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.