Sparse functional regression models: minimax rates and contamination
In functional linear regression and functional generalized linear regression models, the effect of the predictor function is usually assumed to be spread across the index space. In this dissertation we consider the sparse functional linear model and the sparse functional generalized linear models (GLM), where the impact of the predictor process on the response is only via its value at one point in the index space, defined as the sensitive point. We are particularly interested in estimating the sensitive point. The minimax rate of convergence for estimating the parameters in sparse functional linear regression is derived. It is shown that the optimal rate for estimating the sensitive point depends on the roughness of the predictor function, which is quantified by a "generalized Hurst exponent". The least squares estimator (LSE) is shown to attain the optimal rate. Also, a lower bound is given on the minimax risk of estimating the parameters in sparse functional GLM, which also depends on the generalized Hurst exponent of the predictor process. The order of the minimax lower bound is the same as that of the weak convergence rate of the maximum likelihood estimator (MLE), given that the functional predictor behaves like a Brownian motion.
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