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     

Consistency of maximum likelihood estimators in a large class of deconvolution models

We consider the maximum likelihood estimator $\hat{F}_n$ of a distribution function in a class of deconvolution models where the known density of the noise variable is of bounded variation. This class of noise densities contains in particular bounded, decreasing densities. The estimator $\hat{F}_n$ is defined, characterized in terms of Fenchel optimality conditions and computed. Under appropriate conditions, various consistency results for $\hat{F}_n$ are derived, including uniform strong consistency. The Canadian Journal of Statistics 41: 98–110; 2013 © 2012 Statistical Society of Canada     

Adaptive estimation of error density in nonparametric regression with small sample size

It has been established recently in Efromovich [2005. Estimation of the density of regression errors. Ann. Statist. 33, 2194–2227] that, under a mild assumption, the error density in a nonparametric regression can be asymptotically estimated with the accuracy of an oracle that knows underlying regression errors. The asymptotic nature of the result, and in particular the used methodology of splitting data for estimating nuisance functions and the error density, does not make an asymptotic estimator, suggested in that article, feasible for practically interesting cases of small sample sizes. This article continues the research and solves two important issues. First, it shows that the asymptotic holds without splitting the data. Second, a data-driven estimator, based on the new asymptotic, is suggested and then tested on real and simulated examples.     

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     

Positive quadrant dependence testing and constrained copula estimation

Positive quadrant dependence is a specific dependence structure that is of practical importance in for example modelling dependencies in insurance and actuarial sciences. This dependence structure imposes a constraint on the copula function. The interest in this paper is to test for positive quadrant dependence. One way to assess the distribution of the test statistics under the null hypothesis of positive quadrant dependence is to resample from a constrained copula. This requires constrained estimation of a copula function. We show that this use of resampling under a constrained copula improves considerably the power performance of existing testing procedures. We propose two resampling procedures, one based on a parametric constrained copula estimation and one relying on nonparametric estimation of a positive quadrant dependence copula, and discuss their properties. The finite‐sample performances of the resulting testing procedures are evaluated via a simulation study that also includes comparisons with existing tests. Finally, a data set of Danish fire insurance claims is tested for positive quadrant dependence. The Canadian Journal of Statistics 41: 36–64; 2013 © 2012 Statistical Society of Canada     

Fully efficient estimation of coefficients of correlation in the presence of imputed survey data

Marginal imputation, that consists of imputing items separately, generally leads to biased estimators of bivariate parameters such as finite population coefficients of correlation. To overcome this problem, two main approaches have been considered in the literature: the first consists of using customary imputation methods such as random hot‐deck imputation and adjusting for the bias at the estimation stage. This approach was studied in Skinner & Rao 2002 . In this paper, we extend the results of Skinner & Rao 2002 to the case of arbitrary sampling designs and three variants of random hot‐deck imputation. The second approach consists of using an imputation method, which preserves the relationship between variables. Shao & Wang 2002 proposed a joint random regression imputation procedure that succeeds in preserving the relationships between two study variables. One drawback of the Shao–Wang procedure is that it suffers from an additional variability (called the imputation variance) due to the random selection of residuals, resulting in potentially inefficient estimators. Following Chauvet, Deville, & Haziza 2011 , we propose a fully efficient version of the Shao–Wang procedure that preserves the relationship between two study variables, while virtually eliminating the imputation variance. Results of a simulation study support our findings. An application using data from the Workplace and Employees Survey is also presented. The Canadian Journal of Statistics 40: 124–149; 2012 © 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