Minimax estimation of a discrete location parameter for a continuous distribution

The subject of this research is the following stochastic model: $Z = \theta + V$. The random variable $Z$ is a measurement of the discrete location parameter $\theta$ in continuous, additive noise $V$. The possible values of $\theta$ are $\{0,\pm u,\pm2u,\...,\pm Nu\},$ where $N$ is a positive integer and $u$ is a positive number. The distribution of $Z$, denoted $F\sb{Z}(\cdot\vert\theta)$, is continuous and increasing on $\Re$ and has a continuous density. In a standard-estimation problem, the distribution $F\sb{Z}(\cdot\vert\theta)$ is known. In a robust-estimation problem, the distribution $F\sb{Z}(\cdot\vert\theta)$ is uncertain: It is an unknown member of an uncertainty class of distributions. In either case, the goal of this research is to find a minimax estimator of the location parameter $\theta$ from an observation $Z$. This goal subsumes the identification of equalizer decision rules and Bayes decision rules for estimating $\theta$. The loss function in the underlying statistical decision problem is the zero-one loss function with error tolerance $e$ (a non-negative integer):(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$L\sb{e}(\theta,\\theta):{=}\cases{0 &\rm if\ \ $\vert\theta-\\theta\vert\le eu$\cr1 &\rm if\ \ $\vert\theta-\\theta\vert\u3e eu$\cr}$$(TABLE/EQUATION ENDS)The loss is independent of the noise distribution. Different additional assumptions define the specific problems considered. Chapters 2, 3, and 4 consider standard estimation. Chapter 2 discusses standard estimation as a problem in statistical decision theory. Chapter 3 assumes that the observable $Z$ has a monotone likelihood ratio. Its decision rules are monotonic. In contrast, chapter 4 assumes that the observable has a Cauchy distribution, which does not have a monotone likelihood ratio. Its decision rules are not monotonic. Chapters 5, 6, and 7 consider robust estimation. Chapter 5 formulates robust estimation as a problem in statistical decision theory. Chapter 6 extends the results for the standard estimation of chapter 3 to robust estimation, and consequently its decision rules are monotonic. Chapter 7 considers a simple robust-estimation problem for which the extension of chapter 6 does not apply. Its decision rules are not monotonic. In all of these problems, the decision rules are nonrandomized step functions. The steps occur at points determined by nonlinear systems of equations in the noise distribution. Chapters 2-7 summarize the main results. Chapters 8-21 give the analysis