Developments of regularized approaches in non-standard paradigm
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This dissertation proposes new regularized approaches to two different topics. The first topic revisits the autocovariance function estimation, a fundamental problem in statistical inference for time series data. We convert the function estimation problem into constrained penalized regression with a generalized penalty that provides us with flexible and accurate estimation, and study the asymptotic properties of the proposed estimator. In case of a nonzero time series, we apply a penalized regression technique to a differenced time series, which does not require a separate detrending procedure. We propose four different data-driven criteria to determine the tuning parameters, whose selection is critical in penalized regression. A simulation study shows that the proposed approach is superior to three existing methods and the tuning parameter selection is effective. We also briefly discuss the extension of the proposed approach to interval-valued time series. The second topic concerns the study of the entire conditional distribution of a response given predictors in a regression setting. A usual approach to address heterogenous data is quantile regression, but it has some drawbacks and is computationally challenging due to the $L_1$ norm. As an alternative to quantile regression, we consider expectile regression that relies on the minimization of the asymmetric $L_2$ norm. We assume that only a small set of predictors is relevant to the response, and develop penalized expectile regression with SCAD and adaptive LASSO penalties. With properly chosen tuning parameters, we show that the proposed estimators enjoy the oracle properties. A numerical study using simulated and real examples demonstrates the superior performance of the proposed penalized expectile regression over penalized quantile regression.