Minimax estimation of a bounded location parameter using multilevel loss

This dissertation considers minimax estimation under multilevel loss of a bounded location parameter $\theta\in\Theta$ for an arbitrary symmetric distribution $F(z - \theta)$ with monotone likelihood ratio. The parameter space is restricted to a symmetric interval $\Theta = \lbrack {-d},d\rbrack.$ The multilevel loss function is a generalization of the zero-one loss function, and can be represented by a linear combination of component zero-one loss functions. The problem is considered using two multilevel loss functions: zero-alpha-one loss; and n-level loss. Minimax solutions under multilevel loss are shown to be restricted to a class of monotone, antisymmetric decision rules with range $\vert\delta\vert\le\lbrack 0,\ d - e\sb1\rbrack.$ The geometry of the zero-alpha-one problem is analyzed to present the overall picture and categorize the possible cases. A solution technique is developed which allows the problem to be represented by a finite-dimensional linear-algebraic expression. The complete solution to the n-level loss problem is presented for the simplest case which has the truncated MLE as a global minimax rule when sufficient conditions are satisfied. The solution is obtained by showing that the truncated MLE is an almost equalizer rule which is Bayes with respect to a least favorable prior. The equations which determine a least favorable prior are derived and presented. These results are shown to be valid for any symmetric distribution with monotone likelihood ratio. Partial solutions are also presented for some other special cases of the general problem and a method to approximate a minimax rule for an arbitrary continuous bowl-shaped loss function is developed. Finally, the behavior of minimax rules and least favorable priors is examined for various shapes of loss functions and sampling distributions. Evidence is presented which suggests that when the scale parameter of the distribution approaches zero, the minimax rules approach the truncated MLE, and the envelope of the least favorable priors approach the minimal Fisher information prior independent of the shape of the loss function and shape of the distribution