Statistical estimation in the presence of possibly incorrect model assumptions

Abstract

The estimation problem of a parameter of interest when some model assumptions may be incorrect is considered. The parameter of interest is defined in a model-independent manner and the estimating procedure selects a model with the smallest mean square error (MSE) as estimated by a proposed MSE estimator. This proposed MSE estimator combines both a nonparametric bootstrap and plug-in estimation in its structure. It requires at least one consistent estimator with a quickly disappearing systematic bias ($\sqrt n$ mean convergence). This estimator is not tied up to a single set of model assumptions (e.g., a class of parametric models), and thus it works across various sets of possibly nonnested classes of statistical models. The derived large sample properties constitute theoretical justification of its use and allow the estimation of the probability of how likely this estimator will have the smallest MSE in a pool of candidate estimators. Multiple simulation studies illustrate the performance of the proposed procedure under various scenarios. A real data example highlights its practical use when a single model is selected from several conceptually different statistical modeling techniques (parametric regression, Cox regression, stratified Cox regression, regression on pseudo-values) and other model selection approaches are not applicable.