Impossibility Results for Bounds Estimation (preliminary and incomplete)

We consider estimation of bounds determined by multiple features of the data, such as bounds arising from two or more moment inequalities. In such problems, conventional estimators of the boundaries are known to be biased, and test statistics do not have standard limit distributions, at least at certain points in the parameter space. We use a limit of experiments framework to give a characterization of all attainable limit distributions of estimators, when the boundary of interest is identified as the minimum or maximum of a finite number of data features. We find that certain desirable properties—local asymptotic unbiasedness and regularity—cannot be achieved by any estimator. The problem is closely related to the problem of estimating the minimum of a set of normal means, and examining this simpler problem can suggest alternative procedures that have good risk properties.