A comparison of nonparametric priors in hierarchical mixture modelling for AFT regression

We will pursue a Bayesian nonparametric approach in the hierarchical mixture modelling of lifetime data in two situations: density estimation, when the distribution is a mixture of parametric densities with a nonparametric mixing measure, and accelerated failure time (AFT) regression modelling, when the same type of mixture is used for the distribution of the error term. The Dirichlet process is a popular choice for the mixing measure, yielding a Dirichlet process mixture model for the error; as an alternative, we also allow the mixing measure to be equal to a normalized inverse-Gaussian prior, built from normalized inverse-Gaussian finite dimensional distributions, as recently proposed in the literature. Markov chain Monte Carlo techniques will be used to estimate the predictive distribution of the survival time, along with the posterior distribution of the regression parameters. A comparison between the two models will be carried out on the grounds of their predictive power and their ability to identify the number of components in a given mixture density.     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

Bayesian inference for semiparametric regression using a Fourier representation

This paper presents the Bayesian analysis of a semiparametric regression model that consists of parametric and nonparametric components. The nonparametric component is represented with a Fourier series where the Fourier coefficients are assumed a priori to have zero means and to decay to 0 in probability at either algebraic or geometric rates. The rate of decay controls the smoothness of the response function. The posterior analysis automatically selects the amount of smoothing that is coherent with the model and data. Posterior probabilities of the parametric and semiparametric models provide a method for testing the parametric model against a non-specific alternative. The Bayes estimator's mean integrated squared error compares favourably with the theoretically optimal estimator for kernel regression.     

Bayesian variable selection for proportional hazards models

The authors consider the problem of Bayesian variable selection for proportional hazards regression models with right censored data. They propose a semi-parametric approach in which a nonparametric prior is specified for the baseline hazard rate and a fully parametric prior is specified for the regression coefficients. For the baseline hazard, they use a discrete gamma process prior, and for the regression coefficients and the model space, they propose a semi-automatic parametric informative prior specification that focuses on the observables rather than the parameters. To implement the methodology, they propose a Markov chain Monte Carlo method to compute the posterior model probabilities. Examples using simulated and real data are given to demonstrate the methodology.     

Semiparametric multiple kernel estimators and model diagnostics for count regression functions

Abstract

This study concerns semiparametric approaches to estimate discrete multivariate count regression functions. The semiparametric approaches investigated consist of combining discrete multivariate nonparametric kernel and parametric estimations such that (i) a prior knowledge of the conditional distribution of model response may be incorporated and (ii) the bias of the traditional nonparametric kernel regression estimator of Nadaraya-Watson may be reduced. We are precisely interested in combination of the two estimations approaches with some asymptotic properties of the resulting estimators. Asymptotic normality results were showed for nonparametric correction terms of parametric start function of the estimators. The performance of discrete semiparametric multivariate kernel estimators studied is illustrated using simulations and real count data. In addition, diagnostic checks are performed to test the adequacy of the parametric start model to the true discrete regression model. Finally, using discrete semiparametric multivariate kernel estimators provides a bias reduction when the parametric multivariate regression model used as start regression function belongs to a neighborhood of the true regression model.     

Bayesian semiparametric inference for the accelerated failure-time model

Bayesian semiparametric inference is considered for a loglinear model. This model consists of a parametric component for the regression coefficients and a nonparametric component for the unknown error distribution. Bayesian analysis is studied for the case of a parametric prior on the regression coefficients and a mixture-of-Dirichlet-processes prior on the unknown error distribution. A Markov-chain Monte Carlo (MCMC) method is developed to compute the features of the posterior distribution. A model selection method for obtaining a more parsimonious set of predictors is studied. The method adds indicator variables to the regression equation. The set of indicator variables represents all the possible subsets to be considered. A MCMC method is developed to search stochastically for the best subset. These procedures are applied to two examples, one with censored data.     

Nonparametric predictive inference for stock returns

In finance, inferences about future asset returns are typically quantified with the use of parametric distributions and single-valued probabilities. It is attractive to use less restrictive inferential methods, including nonparametric methods which do not require distributional assumptions about variables, and imprecise probability methods which generalize the classical concept of probability to set-valued quantities. Main attractions include the flexibility of the inferences to adapt to the available data and that the level of imprecision in inferences can reflect the amount of data on which these are based. This paper introduces nonparametric predictive inference (NPI) for stock returns. NPI is a statistical approach based on few assumptions, with inferences strongly based on data and with uncertainty quantified via lower and upper probabilities. NPI is presented for inference about future stock returns, as a measure for risk and uncertainty, and for pairwise comparison of two stocks based on their future aggregate returns. The proposed NPI methods are illustrated using historical stock market data.     

