Bayesian spectral analysis models for quantile regression with Dirichlet process mixtures

This paper presents a Bayesian analysis of partially linear additive models for quantile regression. We develop a semiparametric Bayesian approach to quantile regression models using a spectral representation of the nonparametric regression functions and the Dirichlet process (DP) mixture for error distribution. We also consider Bayesian variable selection procedures for both parametric and nonparametric components in a partially linear additive model structure based on the Bayesian shrinkage priors via a stochastic search algorithm. Based on the proposed Bayesian semiparametric additive quantile regression model referred to as BSAQ, the Bayesian inference is considered for estimation and model selection. For the posterior computation, we design a simple and efficient Gibbs sampler based on a location-scale mixture of exponential and normal distributions for an asymmetric Laplace distribution, which facilitates the commonly used collapsed Gibbs sampling algorithms for the DP mixture models. Additionally, we discuss the asymptotic property of the sempiparametric quantile regression model in terms of consistency of posterior distribution. Simulation studies and real data application examples illustrate the proposed method and compare it with Bayesian quantile regression methods in the literature.     

Bayesian analysis of generalized elliptical semi-parametric models

In this paper, we study the statistical inference based on the Bayesian approach for regression models with the assumption that independent additive errors follow normal, Student-t, slash, contaminated normal, Laplace or symmetric hyperbolic distribution, where both location and dispersion parameters of the response variable distribution include nonparametric additive components approximated by B-splines. This class of models provides a rich set of symmetric distributions for the model error. Some of these distributions have heavier or lighter tails than the normal as well as different levels of kurtosis. In order to draw samples of the posterior distribution of the interest parameters, we propose an efficient Markov Chain Monte Carlo (MCMC) algorithm, which combines Gibbs sampler and Metropolis–Hastings algorithms. The performance of the proposed MCMC algorithm is assessed through simulation experiments. We apply the proposed methodology to a real data set. The proposed methodology is implemented in the R package BayesGESM using the function gesm().     

Nonparametric Additive Instrumental Variable Estimator: A Group Shrinkage Estimation Perspective

In this article, we study a nonparametric approach regarding a general nonlinear reduced form equation to achieve a better approximation of the optimal instrument. Accordingly, we propose the nonparametric additive instrumental variable estimator (NAIVE) with the adaptive group Lasso. We theoretically demonstrate that the proposed estimator is root-n consistent and asymptotically normal. The adaptive group Lasso helps us select the valid instruments while the dimensionality of potential instrumental variables is allowed to be greater than the sample size. In practice, the degree and knots of B-spline series are selected by minimizing the BIC or EBIC criteria for each nonparametric additive component in the reduced form equation. In Monte Carlo simulations, we show that the NAIVE has the same performance as the linear instrumental variable (IV) estimator for the truly linear reduced form equation. On the other hand, the NAIVE performs much better in terms of bias and mean squared errors compared to other alternative estimators under the high-dimensional nonlinear reduced form equation. We further illustrate our method in an empirical study of international trade and growth. Our findings provide a stronger evidence that international trade has a significant positive effect on economic growth.     

Inference in generalized additive mixed modelsby using smoothing splines

Generalized additive mixed models are proposed for overdispersed and correlated data, which arise frequently in studies involving clustered, hierarchical and spatial designs. This class of models allows flexible functional dependence of an outcome variable on covariates by using nonparametric regression, while accounting for correlation between observations by using random effects. We estimate nonparametric functions by using smoothing splines and jointly estimate smoothing parameters and variance components by using marginal quasi-likelihood. Because numerical integration is often required by maximizing the objective functions, double penalized quasi-likelihood is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the method proposed is that it allows us to make systematic inference on all model components within a unified parametric mixed model framework and can be easily implemented by fitting a working generalized linear mixed model by using existing statistical software. A bias correction procedure is also proposed to improve the performance of double penalized quasi-likelihood for sparse data. We illustrate the method with an application to infectious disease data and we evaluate its performance through simulation.     

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.     

Estimation and Variable Selection for Semiparametric Additive Partial Linear Models (SS-09-140)

Semiparametric additive partial linear models, containing both linear and nonlinear additive components, are more flexible compared to linear models, and they are more efficient compared to general nonparametric regression models because they reduce the problem known as "curse of dimensionality". In this paper, we propose a new estimation approach for these models, in which we use polynomial splines to approximate the additive nonparametric components and we derive the asymptotic normality for the resulting estimators of the parameters. We also develop a variable selection procedure to identify significant linear components using the smoothly clipped absolute deviation penalty (SCAD), and we show that the SCAD-based estimators of non-zero linear components have an oracle property. Simulations are performed to examine the performance of our approach as compared to several other variable selection methods such as the Bayesian Information Criterion and Least Absolute Shrinkage and Selection Operator (LASSO). The proposed approach is also applied to real data from a nutritional epidemiology study, in which we explore the relationship between plasma beta-carotene levels and personal characteristics (e.g., age, gender, body mass index (BMI), etc.) as well as dietary factors (e.g., alcohol consumption, smoking status, intake of cholesterol, etc.).     

