Estimation and Classification Problems under Equality Restrictions on Parameters

Abstract

This thesis deals with the problem of estimation and classification for the parametric stochastic model under the equality restriction on the model parameters. A parametric classification problem is a supervised learning problem in machine learning. The key characteristic of parametric classification problems is that they assume that the input data comes from a specific distribution and that the relationship between the input features and the output classes can be captured using a fixed set of parameters. In this thesis, our primary focus is to estimate the unknown model parameters and study the corresponding classification procedures for the different parametric models. In Chapter 1, we briefly introduce the general problem of classification and estimation of unknown parameters. We provide a comprehensive literature review on classification and estimating the unknown model parameters under equality restriction. In Chapter 2, we discuss some fundamental notations, terminology, and results for the estimation and classification problem that helps to frame the rest of the chapters. In Chapter 3, we revisit the problems of estimation and classification for k(_ 2) normal populations with common mean and ordered variances. In the literature, authors have proposed classification rules based on the Graybill-Deal estimator or its improved version of a common mean under ordered variances. Still, the maximum likelihood estimator (MLE), whose closed-form expression does not exist, is not used for classification purposes. We have proposed the plug-in type restricted version of MLEs of model parameters and utilize these estimators to construct plug-in type classification rules. More importantly, a simulation study has been carried out to numerically compare the performances for all the classification rules, including the existing ones. It has been observed that the classification rules, which are based on the MLE and its restricted version, perform quite satisfactorily (if not the best) compared to other rules. This study is extended to several normal populations with a common mean and ordered variances. To show the practical implication of the proposed methodology, we have considered real-life datasets and obtained the rules’ accuracy. Chapter 4 studies the estimation and classification problem for two inverse Gaussian populations under the equality restriction on model parameters. We have considered the two different situations when the mean parameter is common and the other when the scale like (dispersion ) parameter is common. Under the common mean setup, we proposed restricted-type MLEs and some plug-in type estimators for a common mean parameter. We proposed several plug-in type classification rules using these estimators under order-restricted scale-like parameters. In the sequel, we numerically compared the risk values of all the estimators, which shows that one of the proposed plug-in types restricted MLE outperforms others, including the Graybil-Deal type estimator of the common mean. Our computational results reveal that the proposed classification rules outperform existing rules regarding the probability of correct classification. A similar study has been done for the case when the dispersion parameter is common and mean parameters are different. We proposed the Bayes estimator for the model parameters using the Markov chain Monte Carlo (MCMC) method, which is missing in the literature. The classification rule based on Bayes estimators outperforms the existing rules in terms of the expected probability of correct classification. Chapter 5 considers the estimation and classification problem for two logistic populations under equality restriction on the location or scale parameters. The existence and uniqueness of MLEs of the common location and scale parameters are proved. Further, We constructed several plug-in type classification rules based on the MLEs and the Bayes estimators of model parameters. Moreover, the oracle property for the rules based on the MLEs is established. More importantly, an in-depth simulation study is carried out to compare the performance of proposed model estimators and the corresponding classification rules. Similarly, we considered the estimation and classification problems for two logistic populations with common scales and different mean parameters. In Chapter 6, we studied the classification procedure when the training samples are type-II censored from two exponential populations with a common location and different scale parameters. We have also considered the case when prior information about ordering scale parameters exists. We have developed the classification procedure to classify a censored observation into one of the two exponential populations. Using the original and improved estimators of the common location parameters, we have proposed several classification rules and compared their performances numerically. In Chapter 7, we have generalised these results for the case when the training samples are progressive type-II censored. In this regard, we obtain several estimators of the common location and derive a sufficient condition for improving these estimators, considering with and without order restriction on scale parameters. Further, we have constructed several classification rules to classify a group of progressive type-II censored samples into one of the exponential populations. In both cases, real-life datasets are used to demonstrate the estimation and classification methodologies. Chapter 8 concludes our findings and discusses some of our future research problems.