Functional linear regression for discretely observed data: from ideal to reality
研究了函数线性回归中，从完全观测协变量函数到离散带噪观测的理论差距，提出池化与样本分割方法，使估计量达到最优收敛速度，并通过模拟和真实数据验证了有效性。
Summary Despite extensive studies on functional linear regression, there exists a fundamental gap in theory between the ideal estimation from fully observed covariate functions and the reality that one can only observe functional covariates discretely with noise. The challenge arises when deriving a sharp perturbation bound for the estimated eigenfunctions in the latter case, which renders existing techniques for functional linear regression not applicable. We use a pooling method to attain the estimated eigenfunctions and propose a sample-splitting strategy to estimate the principal component scores, which facilitates the theoretical treatment for discretely observed data. The slope function is estimated by approximated least squares, and we show that the resulting estimator attains the optimal convergence rates for both estimation and prediction when the number of measurements per subject reaches a certain magnitude of the sample size. This phase transition phenomenon differs from the known results for the pooled mean and covariance estimation, and reveals the elevated difficulty in estimating the regression function. Numerical experiments, using simulated and real data examples, yield favourable results when compared with existing methods.