Assessing additivity in nonparametric models —A kernel‐based method

In this article, we address the testing problem for additivity in nonparametric regression models. We develop a kernel‐based consistent test of a hypothesis of additivity in nonparametric regression, and establish its asymptotic distribution under a sequence of local alternatives. Compared to other existing kernel‐based tests, the proposed test is shown to effectively ameliorate the influence from estimation bias of the additive component of the nonparametric regression, and hence increase its efficiency. Most importantly, it avoids the tuning difficulties by using estimation‐based optimal criteria, while there is no direct tuning strategy for other existing kernel‐based testing methods. We discuss the usage of the new test and give numerical examples to demonstrate the practical performance of the test. The Canadian Journal of Statistics 39: 632–655; 2011. © 2011 Statistical Society of Canada     

Self‐consistent estimation of mean response functions and their derivatives

Many methods have been developed for the nonparametric estimation of a mean response function, but most of these methods do not lend themselves to simultaneous estimation of the mean response function and its derivatives. Recovering derivatives is important for analyzing human growth data, studying physical systems described by differential equations, and characterizing nanoparticles from scattering data. In this article the authors propose a new compound estimator that synthesizes information from numerous pointwise estimators indexed by a discrete set. Unlike spline and kernel smooths, the compound estimator is infinitely differentiable; unlike local regression smooths, the compound estimator is self‐consistent in that its derivatives estimate the derivatives of the mean response function. The authors show that the compound estimator and its derivatives can attain essentially optimal convergence rates in consistency. The authors also provide a filtration and extrapolation enhancement for finite samples, and the authors assess the empirical performance of the compound estimator and its derivatives via a simulation study and an application to real data. The Canadian Journal of Statistics 39: 280–299; 2011 © 2011 Statistical Society of Canada     

Sequential design for nonparametric inference

The performance of nonparametric function estimates often depends on the choice of design points. Based on the mean integrated squared error criterion, we propose a sequential design procedure that updates the model knowledge and optimal design density sequentially. The methodology is developed under a general framework covering a wide range of nonparametric inference problems, such as conditional mean and variance functions, the conditional distribution function, the conditional quantile function in quantile regression, functional coefficients in varying coefficient models and semiparametric inferences. Based on our empirical studies, nonparametric inference based on the proposed sequential design is more efficient than the uniform design and its performance is close to the true but unknown optimal design. The Canadian Journal of Statistics 40: 362–377; 2012 © 2012 Statistical Society of Canada     

Improving variance function estimation in semiparametric longitudinal data analysis

We propose an efficient and robust method for variance function estimation in semiparametric longitudinal data analysis. The method utilizes a local log‐linear approximation for the variance function and adopts a generalized estimating equation approach to account for within subject correlations. We show theoretically and empirically that our method outperforms estimators using working independence that ignores the correlations. The Canadian Journal of Statistics 39: 656–670; 2011. © 2011 Statistical Society of Canada     

Nonparametric regression with cross‐classified responses

In this article, we develop regression models with cross‐classified responses. Conditional independence structures can be explored/exploited through the selective inclusion/exclusion of terms in a certain functional ANOVA decomposition, and the estimation is done nonparametrically via the penalized likelihood method. A cohort of computational and data analytical tools are presented, which include cross‐validation for smoothing parameter selection, Kullback–Leibler projection for model selection, and Bayesian confidence intervals for odds ratios. Random effects are introduced to model possible correlations such as those found in longitudinal and clustered data. Empirical performances of the methods are explored in simulation studies of limited scales, and a real data example is presented using some eyetracking data from linguistic studies. The techniques are implemented in a suite of R functions, whose usage is briefly described in the appendix. The Canadian Journal of Statistics 39: 591–609; 2011. © 2011 Statistical Society of Canada     

Testing heteroscedasticity in nonlinear and nonparametric regressions

This article presents new nonparametric tests for heteroscedasticity in nonlinear and nonparametric regression models. The tests have an asymptotic standard normal distribution under the null hypothesis of homoscedasticity and are robust against any form of heteroscedasticity. A Monte Carlo simulation with critical values obtained from the wild bootstrap procedure is provided to asses the finite sample performances of the tests. A real application of testing interest rate volatility functions illustrates the usefulness of the tests proposed. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Regression analysis for a summed missing data problem under an outcome‐dependent sampling scheme

In this paper, we consider a regression analysis for a missing data problem in which the variables of primary interest are unobserved under a general biased sampling scheme, an outcome‐dependent sampling (ODS) design. We propose a semiparametric empirical likelihood method for accessing the association between a continuous outcome response and unobservable interesting factors. Simulation study results show that ODS design can produce more efficient estimators than the simple random design of the same sample size. We demonstrate the proposed approach with a data set from an environmental study for the genetic effects on human lung function in COPD smokers. The Canadian Journal of Statistics 40: 282–303; 2012 © 2012 Statistical Society of Canada