Estimating Parameters under Equality and Inequality Restrictions

Abstract

The problem of estimating statistical parameters under equality or inequality (order) restrictions has received considerable attention by several researchers due to its vast applications in various physical, industrial and biological experiments. For example, the problem of estimating the common mean of two normal populations when the variances are unknown has a long history and is popularly known as “common mean problem”. This problem is also referred as Meta-Analysis, where samples (data) from multiple sources are combined with a common objective. The “common mean problem” has its origin in the recovery of inter-block information when dealing with Balanced Incomplete Block Designs (BIBDs) problems. In this thesis, we study problem of estimating parameters and quantiles of two or more normal and exponential populations when the parameters are equal or ordered from decision theoretic point of view.

In Chapter 1, we give the motivation and do a detailed review of literature for the following problems. In Chapter 2, we discuss some basic definitions and decision theoretic results which are useful in developing the subsequent chapters. In Chapter 3, the problem of estimating the
common mean of two normal populations has been considered when it is known a priori that the variances are ordered. Under order restriction on the variances, some new alternative estimators have been proposed including one that uses the maximum likelihood estimator (MLE). These new estimators beat some of the existing popular estimators in terms of stochastic domination as well as Pitman measure of closeness criterion. In Chapter 4, we have considered the
problem of estimating quantiles for k( 2) normal populations with a common mean. A general result has been proved which helps in obtaining better estimators. Introducing the principle of invariance, sufficient conditions for improving estimators in certain equivariant classes have been derived. As a consequence some complete class results have been proved. A detailed simulation study has been carried out in order to numerically compare the performances of all the proposed estimators for the cases k = 3 and 4: A similar type of result has also been obtained
for estimating the quantile vector. In Chapter 5, we deal with the problem of estimating quantiles and ordered scales of two exponential populations under equality assumption on
the location parameters using type-II censored samples. First, we consider the estimation of quantiles of first population when type-II censored samples are available from two exponential populations. Sufficient conditions for improving equivariant estimators have been derived
and as a consequence improved estimators have been obtained. A detailed simulation study has been carried out to compare the performances of improved estimators along with some
of the existing ones. Further, we deal with the problem of estimating vector of ordered scale parameters. Under order restriction on the scale parameters, we derive the restricted maximum likelihood estimator for the vector parameter. We obtain classes of equivariant estimators and prove some inadmissibility results. Consequently, improved estimators have been derived. Finally a numerical comparison has been done among all the proposed estimators.
In Chapter 6, the problem of estimating ordered quantiles of two exponential populations is considered assuming equality of location parameters. Under order restriction, we propose
new estimators which are the isotonized version of some baseline estimators. A sufficient condition for improving equivariant estimators are derived under order restriction on quantiles.Consequently, estimators improving upon the baseline estimators are derived. Further, the
problem of estimating ordered quantiles of two exponential populations is considered assuming equality of the scale parameters using type-II censored samples. Under order restrictions on the quantiles, isotonized version of some existing estimators have been proposed. Bayes estimators
have been derived for the quantiles assuming order restriction on the quantiles. In Chapter 7, we consider the estimation of the common scale parameter of two exponential populations when the location parameters satisfy a simple ordering. Bayes estimators using uniform prior and a
conditional inverse gamma prior have been obtained. Finally all the derived estimators have been numerically compared along with some of the existing estimators. In Chapter 8, we give an overall conclusion of the results obtained in the thesis and discuss some of our future research work.