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 (FDA) 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.

翻译：积分计算是函数数据分析中的一项基本任务，函数数据通常被视为平方可积函数空间中的随机元。借鉴蒙特卡洛积分领域的最新进展，我们提出了针对一元和多元随机函数积分的有效无偏估计与推断方法。研究并展示了该方法在涉及随机设计点的函数数据分析（FDA）若干关键问题中的应用。在无噪声情况下，所提估计量的收敛速度优于样本均值及常规数值积分算法。此外，无论在有噪声还是无噪声场景中，所提估计量通常能提供更优的覆盖概率及更短的置信区间与预测区间，从而促进有效的统计推断。