Sharp oracle inequalities and slope heuristic for specification probabilities estimation in discrete random fields

TL;DR

This paper develops model selection procedures with sharp oracle inequalities for estimating specification probabilities in discrete random fields, supported by theoretical results and practical applications.

Contribution

It introduces new estimation methods with proven sharp oracle inequalities and validates the slope heuristic for penalty calibration in this context.

Findings

Estimators satisfy sharp oracle inequalities in $L_2$-risk.

The slope heuristic effectively calibrates penalties.

Methods perform well in simulation and real neuronal data.

Abstract

We study the problem of estimating the one-point specification probabilities in non-necessary finite discrete random fields from partially observed independent samples. Our procedures are based on model selection by minimization of a penalized empirical criterion. The selected estimators satisfy sharp oracle inequalities in $L_2$-risk. We also obtain theoretical results on the slope heuristic for this problem, justifying the slope algorithm to calibrate the leading constant in the penalty. The practical performances of our methods are investigated in two simulation studies. We illustrate the usefulness of our approach by applying the methods to a multi-unit neuronal data from a rat hippocampus.