# Robust location estimation for MLR and non-MLR distributions

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## Abstract

We study the problem of estimating an unknown parameter $\theta$ from an observation of a random variable $Z = \theta + V$. This is the location data model; V is random noise with absolutely continuous distribution F, independent of $\theta$. The distribution F belongs to a given uncertainty class of distributions ${\cal F}$, $\vert{\cal F}\vert\geq 1$. We seek robust minimax decision rules for estimating the location parameter $\theta$. 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 ${f(x + 2e)}\over{f(x)}$, 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 ${\cal M}$-problems (${\cal M}$-distributions) are characterized by a strictly monotone decreasing ratio; the minimax rules for ${\cal M}$-problems are admissible. They are monotone nondecreasing with a very simple structure--continuous, piecewise-linear. The class of ${\cal M}$-problems includes, but is not limited to, the distributions with monotone likelihood ratio (MLR) and non-MLR mixtures of normal distributions. (II) Type ${\cal N M}$-problems $({\cal N M}$-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. ^

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