Discrete Nonparametric Kernel and Parametric Methods for the Modeling of Pavement Deterioration

This article is concerned with one discrete nonparametric kernel and two parametric regression approaches for providing the evolution law of pavement deterioration. The first parametric approach is a survival data analysis method; and the second is a nonlinear mixed-effects model. The nonparametric approach consists of a regression estimator using the discrete associated kernels. Some asymptotic properties of the discrete nonparametric kernel estimator are shown as, in particular, its almost sure consistency. Moreover, two data-driven bandwidth selection methods are also given, with a new theoretical explicit expression of optimal bandwidth provided for this nonparametric estimator. A comparative simulation study is realized with an application of bootstrap methods to a measure of statistical accuracy.     

Estimation of regression parameters in a semiparametric transformation model

A semiparametric approach to model skewed/heteroscedastic regression data is discussed. We work with a semiparametric transform-both-sides regression model, which contains a parametric regression function and a nonparametric transformation. This model is adequate when the relationship between the median response and the explanatory variable has been specified by a theoretical result or a previous empirical study. The transform-both-sides model with a parametric transformation has been studied extensively and applied successfully to a number data sets. Allowing a nonparametric transformation function increases the flexibility of the model. In this article, we estimate the nonparametric transformation function by the conditional kernel density approach developed by Wang and Ruppert (1995), and then use a pseudo-maximum likelihood estimator to estimate the regression parameters. This estimate of the regression parameters has not been studied previously. In this article, the asymptotic distribution of this pseudo-MLE is derived. We also show that when σ, the standard deviation of the error, goes to zero (small σ asymptotics), this estimator is adaptive. Adaptive means that the regression parameters are estimated as precisely as when the transformation is known exactly. A similar result holds in the parametric approaches of Carroll and Ruppert (1984) and Ruppert and Aldershof (1989). Simulated and real examples are provided to illustrate the performance of the proposed estimator for finite sample size.     

Nonparametric modeling of clustered customer survival data

We incorporate a random clustering effect into the nonparametric version of Cox Proportional Hazards model to characterize clustered survival data. The simulation studies provide evidence that clustered survival data can be better characterized through a nonparametric model. Predictive accuracy of the nonparametric model is affected by number of clusters and distribution of the random component accounting for clustering effect. As the functional form of the covariate departs from linearity, the nonparametric model is becoming more advantageous over the parametric counterpart. Finally, nonparametric is better than parametric model when data are highly heterogenous and/or there is misspecification error.     

Prediction bands for the EDF of a future sample

We develop both nonparametric and parametric methods for obtaining prediction bands for the empirical distribution function (EDF) of a future sample. These methods yield simultaneous prediction intervals for all order statistics of the future sample, and they also correspond to tests for the two-sample problem. The nonparametric prediction bands correspond to the two-sample Kolmogorov-Smirnov test and related nonparametric tests, but the parametric prediction bands correspond to entirely new parametric two-sample tests. The parametric prediction bands tend to outperform the nonparametric bands when the parametric assumptions hold, but they may have true coverage probabilities well below their nominal levels when the parametric assumptions fail. A new computational algorithm is used to obtain critical values in the nonparametric case.     

Estimating the error distribution function in semiparametric additive regression models

We consider semiparametric additive regression models with a linear parametric part and a nonparametric part, both involving multivariate covariates. For the nonparametric part we assume two models. In the first, the regression function is unspecified and smooth; in the second, the regression function is additive with smooth components. Depending on the model, the regression curve is estimated by suitable least squares methods. The resulting residual-based empirical distribution function is shown to differ from the error-based empirical distribution function by an additive expression, up to a uniformly negligible remainder term. This result implies a functional central limit theorem for the residual-based empirical distribution function. It is used to test for normal errors.     

A Bayesian Non-parametric Approach to Survival Analysis Using Polya Trees

This paper presents a Bayesian non-parametric approach to survival analysis based on arbitrarily right censored data. The analysis is based on posterior predictive probabilities using a Polya tree prior distribution on the space of probability measures on [0, ∞). In particular we show that the estimate generalizes the classical Kaplanndash;Meier non-parametric estimator, which is obtained in the limiting case as the weight of prior information tends to zero.     

On predictive distributions and Bayesian networks

In this paper we are interested in discrete prediction problems for a decision-theoretic setting, where the task is to compute the predictive distribution for a finite set of possible alternatives. This question is first addressed in a general Bayesian framework, where we consider a set of probability distributions defined by some parametric model class. Given a prior distribution on the model parameters and a set of sample data, one possible approach for determining a predictive distribution is to fix the parameters to the instantiation with the maximum a posteriori probability. A more accurate predictive distribution can be obtained by computing the evidence (marginal likelihood), i.e., the integral over all the individual parameter instantiations. As an alternative to these two approaches, we demonstrate how to use Rissanen's new definition of stochastic complexity for determining predictive distributions, and show how the evidence predictive distribution with Jeffrey's prior approaches the new stochastic complexity predictive distribution in the limit with increasing amount of sample data. To compare the alternative approaches in practice, each of the predictive distributions discussed is instantiated in the Bayesian network model family case. In particular, to determine Jeffrey's prior for this model family, we show how to compute the (expected) Fisher information matrix for a fixed but arbitrary Bayesian network structure. In the empirical part of the paper the predictive distributions are compared by using the simple tree-structured Naive Bayes model, which is used in the experiments for computational reasons. The experimentation with several public domain classification datasets suggest that the evidence approach produces the most accurate predictions in the log-score sense. The evidence-based methods are also quite robust in the sense that they predict surprisingly well even when only a small fraction of the full training set is used.     

