Robust location estimation for MLR and non-MLR distributions.

摘要

We study the problem of estimating an unknown parameter {dollar}theta{dollar} from an observation of a random variable {dollar}Z = theta + V{dollar}. This is the location data model; V is random noise with absolutely continuous distribution F, independent of {dollar}theta{dollar}. The distribution F belongs to a given uncertainty class of distributions {dollar}{lcub}cal F{rcub}{dollar}, {dollar}vert{lcub}cal F{rcub}vertgeq 1{dollar}. We seek robust minimax decision rules for estimating the location parameter {dollar}theta{dollar}. The parameter space is restricted--a known compact interval. The minimax risk is evaluated with respect to a zero-one loss function with a given error-tolerance e. The zero-one loss uniformly penalizes estimates which differ from the true parameter by more than the threshold e (these are unacceptable errors). The minimax criterion with zero-one loss is suitable for modeling problems for which it is desirable to minimize the maximum probability to getting unacceptable errors. As a consequence of this approach we obtain fixed size confidence intervals with highest probability of coverage.; We consider the distribution-dependent function {dollar}{lcub}f(x + 2e){rcub}over{lcub}f(x){rcub}{dollar}, where e is the error-tolerance and f is the noise density. We distinguish two different types of problems (involving two different types of distributions) based on the behavior of this ratio: (I) Type {dollar}{lcub}cal M{rcub}{dollar}-problems ({dollar}{lcub}cal M{rcub}{dollar}-distributions) are characterized by a strictly monotone decreasing ratio; the minimax rules for {dollar}{lcub}cal M{rcub}{dollar}-problems are admissible. They are monotone nondecreasing with a very simple structure--continuous, piecewise-linear. The class of {dollar}{lcub}cal M{rcub}{dollar}-problems includes, but is not limited to, the distributions with monotone likelihood ratio (MLR) and non-MLR mixtures of normal distributions. (II) Type {dollar}{lcub}cal N M{rcub}{dollar}-problems {dollar}({lcub}cal N M{rcub}{dollar}-distributions) are characterized by nonmonotone ratios; the minimax rules for these problems are in general nonmonotone.; The problem domain of low-level sensor fusion provides the motivation for our research. We examine sensor fusion problems for location data models using statistical decision theory. The decision-theoretic results we obtain are used for: (i) a robust test of the hypothesis that data from different sensors are consistent; and (ii) a robust procedure for combining the data which pass this preliminary consistency test.