Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach

We study statistical inference and distributionally robust solution methods for stochastic optimization problems, focusing on confidence intervals for optimal values and solutions that achieve exact coverage asymptotically. We develop a generalized empirical likelihood framework—based on distributional uncertainty sets constructed from nonparametric f-divergence balls—for Hadamard differentiable functionals, and in particular, stochastic optimization problems. As consequences of this theory, we provide a principled method for choosing the size of distributional uncertainty regions to provide one- and two-sided confidence intervals that achieve exact coverage. We also give an asymptotic expansion for our distributionally robust formulation, showing how robustification regularizes problems by their variance. Finally, we show that optimizers of the distributionally robust formulations we study enjoy (essentially) the same consistency properties as those in classical sample average approximations. Our general approach applies to quickly mixing stationary sequences, including geometrically ergodic Harris recurrent Markov chains.

Info-metric Methods for the Estimation of Models with Group-Specific Moment Conditions

Motivated by empirical analyses in economics using repeated cross-sectional data, we propose info-metric methods (IM) for estimation of the parameters of statistical models based on the information in population moment conditions that hold at group level. The info-metric estimation can be viewed as the primary approach to a constrained optimization. The estimators can also be obtained via the dual approach to this optimization, known as generalized empirical likelihood (GEL). In Andrews, Hall, Khatoon and Lincoln (2019), we provide a comprehensive framework for inference based on GEL with the grouped-specific moment conditions. In this chapter, we compare the computational requirements of the primary and dual approaches. We also describe the IM/GEL inference framework in the context of a linear regression model that is estimated using the information that the mean of the error is zero for each group. For the latter setting, we use analytical arguments and a small simulation study to compare the properties of IM/GEL-based inferences to those of inferences based on certain extant methods. The IM/GEL methods are illustrated through an application to estimation of the returns to education in which the groups are defined via information on family background.

Generalized Empirical Likelihood-Based Kernel Estimation of Spatially Similar Densities

This study concerns the estimation of spatially similar densities, each with a small number of observations. To achieve flexibility and improved efficiency, we propose kernel-based estimators that are refined by generalized empirical likelihood probability weights associated with spatial moment conditions. We construct spatial moments based on spline basis functions that facilitate desirable local customization. Monte Carlo simulations demonstrate the good performance of the proposed method. To illustruate its usefulness, we apply this method to the estimation of crop yield distributions that are known to be spatically similar.

EFFICIENT TWO-STEP GENERALIZED EMPIRICAL LIKELIHOOD ESTIMATION AND TESTS WITH MARTINGALE DIFFERENCES

This paper considers two-step generalized empirical likelihood (GEL) estimation and tests with martingale differences when there is a computationally simple $\sqrt n-$ consistent estimator of nuisance parameters or the nuisance parameters can be eliminated with an estimating function of parameters of interest. As an initial estimate might have asymptotic impact on final estimates, we propose general C(α)-type transformed moments to eliminate the impact, and use them in the GEL framework to construct estimation and tests robust to initial estimates. This two-step approach can save computational burden as the numbers of moments and parameters are reduced. A properly constructed two-step GEL (TGEL) estimator of parameters of interest is asymptotically as efficient as the corresponding joint GEL estimator. TGEL removes several higher-order bias terms of a corresponding two-step generalized method of moments. Our moment functions at the true parameters are martingales, thus they cover some spatial and time series models. We investigate tests for parameter restrictions in the TGEL framework, which are locally as powerful as those in the joint GEL framework when the two-step estimator is efficient.