APPLICATION OF SEMIPARAMETRIC REGRESSION MODELS IN THE ANALYSIS OF CAPTURE-RECAPTURE EXPERIMENTS

If the capture probabilities in a capture‐recapture experiment depend on covariates, parametric models may be fitted and the population size may then be estimated. Here a semiparametric model for the capture probabilities that allows both continuous and categorical covariates is developed. Kernel smoothing and profile estimating equations are used to estimate the nonparametric and parametric components. Analytic forms of the standard errors are derived, which allows an empirical bias bandwidth selection procedure to be used to estimate the bandwidth. The method is evaluated in simulations and is applied to a real data set concerning captures of Prinia flaviventris, which is a common bird species in Southeast Asia.     

Nonparametric estimation of a switching regression model

High-dimensional data often exhibit multi-collinearity, leading to unstable regression coefficients. To address sample selection bias and problems associated with high dimensionality, principal components were extracted and used as predictors in a switching regression model. Since principal component regression often results to decline in predictive ability due to the selection of few principal components, we formulate the model with nonparametric function of principal components in lieu of individual predictors. Simulation studies indicated better predictive ability for nonparametric principal component switching regression over the parametric counterpart while mitigating the adverse effects of multi-collinearity and high dimensionality.     

General purpose multiply robust data integration procedures for handling nonprobability samples

In recent years, there has been an increased interest in combining probability and nonprobability samples. Nonprobability sample are cheaper and quicker to conduct but the resulting estimators are vulnerable to bias as the participation probabilities are unknown. To adjust for the potential bias, estimation procedures based on parametric or nonparametric models have been discussed in the literature. However, the validity of the resulting estimators relies heavily on the validity of the underlying models. Also, nonparametric approaches may suffer from the curse of dimensionality and poor efficiency. We propose a data integration approach by combining multiple outcome regression models and propensity score models. The proposed approach can be used for estimating general parameters including totals, means, distribution functions, and percentiles. The resulting estimators are multiply robust in the sense that they remain consistent if all but one model are misspecified. The asymptotic properties of point and variance estimators are established. The results from a simulation study show the benefits of the proposed method in terms of bias and efficiency. Finally, we apply the proposed method using data from the Korea National Health and Nutrition Examination Survey and data from the National Health Insurance Sharing Services.     

A Nonparametric Frailty Model for Clustered Survival Data

Clayton-type counting process formulations for survival data and parametric gamma models for cluster-specific frailty quantities are now routinely applied in analyses of clustered survival data. On the other hand, although nonparametric frailty models have been studied, they are not used much in practice. In this article, the distribution of the frailty terms is assumed to be an unknown random variable. The unknown frailty distribution is then modelled completely with a Dirichlet process prior. This prior assigns cluster units into sub-classes whose members have the same random frailty effect. The Gibbs sampler algorithm is used for computing posterior parameter estimates of the fixed effect hazards regression and the frailty distribution. The methodology is used to analyze community-clustered child survival in sub-Saharan Africa. The results show that the communities could be separated into fewer distinct classes of risk of childhood mortality; the fewer classes could be studied easily in order to provide useful guidance on the more effective use of resources for child health intervention programmes.     

Bayesian Semiparametric Modelling in Quantile Regression

Abstract. We propose a Bayesian semiparametric methodology for quantile regression modelling. In particular, working with parametric quantile regression functions, we develop Dirichlet process mixture models for the error distribution in an additive quantile regression formulation. The proposed non‐parametric prior probability models allow the shape of the error density to adapt to the data and thus provide more reliable predictive inference than models based on parametric error distributions. We consider extensions to quantile regression for data sets that include censored observations. Moreover, we employ dependent Dirichlet processes to develop quantile regression models that allow the error distribution to change non‐parametrically with the covariates. Posterior inference is implemented using Markov chain Monte Carlo methods. We assess and compare the performance of our models using both simulated and real data sets.     

Asymptotic distributions of error density and distribution function estimators in nonparametric regression

This paper considers the problem of estimating the error density and distribution functions in nonparametric regression models. The asymptotic distribution of a suitably standardized density estimator at a fixed point is shown to be normal while that of the maximum of a suitably normalized deviation of the density estimator from the true density function is the same as in the case of the one sample set up. Finally, the standardized residual empirical process is shown to be uniformly close to the similarly standardized empirical process of the errors. This paper thus generalizes some of the well known results about the residual density estimators and the empirical process in parametric regression models to nonparametric regression models, thereby enhancing the domain of their applications.