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Functional Linear Regression for Discretely Observed Data: From Ideal to Reality
In this work, we give a new insight into estimation and prediction issues in functional linear regression (FLR) when the covariate process is discretely observed with noise. Without the fully observed functional data, it is difficult to derive a sharp bound for the estimated eigenfunctions, which makes the existing techniques for FLR unfeasible. We use pooling method to attain the estimated eigenfunctions without pre-smoothing each curve and propose a sample-splitting approach to estimate the component scores, which is novel for treating discretely observed data and facilitate the theoretical analysis. We then obtain the estimated slope function by the approximated least squared method. We show that the proposed method attains the optimal convergence rate as if the curves are fully observed for slope estimation and prediction error when the number of measurements per subject reach the magnitude which is determined the smoothness of the covariance and slope function, where is the number of subjects. This phase transition of the convergence rate is always between and, which differs from the phase transition of the pooled mean and covariance estimation at and reveals the elevated difficulties in estimating the slope function. We also evaluate the numerical performance of our proposed method using simulated and real data examples, yielding similar or favourable results when contrasted to comparable methods.
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