Rate accelerated inference for integrals of multivariate random functions

TL;DR

This paper introduces accelerated, unbiased Monte Carlo-based methods for estimating integrals of multivariate random functions, improving convergence and inference accuracy in functional data analysis.

Contribution

It presents novel unbiased estimation procedures that outperform traditional methods in speed and accuracy, applicable to noisy and noiseless functional data.

Findings

Faster convergence than sample mean and traditional algorithms.

Improved coverage with shorter confidence intervals.

Effective inference in noisy and noiseless settings.

Abstract

The computation of integrals is a fundamental task in the analysis of functional data, which are typically considered as random elements in a space of squared integrable functions. Borrowing ideas from recent advances in the Monte Carlo integration literature, we propose effective unbiased estimation and inference procedures for integrals of uni- and multivariate random functions. Several applications to key problems in functional data analysis involving random design points are studied and illustrated. In the absence of noise, the proposed estimates converge faster than the sample mean and the usual algorithms for numerical integration. Moreover, the proposed estimator facilitates effective inference by generally providing better coverage with shorter confidence and prediction intervals, in both noisy and noiseless setups.