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Minimax estimation and testing for moment condition models via large deviations

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This paper studies asymptotically optimal estimation and testing procedures for moment condition models using the theory of large deviations (LD). Minimax risk estimation and testing are discussed in details. The aim of the paper is three-fold. First, it studies a moment condition model by treating it as a statistical experiment in Le Cam's sense, and investigates its large deviation properties. Second, it develops a new minimax estimator for the model by considering Bahadur's large deviation efficiency criterion. The estimator can be regarded as a robustified version of the conventional empirical likelihood estimator. Third, it considers a Chernoff-type risk for parametric testing in the model, which is concerned with the LD probabilities of type I errors and type II errors. It is shown that the empirical likelihood ratio test is asymptotically minimax in this context.

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... n = 1{|| # n − θ 0 || > c} depends on θ 0 and F 0 , therefore the worst case scenario is given by the pair (allowed in the model, Equation (1)) that maximizes Pr{A n }. Suppose an estimator # n minimizes this worst case probability, thereby achieving minimaxity. The limit inferior of the minimax probability provides an asymptotic minimax criterion. Kitamura and Otsu (2005) show that an estimator that attains the lower bound of the asymptotic minimax criterion can be obtained from the EL objective function (θ) in Equation (2) as follows, Calculating # ld in practice is straightforward. If the dimension of θ is high, it is also possible to focus on a low dimensional sub-vector of θ and obtain a large deviat ...
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