Simultaneous confidence bands for the functional mean of convex curves

Abstract

Simultaneous confidence bands for the mean of convex curves in a functional, design-based approach are presented. Specifically, a constrained functional Horvitz-Thompson estimator is proposed to address the constrained nature of the data. Indeed, a naive direct application of the functional Horvitz-Thompson estimator on the smoothed trajectories may lead to non-convex estimates. As a consequence, convex functions are defined as solutions of differential equations. Thus, data are embedded in the Hilbert space of square integrable functions, where ‘standard’ functional data analysis methodologies can be coherently interpreted. An approximation of the variance of the estimator is derived using the delta method. The coverage level of the estimated bands is evaluated through a simulation study. The proposed method is employed on a real dataset concerning the concentration of tourists daily expenditure in the Majella National Park, Central Italy.