A high-dimensional focused information criterion

The focused information criterion for model selection is constructed to select the model that best estimates a particular quantity of interest, the focus, in terms of mean squared error. We extend this focused selection process to the high-dimensional regression setting with potentially a larger number of parameters than the size of the sample. We distinguish two cases: (i) the case where the considered submodel is of low-dimension and (ii) the case where it is of high-dimension. In the former case, we obtain an alternative expression of the low-dimensional focused information criterion that can directly be applied. In the latter case we use a desparsified estimator that allows us to derive the mean squared error of the focus estimator. We illustrate the performance of the high-dimensional focused information criterion with a numerical study and a real dataset.