Sample Splitting as an M-Estimator with Application to Physical Activity Scoring

Sample splitting is widely used in statistical applications, including classically in classification and more recently for inference post model selection. Motivating by problems in the study of diet, physical activity, and health, we consider a new application of sample splitting. Physical activity researchers wanted to create a scoring system to quickly assess physical activity levels. A score is created using a large cohort study. Then, using the same data, this score serves as a covariate in a model for the risk of disease or mortality. Since the data are used twice in this way, standard errors and confidence intervals from fitting the second model are not valid. To allow for proper inference, sample splitting can be used. One builds the score with a random half of the data and then uses the score when fitting a model to the other half of the data. We derive the limiting distribution of the estimators. An obvious question is what happens if multiple sample splits are performed. We show that as the number of sample splits increases, the combination of multiple sample splits is effectively equivalent to solving a set of estimating equations.