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Abstract: We consider the estimation problem of conditional quantile when multi-dimensional covariates are involved. To overcome the “curse of dimensionality” yet retain model flexibility, we propose two partially linear models for conditional quantiles: partially linear single-index models (QPLSIM) and partially linear additive models (QPLAM). The unknown univariate functions are estimated by penalized splines. An approximate iteratively reweighted penalized least square algorithm is developed. To facilitate model comparisons, we develop effective model degrees of freedom for penalized spline conditional quantiles. Two smoothing parameter selection criteria, Generalized Approximate Cross-validation (GACV) and Schwartz-type Information Criterion (SIC) are studied. Some asymptotic properties are established. Finite sample properties are investigated through simulation studies. Application to the Boston Housing data demonstrates the success of the proposed approach. Both simulations and real applications show encouraging results of the proposed estimators. [Copyright &y& Elsevier]

Abstract: The functional coefficient regression models assume that the regression coefficients vary with some “threshold” variable, providing appreciable flexibility in capturing the underlying dynamics in data and avoiding the so-called “curse of dimensionality” in multivariate nonparametric estimation. We first investigate the estimation, inference, and forecasting for the functional coefficient regression models with dependent observations via penalized splines. The P-spline approach, as a direct ridge regression shrinkage type global smoothing method, is computationally efficient and stable. With established fixed-knot asymptotics, inference is readily available. Exact inference can be obtained for fixed smoothing parameter , which is most appealing for finite samples. Our penalized spline approach gives an explicit model expression, which also enables multi-step-ahead forecasting via simulations. Furthermore, we examine different methods of choosing the important smoothing parameter : modified multi-fold cross-validation (MCV), generalized cross-validation (GCV), and an extension of empirical bias bandwidth selection (EBBS) to P-splines. In addition, we implement smoothing parameter selection using mixed model framework through restricted maximum likelihood (REML) for P-spline functional coefficient regression models with independent observations. The P-spline approach also easily allows different smoothness for different functional coefficients, which is enabled by assigning different penalty accordingly. We demonstrate the proposed approach by both simulation examples and a real data application. [Copyright &y& Elsevier]