Estimating nonlinear additive models with nonstationarities and correlated errors

In this paper, we study a nonparametric additive regression model suitable for a wide range of time series applications. Our model includes a periodic component, a deterministic time trend, various component functions of stochastic explanatory variables, and an AR(p) error process that accounts for serial correlation in the regression error. We propose an estimation procedure for the nonparametric component functions and the parameters of the error process based on smooth backfitting and quasimaximum likelihood methods. Our theory establishes convergence rates and the asymptotic normality of our estimators. Moreover, we are able to derive an oracle‐type result for the estimators of the AR parameters: Under fairly mild conditions, the limiting distribution of our parameter estimators is the same as when the nonparametric component functions are known. Finally, we illustrate our estimation procedure by applying it to a sample of climate and ozone data collected on the Antarctic Peninsula.     

Positive quadrant dependence tests for copulas

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada     

The superharmonic condition for simultaneous estimation of means in exponential familles

The mean vector associated with several independent variates from the exponential subclass of Hudson (1978) is estimated under weighted squared error loss. In particular, the formal Bayes and “Stein-like” estimators of the mean vector are given. Conditions are also given under which these estimators dominate any of the “natural estimators”. Our conditions for dominance are motivated by a result of Stein (1981), who treated the N_p (θ, I) case with p ≥ 3. Stein showed that formal Bayes estimators dominate the usual estimator if the marginal density of the data is superharmonic. Our present exponential class generalization entails an elliptic differential inequality in some natural variables. Actually, we assume that each component of the data vector has a probability density function which satisfies a certain differential equation. While the densities of Hudson (1978) are particular solutions of this equation, other solutions are not of the exponential class if certain parameters are unknown. Our approach allows for the possibility of extending the parametric Stein-theory to useful nonexponential cases, but the problem of nuisance parameters is not treated here.     

Nonparametric hierarchical Bayes analysis of binomial data via Bernstein polynomial priors

For binomial data analysis, many methods based on empirical Bayes interpretations have been developed, in which a variance‐stabilizing transformation and a normality assumption are usually required. To achieve the greatest model flexibility, we conduct nonparametric Bayesian inference for binomial data and employ a special nonparametric Bayesian prior—the Bernstein–Dirichlet process (BDP)—in the hierarchical Bayes model for the data. The BDP is a special Dirichlet process (DP) mixture based on beta distributions, and the posterior distribution resulting from it has a smooth density defined on [0, 1]. We examine two Markov chain Monte Carlo procedures for simulating from the resulting posterior distribution, and compare their convergence rates and computational efficiency. In contrast to existing results for posterior consistency based on direct observations, the posterior consistency of the BDP, given indirect binomial data, is established. We study shrinkage effects and the robustness of the BDP‐based posterior estimators in comparison with several other empirical and hierarchical Bayes estimators, and we illustrate through examples that the BDP‐based nonparametric Bayesian estimate is more robust to the sample variation and tends to have a smaller estimation error than those based on the DP prior. In certain settings, the new estimator can also beat Stein's estimator, Efron and Morris's limited‐translation estimator, and many other existing empirical Bayes estimators. The Canadian Journal of Statistics 40: 328–344; 2012 © 2012 Statistical Society of Canada     

Semiparametric estimation for count data through weighted distributions

This paper is concerned with semiparametric discrete kernel estimators when the unknown count distribution can be considered to have a general weighted Poisson form. The estimator is constructed by multiplying the Poisson estimate with a nonparametric discrete kernel-type estimate of the Poisson weight function. Comparisons are then carried out with the ordinary discrete kernel probability mass function estimators. The Poisson weight function is thus a local multiplicative correction factor, and is considered as the uniform measure to detect departures from the equidispersed Poisson distribution. In this way, the effects of dispersion and zero-proportion with respect to the standard Poisson distribution are also minimized. This method of estimation is also applied to the weighted binomial form for the count distribution having a finite support. The proposed estimators, in addition to being simple, easy-to-implement and effective, also outperform the competing nonparametric and parametric estimators in finite-sample situations. Two examples illustrate this new semiparametric estimation.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.     

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     

Semiparametric regression under long-range dependent errors

In this paper, we consider a semiparametric regression model under long-range dependent errors. By approximating the nonparametric component by a finite series sum, we construct consistent estimators for both parametric and nonparametric components. Meanwhile, convergence rates for the consistent estimators are also investigated. Additionally, an optimal truncation parameter selection procedure is proposed.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

Estimation and testing in generalized partial linear models—A comparative study

A particular semiparametric model of interest is the generalized partial linear model (GPLM) which extends the generalized linear model (GLM) by a nonparametric component.The paper reviews different estimation procedures based on kernel methods as well as test procedures on the correct specification of this model (vs. a parametric generalized linear model). Simulations and an application to a data set on East–West German migration illustrate similarities and dissimilarities of the estimators and test statistics.     

Nonparametric measures of intraclass correlation

Three nonparametric measures of intraclass correlation based on the notion of concordance are considered. Their unbiased estimators and nonparametric tests based on the estimators are studied and it is shown that an analogue of the Kendall's tau provides small variance estimator and relatively powerful test. Furthermore, the approximate variance of the estimator is given when the correlation is small in the normal model.     

