| Abstract: | A reduced-rank mixed effects model is developed for robust modeling of paired observed func-tional data. In this model, the curves for each functional variable are summarized using a few functional principal components, and the association of the two functional variables is modeled through the association of the principal component scores. Multivariate scale mixture of normal distributions is used to model the principal component scores and the measurement errors in order to handle outlying observations and achieve robust inference. The mean functions and principal component functions are modeled using splines and roughness penalties are applied to avoid over-fitting. An EM algorithm is developed for computation of model fitting and prediction. A simulation study shows that the proposed method outperforms an existing method which is not designed for robust estimation. The effectiveness of the proposed method is illustrated in an application of fitting multi-band light curves of Type Ia supernovae. A varying density coefficient model is developed on the Bayes space dealing with the probability density function responses. The varying coefficients in the corresponding model are random elements of the Bayes space. Firstly, the concept of random probability density function is introduced by inducing a random element in a generalized Hilbert space by a random variable. Then, the density function is regarded as the element in Bayes space, so the addition and multiplication in Bayes space can be used to define the varying coefficient model in Bayes space. Furthermore, a spline based parameter estimation method is proposed for estimating the varying density coefficients and the asymptotic theory for estimators is well established. One adaptive tuning parameter selection method is proposed as well to be applied in the simulation and real data analysis. A simulation study shows that the proposed method outperforms an existing method both on model accuracy and computation cost. The effectiveness of the proposed method is illustrated in an application of analyzing the age structure on three demographic covariates, assuming the coefficients are influenced by the logarithm of GDP per capita. The electronic version of this dissertation is accessible from https://hdl.handle.net/1969.1/197315 |
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