Statistical Estimation Problems on Restricted Parameters

This book integrates the fundamental theory by the decision theoretic approach to estimate parameters and to predict density when parameters are restricted. It discusses methods to construct the estimators, improving upon the ones usually employed, whether or not the usual ones satisfy the restriction. The topics addressed in the book include the shrinkage estimators of the normal means when the means are restricted by linear inequalities including order restriction. It also investigates how the dimension of the parameter space plays an important role when the restricted maximum likelihood estimation (MLE) is compared with the usual MLE for estimating any linear function of the nonnegative normal means. Further, it presents recent results on the estimation of the common mean and the ordered means of two normal distributions in the presence and absence of order restriction on the variances under the Pitman closeness criterion. In addition to normal distribution, the book discusses the estimation of linear functions of the ordered scale parameters of two gamma distributions and those of the ordered means of two Poisson distributions. Finally, the predictive density estimation for two ordered normal means is discussed under alpha-divergence loss and stochastic domination, and the Pitman closeness domination results are given.