Estimating Quantiles and Common Parameter in Certain Stochastic Models

Abstract

The problem of estimation of a function of parameters or particularly quantiles has received considerable attention by several researchers in the recent past, due to its practical applications and the theoretical challenges involve in it from a decision theoretic point of view. Specifically the problem has been studied extensively when the underlying distribution is either a two parameter exponential or normal. Exponential quantiles are widely used in the study of reliability, life-testing, survival analysis and related areas.Further quantiles are used to characterize the distributions and hence help in comparing various populations. Similarly, the problem of estimating common parameter has a long history in the literature of statistical inference.Particularly in the case of normal and exponential the problem has been studied extensively. However, apart from normal and exponential a little attention has been paid in estimating quantiles and common parameter.For example, most of the practical data follow
life time distribution such as logistic, gamma and inverse Gaussian etc. Keeping in view of the above we study the following problems.
In Chapter 1, we discuss the literature review on the related subjects and give a motivation for studying the problem under consideration. Also the summery of the results obtained in the thesis has been discussed there.
In Chapter 2, we discuss some basic results from decision
theoretic as well as classical point of view on estimation theory.
In Chapter 3, the problem of estimating quantiles of two normal distributions has been considered with common mean and ordered variances. Exploiting the prior information
regarding the ordering of the variances, several improved estimators for the common mean have been proposed.Using these new improved estimators for the common mean,improved
estimators for the quantiles have been derived. These improved estimators beat their unrestricted counterparts in terms of the risk values. Moreover, a plug-in type restricted MLE for the quantile has been proposed, which was shown to dominate the MLE with respect to the quadratic loss. Further, using invariance principle, equivariant estimators have been derived and an inadmissility result has been proved there. Further, improved estimators have been
derived for this class of estimators. Finally, all the improved estimators have been compared through a Monte-Carlo simulation method and the recommendations have been made there.The percentage of risk reduction after considering the order restriction on the variances hasbeen noticed which are quite significant.
In Chapter 4, we have considered the simultaneous estimation of quantiles of two normal populations with a common mean and the variances are ordered.
In Chapter 5, the problem of estimation of common scale and shape parameter of two gamma populations have been considered. It has been noticed that the MLE does not have
a closed form. The MLE of the parameters have been obtained by solving a system of equations using certain numerical techniques. Further asymptotic confidence intervals have
been obtained using the Fisher information matrix. Moreover, we have derived certain Bayes estimators using various priors such as vague prior, Jeffreys prior and the conjugate prior. It has also been observed that the closed form of these Bayes estimators do not exist too. Using certain approximations for the ratios of the integrals,we have obtained the approximate Bayes estimators with respect to the squared error loss using all these priors. Finally all the proposed estimators have been compared through their bais and the mean squared errors.
In Chapter 6, the problem of estimating common dispersion parameter of several inverse Gaussian distributions has been considered when the location parameters are unknown. We
derive the maximum likelihood estimator (MLE) of the associated parameters. Using the fisher information matrix, an asymptotic confidence interval for the common dispersion parameter as well as other location parameters has been obtained.Further Bayes estimators with respect to
non-informative, Jeffrey’s and conjugate priors have been considered. We observe that unlike the MLE, the closed form of these Bayes estimators do not exist. Using certain approximations for the ratio of the integrals, approximate Bayes estimators have been obtained. All the proposed estimators are compared in terms of their bias and mean squared errors (MSEs) through simulation in the case of two and three populations.Finally a real data set has been
considered to demonstrate the potential application of our model.
In Chapter 7, we consider the estimation of the common scale and quantiles of two logistic distributions. First we consider the estimation of common scale parameter of two logistic distributions when the location parameters are unknown. The MLE as well as the asymptotic confidence intervals for the associated parameters have been derived numerically. Bayes estimators with respect to various priors using squared error and linex loss functions have been derived. Finally all these estimators have been compared through Monte-Carlo simulation method numerically using bias and the mean squared errors.
Chapter 8,concludes the results obtained in the thesis and discusses some of our future research works.