Nonparametric Bayesian survival analysis using mixtures of Weibull distributions

Bayesian nonparametric methods have been applied to survival analysis problems since the emergence of the area of Bayesian nonparametrics. However, the use of the flexible class of Dirichlet process mixture models has been rather limited in this context. This is, arguably, to a large extent, due to the standard way of fitting such models that precludes full posterior inference for many functionals of interest in survival analysis applications. To overcome this difficulty, we provide a computational approach to obtain the posterior distribution of general functionals of a Dirichlet process mixture. We model the survival distribution employing a flexible Dirichlet process mixture, with a Weibull kernel, that yields rich inference for several important functionals. In the process, a method for hazard function estimation emerges. Methods for simulation-based model fitting, in the presence of censoring, and for prior specification are provided. We illustrate the modeling approach with simulated and real data.     

GMM nonparametric correction methods for logistic regression with error‐contaminated covariates and partially observed instrumental variables

We consider logistic regression with covariate measurement error. Most existing approaches require certain replicates of the error‐contaminated covariates, which may not be available in the data. We propose generalized method of moments (GMM) nonparametric correction approaches that use instrumental variables observed in a calibration subsample. The instrumental variable is related to the underlying true covariates through a general nonparametric model, and the probability of being in the calibration subsample may depend on the observed variables. We first take a simple approach adopting the inverse selection probability weighting technique using the calibration subsample. We then improve the approach based on the GMM using the whole sample. The asymptotic properties are derived, and the finite sample performance is evaluated through simulation studies and an application to a real data set.     

Bayesian model comparison for compartmental models with applications in positron emission tomography

We develop strategies for Bayesian modelling as well as model comparison, averaging and selection for compartmental models with particular emphasis on those that occur in the analysis of positron emission tomography (PET) data. Both modelling and computational issues are considered. Biophysically inspired informative priors are developed for the problem at hand, and by comparison with default vague priors it is shown that the proposed modelling is not overly sensitive to prior specification. It is also shown that an additive normal error structure does not describe measured PET data well, despite being very widely used, and that within a simple Bayesian framework simultaneous parameter estimation and model comparison can be performed with a more general noise model. The proposed approach is compared with standard techniques using both simulated and real data. In addition to good, robust estimation performance, the proposed technique provides, automatically, a characterisation of the uncertainty in the resulting estimates which can be considerable in applications such as PET.     

A bayesian estimator for the dependence function of a bivariate extreme‐value distribution

Any continuous bivariate distribution can be expressed in terms of its margins and a unique copula. In the case of extreme‐value distributions, the copula is characterized by a dependence function while each margin depends on three parameters. The authors propose a Bayesian approach for the simultaneous estimation of the dependence function and the parameters defining the margins. They describe a nonparametric model for the dependence function and a reversible jump Markov chain Monte Carlo algorithm for the computation of the Bayesian estimator. They show through simulations that their estimator has a smaller mean integrated squared error than classical nonparametric estimators, especially in small samples. They illustrate their approach on a hydrological data set.     

Additive nonparametric regression with autocorrelated errors

A Bayesian approach is presented for nonparametric estimation of an additive regression model with autocorrelated errors. Each of the potentially non-linear components is modelled as a regression spline using many knots, while the errors are modelled by a high order stationary autoregressive process parameterized in terms of its autocorrelations. The distribution of significant knots and partial autocorrelations is accounted for using subset selection. Our approach also allows the selection of a suitable transformation of the dependent variable. All aspects of the model are estimated simultaneously by using the Markov chain Monte Carlo method. It is shown empirically that the approach proposed works well on several simulated and real examples.     

Bayesian nonparametric modelling of the link function in the single-index model using a Bernstein–Dirichlet process prior

This paper proposes the use of the Bernstein–Dirichlet process prior for a new nonparametric approach to estimating the link function in the single-index model (SIM). The Bernstein–Dirichlet process prior has so far mainly been used for nonparametric density estimation. Here we modify this approach to allow for an approximation of the unknown link function. Instead of the usual Gaussian distribution, the error term is assumed to be asymmetric Laplace distributed which increases the flexibility and robustness of the SIM. To automatically identify truly active predictors, spike-and-slab priors are used for Bayesian variable selection. Posterior computations are performed via a Metropolis-Hastings-within-Gibbs sampler using a truncation-based algorithm for stick-breaking priors. We compare the efficiency of the proposed approach with well-established techniques in an extensive simulation study and illustrate its practical performance by an application to nonparametric modelling of the power consumption in a sewage treatment plant.     

Semiparametric Bayesian inference for regression models

This paper presents a method for Bayesian inference for the regression parameters in a linear model with independent and identically distributed errors that does not require the specification of a parametric family of densities for the error distribution. This method first selects a nonparametric kernel density estimate of the error distribution which is unimodal and based on the least-squares residuals. Once the error distribution is selected, the Metropolis algorithm is used to obtain the marginal posterior distribution of the regression parameters. The methodology is illustrated with data sets, and its performance relative to standard Bayesian techniques is evaluated using simulation results.     

