Bayesian regression on non-parametric mixed-effect models with shape-restricted Bernstein polynomials

We develop a Bayesian estimation method to non-parametric mixed-effect models under shape-constrains. The approach uses a hierarchical Bayesian framework and characterizations of shape-constrained Bernstein polynomials (BPs). We employ Markov chain Monte Carlo methods for model fitting, using a truncated normal distribution as the prior for the coefficients of BPs to ensure the desired shape constraints. The small sample properties of the Bayesian shape-constrained estimators across a range of functions are provided via simulation studies. Two real data analysis are given to illustrate the application of the proposed method.     

Two stage least squares estimation in structural cointegration models

This paper considers single-equation estimation of simultaneous equation models with integrated processes. The aim of the paper is to investigate asymptotic and small sample properties of some estimators in this framework. We deal with two groups of estimators: such that originally were designated for reduced form estimation and such for simultaneous equation models. In the first group we deal with Least Squares and Fully Modified Least Squares. The second group comprises Two Stage Least Squares and two modifications of it. The asymptotic analysis in section 2 shows that it is true that all estimators are super-consistent in this context but in principle, only the methods of the second group enable valid inference. Section 3 presents the results of a simulation study which is designed for specific problems of simultaneous equation models. This paper was presented at the European Meeting of the Econometric Society in Istanbul, 1996. The author thanks an anonymous referee for helpful suggestions.     

A New Kernel Distribution Function Estimator Based on a Non-parametric Transformation of the Data

Abstract.  A new kernel distribution function (df) estimator based on a non-parametric transformation of the data is proposed. It is shown that the asymptotic bias and mean squared error of the estimator are considerably smaller than that of the standard kernel df estimator. For the practical implementation of the new estimator a data-based choice of the bandwidth is proposed. Two possible areas of application are the non-parametric smoothed bootstrap and survival analysis. In the latter case new estimators for the survival function and the mean residual life function are derived.     

Non‐parametric estimation for time-dependent AUC

The area under the receiver operating characteristic (ROC) curve (AUC) is one of the commonly used measure to evaluate or compare the predictive ability of markers to the disease status. Motivated by an angiographic coronary artery disease (CAD) study, our objective is mainly to evaluate and compare the performance of several baseline plasma levels in the prediction of CAD-related vital status over time. Based on censored survival data, the non-parametric estimators are proposed for the time-dependent AUC. The limiting Gaussian processes of the estimators and the estimated asymptotic variance–covariance functions enable us to further construct confidence bands and develop testing procedures. Applications and finite sample properties of the proposed estimation methods and inference procedures are demonstrated through the CAD-related death data from the British Columbia Vital Statistics Agency and Monte Carlo simulations.     

Testing and Merging Information for Effect Size Estimation

A large-sample test for testing the equality of two effect sizes is presented. The null and non-null distributions of the proposed test statistic are derived. Further, the problem of estimating the effect size is considered when it is a priori suspected that two effect sizes may be close to each other. The combined data from all the samples leads to more efficient estimator of the effect size. We propose a basis for optimally combining estimation problems when there is uncertainty concerning the appropriate statistical model-estimator to use in representing the sampling process. The objective here is to produce natural adaptive estimators with some good statistical properties. In the context of two bivariate statistical models, the expressions for the asymptotic mean squared error of the proposed estimators are derived and compared with the parallel expressions for the benchmark estimators. We demonstrate that the suggested preliminary test estimator has superior asymptotic mean squared error performance relative to the benchmark and pooled estimators. A simulation study and application of the methodology to real data are presented.     

Polynomial Spline Estimation for a Generalized Additive Coefficient Model

Abstract.  We study a semiparametric generalized additive coefficient model (GACM), in which linear predictors in the conventional generalized linear models are generalized to unknown functions depending on certain covariates, and approximate the non-parametric functions by using polynomial spline. The asymptotic expansion with optimal rates of convergence for the estimators of the non-parametric part is established. Semiparametric generalized likelihood ratio test is also proposed to check if a non-parametric coefficient can be simplified as a parametric one. A conditional bootstrap version is suggested to approximate the distribution of the test under the null hypothesis. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed methods. We further apply the proposed model and methods to a data set from a human visceral Leishmaniasis study conducted in Brazil from 1994 to 1997. Numerical results outperform the traditional generalized linear model and the proposed GACM is preferable.     

Statistic inference for a single-index ARCH-M model

For a single-index autoregressive conditional heteroscedastic (ARCH-M) model, estimators of the parametric and non parametric components are proposed by the profile likelihood method. The research results had shown that all the estimators have consistency and the parametric estimators have asymptotic normality. We extend this line of research by deriving the asymptotic normality of the non parametric estimator. Based on the asymptotic properties, we propose Wald statistic and generalized likelihood ratio statistic to investigate the testing problems for ARCH effect and goodness of fit, respectively. A simulation study is conducted to evaluate the finite-sample performance of the proposed estimation methodology and testing procedure.     

