Polynomial Spline Estimation for A Generalized Additive Coefficient Model

Generalized Additive Model Reveals Nonlinear Trade-Offs/Synergies between Relationships of Ecosystem Services for Mountainous Areas of Southwest China

Ecosystem services (ESs) are an essential link between ecosystems and human well-being, and trade-offs/synergies happen in ESs at different temporal and spatial scales. It is crucial to explore patterns of trade-offs/synergies among ESs, and their nonlinear relationships with changes in ESs. The primary objective of this study was to evaluate five ESs in 2000 and 2018: namely, water yield, food production, carbon sequestration, soil conservation, and habitat quality in mountainous regions of Southwest China. The mean values of the five ESs increased by 365.8 m3/ha, 13.92 t/hm2, 497.09 TgC/yr2, 138.48 t/km2, and 0.002, respectively. Using spatial statistics and analysis, an ES trade-off synergy model (ESTD) was constructed for the five ESs change values. Overall, soil conservation has a trade-off with all five ESs, except habitat quality; this trade-off is increasing slightly. Water yield is in synergy with all ESs except soil conservation, with decreasing synergy; habitat quality is in synergy with all ESs except food production, with increasing synergy. Finally, the nonlinear relationship between the value of the change in the ES and ESTD was analyzed using a generalized additive model. Changes in water yield showed the greatest impact on ESTD except for food production, wherein changes in all three ESs had minimal impacts on ESTD. Food production dominates its trade-offs/synergies relationship with soil conservation; carbon sequestration is the dominant player in its trade-offs/synergies relationship with soil conservation. Habitat quality has a secondary position of influence, except in the trade-offs/synergies involving food production. By exploring the drivers of trade-offs/synergies among ESs, this study can provide guidance for the effective implementation of policies related to ecological protection and restoration.

Penalized estimation for proportional hazards models with current status data

We provide a simple and practical, yet flexible, penalized estimation method for a Cox proportional hazards model with current status data. We approximate the baseline cumulative hazard function by monotone B-splines and use a hybrid approach based on the Fisher-scoring algorithm and the isotonic regression to compute the penalized estimates. We show that the penalized estimator of the nonparametric component achieves the optimal rate of convergence under some smooth conditions and that the estimators of the regression parameters are asymptotically normal and efficient. Moreover, a simple variance estimation method is considered for inference on the regression parameters. We perform 2 extensive Monte Carlo studies to evaluate the finite-sample performance of the penalized approach and compare it with the 3 competing R packages: C1.coxph, intcox, and ICsurv. A goodness-of-fit test and model diagnostics are also discussed. The methodology is illustrated with 2 real applications.

Fitting Cox Models with Doubly Censored Data Using Spline-Based Sieve Marginal Likelihood

In some applications, the failure time of interest is the time from an originating event to a failure event, while both event times are interval censored. We propose fitting Cox proportional hazards models to this type of data using a spline-based sieve maximum marginal likelihood, where the time to the originating event is integrated out in the empirical likelihood function of the failure time of interest. This greatly reduces the complexity of the objective function compared with the fully semiparametric likelihood. The dependence of the time of interest on time to the originating event is induced by including the latter as a covariate in the proportional hazards model for the failure time of interest. The use of splines results in a higher rate of convergence of the estimator of the baseline hazard function compared with the usual nonparametric estimator. The computation of the estimator is facilitated by a multiple imputation approach. Asymptotic theory is established and a simulation study is conducted to assess its finite sample performance. It is also applied to analyzing a real data set on AIDS incubation time.

Estimation and Inference in Generalized Additive Coefficient Models for Nonlinear Interactions with High-Dimensional Covariates

In the low-dimensional case, the generalized additive coefficient model (GACM) proposed by Xue and Yang [Statist. Sinica16 (2006) 1423-1446] has been demonstrated to be a powerful tool for studying nonlinear interaction effects of variables. In this paper, we propose estimation and inference procedures for the GACM when the dimension of the variables is high. Specifically, we propose a groupwise penalization based procedure to distinguish significant covariates for the "large p small n" setting. The procedure is shown to be consistent for model structure identification. Further, we construct simultaneous confidence bands for the coefficient functions in the selected model based on a refined two-step spline estimator. We also discuss how to choose the tuning parameters. To estimate the standard deviation of the functional estimator, we adopt the smoothed bootstrap method. We conduct simulation experiments to evaluate the numerical performance of the proposed methods and analyze an obesity data set from a genome-wide association study as an illustration.