Results

      Articles



      Papers presented at International Conferences




Conferences and Workshops organized in the framework of the project and with support from the project




                                                                                                                                                  

 SYNOPSIS



The purpose of the project was to introduce robust versions of the empirical likelihood method, as well as other new statistical methods for various semiparametric models and especially for moment condition models and two sample density ratio models. The theme proposed in this project addresses an important issue in statistics and econometrics, namely the assessment of the sensitivity of statistical methods to possible deviations from the assumed model and the use of robust methods. For example, in econometrics, imposing restrictions without assuming underlying distributions to modelize complex realities is a valuable methodological tool. However, if some restrictions were not correctly specified or the data set contains outliers, the usual statistical methods for correctly specified moment condition models may give completely erroneous results. The use of robust methods has much practical importance, in order to ensure accurate conclusions of the statistical analysis and for improving knowledge from “non-perfect” data sets. 

 Some of the new statistical methods proposed through this project use information measures, namely divergences and entropy measures. Statistical methods based on divergence minimization extend the likelihood paradigm and often have the advantage to provide a trade-off between efficiency and robustness. In the framework of this project, using duality techniques to the divergence optimization, we obtained large classes of estimators and test statistics for moment condition models and for two sample density ratio models. The duality approach has the advantage to avoid any smoothing or grouping technique which would be necessary for a more direct divergence minimization approach for the same problem. The proposed new statistical methods represent attractive alternatives to the classical empirical likelihood method. They have robustness properties, as well as other asymptotic properties including consistency and may also be efficient when the model is correctly specified. 

For the first stage of the project, our objective was related to the developement and the theoretical study  of new statistical methods for semiparametric models. Empirical likelihood (EL) is a powerful and currently widely used approach for developing efficient nonparametric statistical methods for various semiparametric models. The EL estimator for moment condition models is preferable to other estimators, due to the higher-order asymptotic properties, but these properties are valid only in the case of the correct specification of the moment conditions. Also, in the case of the presence of outliers in the sample, the EL estimator may give completely erroneous results. We proposed robust versions of the EL estimator for moment condition models and proved their asymptotic properties including consistency and asymptotic normality. The robust versions of the EL estimator are based on using truncated orthogonality functions. The truncated orthogonality function is constructed using the multivariate Huber function, such that the original restrictions of the model and the new ones, based on truncated function, are satisfied for the same value of the parameter of interest. We proved the robustness of the new estimators by using the influence function approach. Extensions of the smooth MD approach for inference in semiparametric partially linear models that can be defined by conditional moment conditions have also been considered. We also studied the estimation and inference for conditional estimating equations models in the context of survey sampling. 

For the second stage of the project, our objective was related to the development and the study of new statistical methods for various semiparametric models, as well as the implementation of these methods using specialized statistical software. We introduced a wide class of robust minimum empirical divergence estimators and tests for moment condition models. We considered truncated orthogonality functions, such that the original restrictions of the model and the new restrictions, based on truncated functions, are satisfied for the same value of the parameter of interest. We defined the moment condition model associated to an empirical version of truncated orthogonality function, model that includes the original reference model. Then we defined a large class of estimators for the parameter of interest, by minimizing an empirical version of the dual form of the divergence between the new model and the true model, corresponding to the data. The class of estimators is indexed by the  -function corresponding to the used divergence and contains some known estimators, as special cases. We proved theoretical properties for the new estimators, including B-robustness and the consistency. We derived new robust tests by using estimators of the used divergence. We also proposed large classes of equivariant robust estimators for moment condition models invariant with respect to additive or multiplicative groups of transformations on the observations space. Within such a class of equivariant robust estimators, we characterized the minimum risk equivariant estimator or the Pitman estimator. We obtained asymptotic approximations for the Pitman estimator, in the case when the moment condition model is invariant with respect to additive groups. We developed new model check procedures using lack-of-fitness statistics and proved some convergence results. We also considered various particular models that can be written using moment equations for which we studied new statistical methods. Models examined in the empirical finance literature, often imply moment conditions that can be used in a straightforward way to estimate the model parameters without making strong assumptions regarding the stochastic properties of observed variables.  For example, the stochastic lognormal volatility (SLV) model offers a powerful alternative to GARCH-type models to explain the well-documented time varying volatility and can be written under the form of moment equations.

In the third stage of the project, we continued studies from the previous stages and developed new robust statistical methods for semiparametric moment condition models. We defined new robust Z-estimators for moment condition models and studied their theoretical properties. We also defined minimum dual divergence estimators for semiparametric two-sample density ratio models, for which we studies robustness properties. Our study reveals that, appropriate choices of the divergence from the Cressie-Read class lead to B-robust estimators through the duality technique, thus providing robust alternatives for the empirical likelihood estimator in this context. We also considered the problem of conditional independence between vectors and proposed a new nonparametric test. The approach is based on writing the conditional independence under the form of a set, usually infinite, of conditional moment equations. Statistical methods for other various semiparametric models have been developed, including conditional frontier models or models for functional data. Quantile frontier models, including the partial, robust alpha-frontiers can be related to moment equations, since the quantiles, conditional or not, can be defined through moment equations as well.

A special attention was given to the numerical aspects, including creating algorithms, implementing the proposed statistical methods and the Monte Carlo simulation studies.