Abstract: We propose a sparse estimation method, termed MIC (Minimum approximated Information Criterion), for generalized linear models (GLM) in fixed dimensions. What is essentially involved in MIC is the approximation of the 𝓁₀ -norm by a continuous unit dent function. A reparameterization step is devised to enforce sparsity in parameter estimates while maintaining the smoothness of the objective function. MIC yields superior performance in sparse estimation by optimizing the approximated information criterion without reducing the search space and is computationally advantageous since no selection of tuning parameters is required. Moreover, the reparameterization tactic leads to valid significance testing results free of post-selection inference. We explore the asymptotic properties of MIC, and illustrate its usage with simulated experiments and empirical examples.

Key words and phrases: Adaptive, breakdown point, least trimmed squares, outliers, penalized regression, robust regression, variable selection.