Jean D. Opsomer

Non‐parametric small area estimation using penalized spline regression

We propose a new small area estimation approach that combines small area random effects with a smooth, nonparametrically specified trend. By using penalized splines as the representation for the nonparametric trend, it is possible to express the small area estimation problem as a mixed effect model regression. This model is readily fitted using existing model fitting approaches such as restricted maximum likelihood. We develop a corresponding bootstrap approach for model inference and estimation of the small area prediction mean squared error. The applicability of the method is demonstrated on a survey of lakes in the Northeastern US.

A Root-n Consistent Backfitting Estimator for Semiparametric Additive Modeling

Abstract We explore additive models that combine both parametric and nonparametric terms and propose a √n-consistent backfitting estimator for the parametric component of the model. The theoretical properties of the estimator are developed for the case with a single nonparametric term and extended to an arbitrary number of nonparametric additive terms. An estimator for the optimal bandwidth making minimal use of asymptotic expressions for bias and variance is proposed, and a fast implementation algorithm for model fitting and bandwidth selection is developed. The practical behavior of the estimator and bandwidth selection is illustrated by simulation experiments.

Model-Assisted Estimation of Forest Resources With Generalized Additive Models

Multiphase surveys are often conducted in forest inventories, with the goal of estimating forested area and tree characteristics over large regions. This article describes how design-based estimation of such quantities, based on information gathered during ground visits of sampled plots, can be made more precise by incorporating auxiliary information available from remote sensing. The relationship between the ground visit measurements and the remote sensing variables is modeled using generalized additive models. Nonparametric estimators for these models are discussed and applied to forest data collected in the mountains of northern Utah. Model-assisted estimators that use the nonparametric regression fits are proposed for these data. The design context of this study is two-phase systematic sampling from a spatial continuum, under which properties of model-assisted estimators are derived. Difficulties with the standard variance estimation approach, which assumes simple random sampling in each phase, are described. An alternative assessment of estimator performance based on a synthetic population is implemented and shows that using the model predictions in a model-assisted survey estimation procedure results in substantial efficiency improvements over current estimation approaches.

Model-assisted estimation for complex surveys using penalised splines

Estimation of finite population totals in the presence of auxiliary information is considered. A class of estimators based on penalised spline regression is proposed. These estimators are weighted linear combinations of sample observations, with weights calibrated to known control totals. They allow straightforward extensions to multiple auxiliary variables and to complex designs. Under standard design conditions, the estimators are design consistent and asymptotically normal, and they admit consistent variance estimation using familiar design-based methods. Data-driven penalty selection is considered in the context of unequal probability sampling designs. Simulation experiments show that the estimators are more efficient than parametric regression estimators when the parametric model is incorrectly specified, while being approximately as efficient when the parametric specification is correct. An example using Forest Health Monitoring survey data from the U.S. Forest Service demonstrates the applicability of the methodology in the context of a two-phase survey with multiple auxiliary variables. Copyright 2005, Oxford University Press.

A Fully Automated Bandwidth Selection Method for Fitting Additive Models

Abstract This article describes a fully automated bandwidth selection method for additive models that is applicable to the widely used backfitting algorithm of Buja, Hastie, and Tibshirani. The proposed plug-in estimator is an extension of the univariate local linear regression estimator of Ruppert, Sheather, and Wand and is shown to achieve the same Op (n –2/7) relative convergence rate for bivariate additive models. If more than two covariates are present, theoretical justification of the method requires independence of the covariates, but simulation experiments show that in practice the method is very robust to violations of this assumption. The proposed bandwidth selection method is compared to cross-validation through simulation experiments. Its practical behavior is demonstrated on a real dataset.

Local Likelihood Estimation in Generalized Additive Models

Generalized additive models are a popular class of multivariate non-parametric regression models, due in large part to the ease of use of the local scoring estimation algorithm. However, the theoretical properties of the local scoring estimator are poorly understood. In this article, we propose a local likelihood estimator for generalized additive models that is closely related to the local scoring estimator fitted by local polynomial regression. We derive the statistical properties of the estimator and show that it achieves the same asymptotic convergence rate as a one-dimensional local polynomial regression estimator. We also propose a wild bootstrap estimator for calculating point-wise confidence intervals for the additive component functions. The practical behaviour of the proposed estimator is illustrated through a simulation experiment. Copyright 2003 Board of the Foundation of the Scandinavian Journal of Statistics..

Selecting the amount of smoothing in nonparametric regression estimation for complex surveys

Model-assisted estimation is a common technique to improve the precision of finite population survey estimators by taking advantage of relationships between the survey variables and the available auxiliary information. Breidt and Opsomer introduced a nonparametric model-assisted estimator based on local polynomial regression, which allows these relationships to be modeled nonparametrically. In this article, we address the issue of how to select the amount of smoothing for the nonparametric regression component of the model-assisted estimator. The proposed smoothing parameter selection method is based on minimizing a type of cross-validation criterion, suitably adjusted for the effect of the finite population setting and the survey design. Asymptotic properties of the method are derived, and simulation experiments show that it works well in practical settings as well.

Bootstrapping for Penalized Spline Regression

We describe and contrast several different bootstrap procedures for penalized spline smoothers. The bootstrap methods considered are variations on existing methods, developed under two different probabilistic frameworks. Under the first framework, penalized spline regression is considered as an estimation technique to find an unknown smooth function. The smooth function is represented in a high-dimensional spline basis, with spline coefficients estimated in a penalized form. Under the second framework, the unknown function is treated as a realization of a set of random spline coefficients, which are then predicted in a linear mixed model. We describe how bootstrap methods can be implemented under both frameworks, and we show theoretically and through simulations and examples that bootstrapping provides valid inference in both cases. We compare the inference obtained under both frameworks, and conclude that the latter generally produces better results than the former. The bootstrap ideas are extended to hypothesis testing, where parametric components in a model are tested against nonparametric alternatives. Datasets and computer code are available in the online supplements.

PLUG-IN BANDWIDTH SELECTOR FOR LOCAL POLYNOMIAL REGRESSION ESTIMATOR WITH CORRELATED ERRORS

Consider the fixed regression model where the error random variables are coming from a strictly stationary, non-white noise stochastic process. In a situation like this, automated bandwidth selection methods for non-parametric regression break down. We present a plug-in method for choosing the smoothing parameter for local least squares estimators of the regression function. The method takes the presence of correlated errors explicitly into account through a parametric correlation function specification. The theoretical performance for the local linear estimator of the regression function is obtained in the case of an AR(1) correlation function. These results can readily be extended to other settings, such as different parametric specifications of the correlation function, derivative estimation and multiple non-parametric regression. Estimators of regression functionals and the error correlation based on local least squares ideas are developed in this article. A simulation study and an analysis with real economic data illustrate the selection method proposed.

Nonparametric and Semiparametric Estimation in Complex Surveys

Publisher Summary This chapter focuses on nonparametric and semi-parametric methods in two important statistical areas: estimation of densities and estimation of regression functions. Both of these areas have applications in survey estimation, for both descriptive and analytical uses. Orthogonal decomposition is a non-parametric regression method with good statistical properties that is applicable in situations where the mean function is not necessarily smooth. Neural networks are a class of methods conceptually related to penalized spline regression, in which the parameters are found by nonlinear regression. The semi-parametric model is particularly useful when some of the covariates in a data set are categorical, which by definition cannot be smoothed. In addition to nonparametric regression for multivariate data, another important extension is for models with more complex mean structures, including nonparametric equivalents of generalized linear models. Nonparametric regression applications require the specification of one or several smoothing parameters such as the bandwidth in kernel regression or the penalty in spline regression.