Robust smoothing

Ordinary smoothing spline models and local polynomial regression are senstive to outliers. My research in this area is to develop robust smoothing techniques. We are the first ones to propose M-type smoothin spline ANOVA for correlatd data. It is based on a robustified likelihood and methods for resistant selection of smoothing parameters are developed. We also proposed empirical likelihood based local polynomial regression, only assuming the error distribution is symmetric.

Lack-of-fit tests and model selection

We use smoothing spline models to perform lack-of-fit tests for simpler models and model selection. In the first two papers below, we built a two-part semiparametric model for semicontinuous data and conducted likelihood ratio tests for proportionality of the two parts. Marginal likelihood that involves high dimentional integrals are approximated with the spherical-radial quadrature. In the next paper, nonlinear mixed effects partial spline models are utilized to detect change points and information based criterions are developed for model selection. Corresponding software are published online. Insufficient diagnostic tools for generalized linear models call for research in the last two papers. We developed likelihood ratio and cross validation score ratio based on lack-of-fit tests and proposed a fast bootstrapping procedure for generating null disributions. Software:

Generalized/nonlinear semiparametric mixed models

Estimation in these models is often hindered by high dimensional integration. In a manuscript in preparetion, we proposed a hierarchical spherical radial quadrature algorithm for multilevel generalized mixed effects models and generalized semiparametric models with basis expansions. The algorithm has complexity linear in the number of clusters and number of random effects per cluster, in contrast to the exponential complexity for the adaptive quadrature. The corresponding R and C code are in preparation for publication. We also proposed to use a pseudo-likelihood method that bypasses the integration problem for a particular nonlinear mixed effects model considered in the second paper. Software:

Variance estimation

We first propposed a difference based variance estimator without estimating the nonparametric function in a nonparametric regression model and proved its asymptotic efficiency. Observing that either the squared residuals in such models, or the squared differences, follow chi-square distributions, we proposed in the second paper methods to estimate variance functions with chi-square repsonses.

Acknowledgements: The above research is partially supported by the National Security Agencey, grant H98230-09-1-0044 (Anna Liu as PI) 2009-2011 and UMASS faculty research grant (Anna Liu as PI) 2005-2008.