A Non-stationary Cox Model

The purpose of this paper is to consider the problem of statistical inference about a hazard rate function that is specified as the product of a parametric regression part and a non-parametric baseline hazard. Unlike Cox's proportional hazard model, the baseline hazard not only depends on the duration variable, but also on the starting date of the phenomenon of interest. We propose a new estimator of the regression parameter which allows for non-stationarity in the hazard rate. We show that it is asymptotically normal at root- n and that its asymptotic variance attains the information bound for estimation of the regression coefficient. We also consider an estimator of the integrated baseline hazard, and determine its asymptotic properties. The finite sample performance of our estimators are studied.     

A non-Bayesian predictive approach for statistical calibration

A non-Bayesian predictive approach for statistical calibration is introduced. This is based on particularizing to the calibration setting the general definition of non-Bayesian (or frequentist) predictive probability density proposed by Harris [Predictive fit for natural exponential families, Biometrika 76 (1989), pp. 675–684]. The new method is elaborated in detail in case of Gaussian linear univariate calibration. Through asymptotic analysis and simulation results with moderate sample size, it is shown that the non-Bayesian predictive estimator of the unknown parameter of interest in calibration (commonly, a substance concentration) favourably compares with previous estimators such as the classical and inverse estimators, especially for extrapolation problems. A further advantage of the non-Bayesian predictive approach is that it provides not only point estimates but also a predictive likelihood function that allows the researcher to explore the plausibility of any possible parameter value, which is also briefly illustrated. Furthermore, the introduced approach offers a general framework that can be applied for calibrating on the basis of any parametric statistical model, so making it potentially useful for nonlinear and non-Gaussian calibration problems.     

NONPARAMETRIC ESTIMATION OF GENEWISE VARIANCE FOR MICROARRAY DATA

Estimation of genewise variance arises from two important applications in microarray data analysis: selecting significantly differentially expressed genes and validation tests for normalization of microarray data. We approach the problem by introducing a two-way nonparametric model, which is an extension of the famous Neyman-Scott model and is applicable beyond microarray data. The problem itself poses interesting challenges because the number of nuisance parameters is proportional to the sample size and it is not obvious how the variance function can be estimated when measurements are correlated. In such a high-dimensional nonparametric problem, we proposed two novel nonparametric estimators for genewise variance function and semiparametric estimators for measurement correlation, via solving a system of nonlinear equations. Their asymptotic normality is established. The finite sample property is demonstrated by simulation studies. The estimators also improve the power of the tests for detecting statistically differentially expressed genes. The methodology is illustrated by the data from MicroArray Quality Control (MAQC) project.     

RANKED SET SAMPLING FROM LOCATION-SCALE FAMILIES OF SYMMETRIC DISTRIBUTIONS

Statistical inference based on ranked set sampling has primarily been motivated by nonparametric problems. However, the sampling procedure can provide an improved estimator of the population mean when the population is partially known. In this article, we consider estimation of the population mean and variance for the location-scale families of distributions. We derive and compare different unbiased estimators of these parameters based on rindependent replications of a ranked set sample of size n.Large sample properties, along with asymptotic relative efficiencies, help identify which estimators are best suited for different location-scale distributions.     

Bridge regression: Adaptivity and group selection

In high-dimensional regression problems regularization methods have been a popular choice to address variable selection and multicollinearity. In this paper we study bridge regression that adaptively selects the penalty order from data and produces flexible solutions in various settings. We implement bridge regression based on the local linear and quadratic approximations to circumvent the nonconvex optimization problem. Our numerical study shows that the proposed bridge estimators are a robust choice in various circumstances compared to other penalized regression methods such as the ridge, lasso, and elastic net. In addition, we propose group bridge estimators that select grouped variables and study their asymptotic properties when the number of covariates increases along with the sample size. These estimators are also applied to varying-coefficient models. Numerical examples show superior performances of the proposed group bridge estimators in comparisons with other existing methods.     

Bayesian inference on extreme value distribution using upper record values

In this paper we address estimation and prediction problems for extreme value distributions under the assumption that the only available data are the record values. We provide some properties and pivotal quantities, and derive unbiased estimators for the location and rate parameters based on these properties and pivotal quantities. In addition, we discuss mean-squared errors of the proposed estimators and exact confidence intervals for the rate parameter. In Bayesian inference, we develop objective Bayesian analysis by deriving non informative priors such as the Jeffrey, reference, and probability matching priors for the location and rate parameters. We examine the validity of the proposed methods through Monte Carlo simulations for various record values of size and present a real data set for illustration purposes.     