Probability density estimation with data missing at random when covariables are present

This paper addresses the problem of the probability density estimation in the presence of covariates when data are missing at random (MAR). The inverse probability weighted method is used to define a nonparametric and a semiparametric weighted probability density estimators. A regression calibration technique is also used to define an imputed estimator. It is shown that all the estimators are asymptotically normal with the same asymptotic variance as that of the inverse probability weighted estimator with known selection probability function and weights. Also, we establish the mean squared error (MSE) bounds and obtain the MSE convergence rates. A simulation is carried out to assess the proposed estimators in terms of the bias and standard error.     

An Asymptotic Characterization of Finite Degree U-statistics With Sample Size-Dependent Kernels: Applications to Nonparametric Estimators and Test Statistics

We provide a simple result on the H-decomposition of a U-statistics that allows for easy determination of its magnitude when the statistic’s kernel depends on the sample size n. The result provides a direct and convenient method to characterize the asymptotic magnitude of semiparametric and nonparametric estimators or test statistics involving high dimensional sums. We illustrate the use of our result in previously studied estimators/test statistics and in a novel nonparametric R² test for overall significance of a nonparametric regression model.     

Generalized estimating equations for mixtures with varying concentrations

A finite mixture model is considered in which the mixing probabilities vary from observation to observation. A parametric model is assumed for one mixture component distribution, while the others are nonparametric nuisance parameters. Generalized estimating equations (GEE) are proposed for the semi‐parametric estimation. Asymptotic normality of the GEE estimates is demonstrated and the lower bound for their dispersion (asymptotic covariance) matrix is derived. An adaptive technique is developed to derive estimates with nearly optimal small dispersion. An application to the sociological analysis of voting results is discussed. The Canadian Journal of Statistics 41: 217–236; 2013 © 2013 Statistical Society of Canada     

Inferences from biased samples with a memory effect

Biased sampling occurs often in observational studies. With one biased sample, the problem of nonparametrically estimating both a target density function and a selection bias function is unidentifiable. This paper studies the nonparametric estimation problem when there are two biased samples that have some overlapping observations (i.e. recaptures) from a finite population. Since an intelligent subject sampled previously may experience a memory effect if sampled again, two general 2-stage models that incorporate both a selection bias and a possible memory effect are proposed. Nonparametric estimators of the target density, selection bias, and memory functions, as well as the population size are developed. Asymptotic properties of these estimators are studied and confidence bands for the selection function and memory function are provided. Our procedures are compared with those ignoring the memory effect or the selection bias in finite sample situations. A nonparametric model selection procedure is also given for choosing a model from the two 2-stage models and a mixture of these two models. Our procedures work well with or without a memory effect, and with or without a selection bias. The paper concludes with an application to a real survey data set.     

Estimation with right‐censored observations under a semi‐Markov model

The semi‐Markov process often provides a better framework than the classical Markov process for the analysis of events with multiple states. The purpose of this paper is twofold. First, we show that in the presence of right censoring, when the right end‐point of the support of the censoring time is strictly less than the right end‐point of the support of the semi‐Markov kernel, the transition probability of the semi‐Markov process is nonidentifiable, and the estimators proposed in the literature are inconsistent in general. We derive the set of all attainable values for the transition probability based on the censored data, and we propose a nonparametric inference procedure for the transition probability using this set. Second, the conventional approach to constructing confidence bands is not applicable for the semi‐Markov kernel and the sojourn time distribution. We propose new perturbation resampling methods to construct these confidence bands. Different weights and transformations are explored in the construction. We use simulation to examine our proposals and illustrate them with hospitalization data from a recent cancer survivor study. The Canadian Journal of Statistics 41: 237–256; 2013 © 2013 Statistical Society of Canada     

Semiparametric statistical inferences for longitudinal data with nonparametric covariance modelling

In this paper, semiparametric modelling for longitudinal data with an unstructured error process is considered. We propose a partially linear additive regression model for longitudinal data in which within-subject variances and covariances of the error process are described by unknown univariate and bivariate functions, respectively. We provide an estimating approach in which polynomial splines are used to approximate the additive nonparametric components and the within-subject variance and covariance functions are estimated nonparametrically. Both the asymptotic normality of the resulting parametric component estimators and optimal convergence rate of the resulting nonparametric component estimators are established. In addition, we develop a variable selection procedure to identify significant parametric and nonparametric components simultaneously. We show that the proposed SCAD penalty-based estimators of non-zero components have an oracle property. Some simulation studies are conducted to examine the finite-sample performance of the proposed estimation and variable selection procedures. A real data set is also analysed to demonstrate the usefulness of the proposed method.     

Statistical inference for a single-index varying-coefficient model

We investigate the estimators of parameters of interest for a single-index varying-coefficient model. To estimate the unknown parameter efficiently, we first estimate the nonparametric component using local linear smoothing, then construct an estimator of parametric component by using estimating equations. Our estimator for the parametric component is asymptotically efficient, and the estimator of nonparametric component has asymptotic normality and optimal uniform convergence rate. Our results provide ways to construct confidence regions for the involved unknown parameters. The finite-sample behavior of the new estimators is evaluated through simulation studies, and applications to two real data are illustrated.     

Estimation for semiparametric varying coefficient models with different smoothing variables under random right censoring

In this paper we study a semiparametric varying coefficient model when the response is subject to random right censoring. The model gives an easy interpretation due to its direct connectivity to the classical linear model and is very flexible since nonparametric functions which accommodates various nonlinear interaction effects between covariates are admitted in the model. We propose estimators for this model using mean-preserving transformation and establish their asymptotic properties. The estimation procedure is based on the profiling and the smooth backfitting techniques. A simulation study is presented to show the reliability of the proposed estimators and an automatic bandwidth selector is given in a data-driven way.