Generalized linear models with covariate measurement error and unknown link function

Generalized linear models (GLMs) with error-in-covariates are useful in epidemiological research due to the ubiquity of non-normal response variables and inaccurate measurements. The link function in GLMs is chosen by the user depending on the type of response variable, frequently the canonical link function. When covariates are measured with error, incorrect inference can be made, compounded by incorrect choice of link function. In this article we propose three flexible approaches for handling error-in-covariates and estimating an unknown link simultaneously. The first approach uses a fully Bayesian (FB) hierarchical framework, treating the unobserved covariate as a latent variable to be integrated over. The second and third are approximate Bayesian approach which use a Laplace approximation to marginalize the variables measured with error out of the likelihood. Our simulation results show support that the FB approach is often a better choice than the approximate Bayesian approaches for adjusting for measurement error, particularly when the measurement error distribution is misspecified. These approaches are demonstrated on an application with binary response.     

On asymptotic properties of Bayesian partially linear models

In this paper, we present large sample properties of a partially linear model from the Bayesian perspective, in which responses are explained by the semiparametric regression model with the additive form of the linear component and the nonparametric component. For this purpose, we investigate asymptotic behaviors of posterior distributions in terms of consistency. Specifically, we deal with a specific Bayesian partially linear regression model with additive noises in which the nonparametric component is modeled using Gaussian process priors. Under the Bayesian partially linear model using Gaussian process priors, we focus on consistency of posterior distribution and consistency of the Bayes factor, and extend these results to generalized additive regression models and study their asymptotic properties. In addition we illustrate the asymptotic properties based on empirical analysis through simulation studies.     

Bayesian fractional polynomials

This paper sets out to implement the Bayesian paradigm for fractional polynomial models under the assumption of normally distributed error terms. Fractional polynomials widen the class of ordinary polynomials and offer an additive and transportable modelling approach. The methodology is based on a Bayesian linear model with a quasi-default hyper-g prior and combines variable selection with parametric modelling of additive effects. A Markov chain Monte Carlo algorithm for the exploration of the model space is presented. This theoretically well-founded stochastic search constitutes a substantial improvement to ad hoc stepwise procedures for the fitting of fractional polynomial models. The method is applied to a data set on the relationship between ozone levels and meteorological parameters, previously analysed in the literature.     

Simultaneous Tolerance Limits and a Unified Approach to some Nonparametric Theory

In this paper nonparametric simultaneous tolerance limits are developed using rectangle probabilities for uniform order statistics. Consideration is given to the handling of censored data, and some comparisons are made with the parametric normal theory. The nonparametric regional estimation techniques of (i) confidence bands for a distribution function, (ii) simultaneous confidence intervals for quantiles and (iii) simultaneous tolerance limits are unified. A Bayesian approach is also discussed.     

A general approach to heteroscedastic linear regression

Our article presents a general treatment of the linear regression model, in which the error distribution is modelled nonparametrically and the error variances may be heteroscedastic, thus eliminating the need to transform the dependent variable in many data sets. The mean and variance components of the model may be either parametric or nonparametric, with parsimony achieved through variable selection and model averaging. A Bayesian approach is used for inference with priors that are data-based so that estimation can be carried out automatically with minimal input by the user. A Dirichlet process mixture prior is used to model the error distribution nonparametrically; when there are no regressors in the model, the method reduces to Bayesian density estimation, and we show that in this case the estimator compares favourably with a well-regarded plug-in density estimator. We also consider a method for checking the fit of the full model. The methodology is applied to a number of simulated and real examples and is shown to work well.     

A flexible approach for multivariate mixed-effects models with non-ignorable missing values

We propose a flexible model approach for the distribution of random effects when both response variables and covariates have non-ignorable missing values in a longitudinal study. A Bayesian approach is developed with a choice of nonparametric prior for the distribution of random effects. We apply the proposed method to a real data example from a national long-term survey by Statistics Canada. We also design simulation studies to further check the performance of the proposed approach. The result of simulation studies indicates that the proposed approach outperforms the conventional approach with normality assumption when the heterogeneity in random effects distribution is salient.     

Hierarchical Generalized Linear Models and Frailty Models with Bayesian Nonparametric Mixing

This paper proposes Bayesian nonparametric mixing for some well-known and popular models. The distribution of the observations is assumed to contain an unknown mixed effects term which includes a fixed effects term, a function of the observed covariates, and an additive or multiplicative random effects term. Typically these random effects are assumed to be independent of the observed covariates and independent and identically distributed from a distribution from some known parametric family. This assumption may be suspect if either there is interaction between observed covariates and unobserved covariates or the fixed effects predictor of observed covariates is misspecified. Another cause for concern might be simply that the covariates affect more than just the location of the mixed effects distribution. As a consequence the distribution of the random effects could be highly irregular in modality and skewness leaving parametric families unable to model the distribution adequately. This paper therefore proposes a Bayesian nonparametric prior for the random effects to capture possible deviances in modality and skewness and to explore the observed covariates' effect on the distribution of the mixed effects.