Quantile regression in functional linear semiparametric model

This paper proposes nonparametric estimation methods for functional linear semiparametric quantile regression, where the conditional quantile of the scalar responses is modelled by both scalar and functional covariates and an additional unknown nonparametric function term. The slope function is estimated using the functional principal component basis and the nonparametric function is approximated by a piecewise polynomial function. The asymptotic distribution of the estimators of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. The asymptotic distribution of the estimator of the unknown nonparametric function is also established. Simulation studies are conducted to investigate the finite-sample performance of the proposed estimators. The proposed methodology is demonstrated by analysing a real data from ADHD-200 sample.     

Inferences in dynamic logit models in semi-parametric setup for repeated binary data

Binary dynamic fixed and mixed logit models are extensively studied in the literature. These models are developed to examine the effects of certain fixed covariates through a parametric regression function as a part of the models. However, there are situations where one may like to consider more covariates in the model but their direct effect is not of interest. In this paper we propose a generalization of the existing binary dynamic logit (BDL) models to the semi-parametric longitudinal setup to address this issue of additional covariates. The regression function involved in such a semi-parametric BDL model contains (i) a parametric linear regression function in some primary covariates, and (ii) a non-parametric function in certain secondary covariates. We use a simple semi-parametric conditional quasi-likelihood approach for consistent estimation of the non-parametric function, and a semi-parametric likelihood approach for the joint estimation of the main regression and dynamic dependence parameters of the model. The finite sample performance of the estimation approaches is examined through a simulation study. The asymptotic properties of the estimators are also discussed. The proposed model and the estimation approaches are illustrated by reanalysing a longitudinal infectious disease data.     

Asymptotic distributions of some test statistics for a linear functional relationship

We consider the testing problems of the structural parameters for the multivariate linear functional relationship model. We treat the likelihood ratio test statistics and the test statistics based on the asymptotic distributions of the maximum likelihood estimators. We derive their asymptotic distributions under each null hypothesis respectively. A simulation study is made to evaluate how we can trust our asymptotic results when the sample size is rather small.     

A Direct Approach to Understanding Posterior Consistency of Bayesian Regression Problems

Previous approaches to establishing posterior consistency of Bayesian regression problems have used general theorems that involve verifying sufficient conditions for posterior consistency. In this article, we consider a direct approach by computing the posterior density explicitly and evaluating its asymptotic behavior. For this purpose, we deal with a sample size dependent prior based on a truncated regression function with increasing sample size, and evaluate the asymptotic properties of the resulting posterior. Based on a concept called posterior density consistency, we attempt to understand posterior consistency. As an application, we illustrate that the posterior density of an orthogonal semiparametric regression model is consistent.     

Parametrically Guided Non-parametric Regression

We present a new approach to regression function estimation in which a non-parametric regression estimator is guided by a parametric pilot estimate with the aim of reducing the bias. New classes of parametrically guided kernel weighted local polynomial estimators are introduced and formulae for asymptotic expectation and variance, hence approximated mean squared error and mean integrated squared error, are derived. It is shown that the new classes of estimators have the very same large sample variance as the estimators in the standard non-parametric setting, while there is substantial room for reducing the bias if the chosen parametric pilot function belongs to a wide neighbourhood around the true regression line. Bias reduction is discussed in light of examples and simulations.     

Plug-in estimators for the mean value and variance functions in delayed renewal processes

Abstract

In this paper, we deal with the problem of estimating the delayed renewal and variance functions in delayed renewal processes. Two parametric plug-in estimators for these functions are proposed and their unbiasedness, asymptotic unbiasedness and consistency properties are investigated. The asymptotic normality of these estimators are established. Further, a method for the computation of the estimators is given. Finally, the performances of the estimators are evaluated for small sample sizes by a simulation study.     

Bayesian regression analysis of data with censored initiating and terminating times: applications to aids

Data with censored initiating and terminating times arises quite frequently in acquired immunodeficiency syndrome (AIDS) epidemiologic studies. Analysis of such data involves a complicated bivariate likelihood, which is difficult to deal with computationally. Bayesian analysis, op the other hand, presents added complexities that have yet to be resolved. By exploiting the simple form of a complete data likelihood and utilizing the power of a Markov Chain Monte Carlo (MCMC) algorithm, this paper presents a methodology for fitting Bayesian regression models to such data. The proposed methods extend the work of Sinha (1997), who considered non-parametric Bayesian analysis of this type of data. The methodology is illustiated with an application to a cohort of HIV infected hemophiliac patients.