Preliminary Draft MINIMAX ESTIMATION AND TESTING FOR MOMENT CONDITION MODELS VIA LARGE DEVIATIONS

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.

[1]  W. Hoeffding.  Asymptotically Optimal Tests for Multinomial Distributions , 1965 .

[2]  Yuichi Kitamura,et al.  An Information-Theoretic Alternative to Generalized Method of Moments Estimation , 1997 .

[3]  G. Sievers.  Estimates of Location: A Large Deviation Comparison , 1978 .

[4]  L. L. Cam,et al.  Asymptotic Methods In Statistical Decision Theory , 1986 .

[5]  Bahadur deficiency of likelihood ratio tests in exponential families , 1981 .

[6]  J. Fu,et al.  On a Theorem of Bahadur on the Rate of Convergence of Point Estimators , 1973 .

[7]  W. Kallenberg.  Chernoff Efficiency and Deficiency , 1982 .

[8]  Yuichi Kitamura,et al.  Testing conditional moment restrictions , 2003 .

[9]  G. Imbens,et al.  Information Theoretic Approaches to Inference in Moment Condition Models , 1995 .

[10]  Y. Nikitin,et al.  Asymptotic Efficiency of Nonparametric Tests , 1995 .

[11]  L. Hansen.  Large Sample Properties of Generalized Method of Moments Estimators , 1982 .

[12]  Anatolii A. Puhalskii,et al.  On large-deviation efficiency in statistical inference , 1998 .

[13]  Yuichi Kitamura,et al.  Empirical Likelihood Based Inference in Conditional Moment Restriction Models , 2004 .

[14]  J. Lawless,et al.  Empirical Likelihood and General Estimating Equations , 1994 .

[15]  Takuhisa Shikimi.  Large deviations for kernel-type empirical distributions , 2002 .

[16]  Whitney K. Newey,et al.  Higher Order Properties of Gmm and Generalized Empirical Likelihood Estimators , 2003 .

[17]  Jonathan M. Borwein,et al.  Partially-Finite Programming in L1 and the Existence of Maximum Entropy Estimates , 1993, SIAM J. Optim..

[18]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[19]  A. A. Pukhal'skii,et al.  On functional principle of large deviations , 1991 .

[20]  Ing Rj Ser.  Approximation Theorems of Mathematical Statistics , 1980 .

[21]  Yuichi Kitamura,et al.  On the Asymptotic Optimality of Empirical Likelihood for Testing Moment Restrictions , 2001 .

[22]  A. Owen.  Empirical likelihood ratio confidence intervals for a single functional , 1988 .

[23]  A. A. Pukhal'skii.  On the Theory of Large Deviations , 1994 .