. Using these conditions, we conclude that the UMRU estimator and natural estimator are inadmissible. Finally, the risk functions of various competing estimators of qL are compared through simulation. iii In Chapter 8, k( 2) independent uniform populations, over the interval (0;qi) and qi > 0 (i = 1; : : : ;k) be an unknown scale parameter, are considered . In this chapter, we consider the problem of estimating the scale parameter qL of the selected uniform population when sample sizes are unequal, and the loss is measured by the generalized stein loss (GSL) function. The uniformly minimum risk unbiased (UMRU) estimator of qL is derived, and two natural estimators of qL are also studied under the generalized stein loss (GSL) function. The natural estimator xN;2 is proved to be the generalized Bayes estimator with respect to a noninformative prior. For k = 2, we give a sufficient condition for inadmissibility of an estimator of qL and show that the UMRU estimator and natural estimator are inadmissible. A simulation study is also carried out for the performance of the risk functions of various competing estimators. Finally some results and discussions are reported. Finally, Chapter 9 presents the summary and concluding remarks of this thesis and the possible directions of the future scope.