Accurate bias estimation with applications to focused model selection

We derive approximations to the bias and squared bias with errors of order $o(1/n)$ where $n$ is the sample size. Our results hold for a large class of estimators, including quantiles, transformations of unbiased estimators, maximum likelihood estimators in (possibly) incorrectly specified models, and functions thereof. Furthermore, we use the approximations to derive estimators of the mean squared error (MSE) which are correct to order $o(1/n)$. Since the variance of many estimators is of order $O(1/n)$, this level of precision is needed for the MSE estimator to properly take the variance into account. We also formulate a new focused information criterion (FIC) for model selection based on the estimators of the squared bias. Lastly, we illustrate the methods on data containing the number of battle deaths in all major inter-state wars between 1823 and the present day. The application illustrates the potentially large impact of using a less-accurate estimator of the squared bias.