High-dimensional Generalized Additive Mixed Model with Longitudinal Data
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The problem of simultaneously estimating and selecting important variables is challenging in the high-dimensional generalized additive mixed model (GAMM) with longitudinal data because of the correlation structures. We give a penalized estimation strategy for efficient variable selection when the number of variables in the linear part and number of components in the non-linear part can grow with the sample size. A Monte Carlo Newton-Raphson algorithm is developed where a Metropolis step for generating random effects is used to apply the proposed double-penalization technique in the high-dimensional GAMM. In addition, to improve efficiency for regression coefficients, the estimation of the working covariance matrix is involved in the proposed iterative algorithm. We further study the asymptotic performance of the resulting estimators and establish the oracle properties. We conduct extensive numerical studies to assess the performance of our proposed estimation strategy and numerically illustrate how efficient it is in selecting significant variables.
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