Divergences and Duality for Estimation and Test under Moment Condition Models

We introduce estimation and test procedures through divergence minimiza- tion for models satisfying linear constraints with unknown parameter. These procedures extend the empirical likelihood (EL) method and share common features with generalized empirical likelihood approach. We treat the problems of existence and characterization of the divergence projections of probability distributions on sets of signed finite measures. We give a precise characterization of duality, for the proposed class of estimates and test statistics, which is used to derive their limiting distributions (including the EL estimate and the EL ratio statistic) both under the null hypotheses and under alterna- tives or misspecification. An approximation to the power function is deduced as well as the sample size which ensures a desired power for a given alternative.

[1]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[2]  Han Hong,et al.  Nonparametric likelihood ratio model selection tests between parametric likelihood and moment condition models , 2007 .

[3]  Keith A. Baggerly,et al.  Empirical likelihood as a goodness-of-fit measure , 1998 .

[4]  Aida Toma,et al.  Robust tests based on dual divergence estimators and saddlepoint approximations , 2010, J. Multivar. Anal..

[5]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[6]  Yuichi Kitamura,et al.  Empirical Likelihood Methods in Econometrics: Theory and Practice , 2006 .

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

[8]  H. White.  Maximum Likelihood Estimation of Misspecified Models , 1982 .

[9]  V. P. Godambe,et al.  An extension of quasi-likelihood estimation , 1989 .

[10]  Shelby J. Haberman,et al.  Adjustment by Minimum Discriminant Information , 1984 .

[11]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[12]  A. Owen.  Empirical Likelihood Ratio Confidence Regions , 1990 .

[13]  Guido W. Imbens,et al.  One-step estimators for over-identified generalized method of moments models , 1997 .

[14]  P. Bertail.  Emirical Likelihood in Some Semiparametric Models , 2006 .

[15]  Nils Lid Hjort,et al.  Extending the Scope of Empirical Likelihood , 2009, 0904.2949.

[16]  M. Broniatowski,et al.  Minimization of divergences on sets of signed measures , 2010, 1003.5457.

[17]  Susanne M. Schennach,et al.  Accompanying document to "Point Estimation with Exponentially Tilted Empirical Likelihood" , 2005, math/0512181.

[18]  Michel Broniatowski,et al.  Parametric estimation and tests through divergences and the duality technique , 2008, J. Multivar. Anal..

[19]  Richard J. Smith.  Alternative Semi-parametric Likelihood Approaches to Generalised Method of Moments Estimation , 1997 .

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

[21]  Anthony Y. C. Kuk,et al.  Median Estimation in the Presence of Auxiliary Information , 1989 .

[22]  A. Keziou.  Dual representation of Φ-divergences and applications , 2003 .

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

[24]  Stephen A. Corcoran,et al.  Bartlett adjustment of empirical discrepancy statistics , 1998 .

[25]  Jiahua Chen,et al.  Empirical likelihood estimation for ?nite populations and the e?ective usage of auxiliary informatio , 1993 .

[26]  Timothy R. C. Read,et al.  Multinomial goodness-of-fit tests , 1984 .

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

[28]  Patrik Guggenberger,et al.  GENERALIZED EMPIRICAL LIKELIHOOD ESTIMATORS AND TESTS UNDER PARTIAL, WEAK, AND STRONG IDENTIFICATION , 2003, Econometric Theory.

[29]  Xiaotong Shen,et al.  Empirical Likelihood , 2002 .

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

[31]  M. Broniatowski,et al.  ESTIMATION AND TESTS FOR MODELS SATISFYING LINEAR CONSTRAINTS WITH UNKNOWN PARAMETER , 2008, 0811.3477.

[32]  Art B. Owen,et al.  Empirical Likelihood for Linear Models , 1991 .