Bayesian estimation on semiparametric models with shape restricted B-splines

In this paper, we develop a Bayesian estimation procedure for semiparametric models under shape constrains. The approach uses a hierarchical Bayes framework and characterizations of shape-constrained B-splines. We employ Markov chain Monte Carlo methods for model fitting, using a truncated normal distribution as the prior for the coefficients of basis functions to ensure the desired shape constraints. The small sample properties of the function estimators are provided via simulation and compared with existing methods. A real data analysis is conducted to illustrate the application of the proposed method.     

Enforcing shape constraints on a probability density estimate using an additive adjustment curve

A method is proposed for shape-constrained density estimation under a variety of constraints, including but not limited to unimodality, monotonicity, symmetry, and constraints on the number of inflection points of the density or its derivative. The method involves computing an adjustment curve that is used to bring a pre-existing pilot estimate into conformance with the specified shape restrictions. The pilot estimate may be obtained using any preferred estimator, and the optimal adjustment can be computed using fast, readily-available quadratic programming routines. This makes the proposed procedure generic and easy to implement.     

Piecewise FARIMA models for long-memory time series

We consider the problem of modelling a long-memory time series using piecewise fractional autoregressive integrated moving average processes. The number as well as the locations of structural break points (BPs) and the parameters of each regime are assumed to be unknown. A four-step procedure is proposed to find out the BPs and to estimate the parameters of each regime. Its effectiveness is shown by Monte Carlo simulations and an application to real traffic data modelling is considered.     

Isotonic L2-projection test for local monotonicity of a hazard

We introduce a test statistic for testing the null hypothesis that the sampling distribution has a strictly increasing hazard rate on a specified interval [a,b]$[a,b]$. It is based on a comparison of the empirical distribution function with a shape-constrained estimate, using the restriction that the hazard is increasing. Its asymptotic (normal) distribution was recently derived in Groeneboom and Jongbloed (submitted for publication). We discuss a bootstrap method for computing the critical values and compare the test, thus obtained, with other recently proposed methods in a simulation study. Moreover, we prove that the bootstrap method works asymptotically. In proving that the (smooth and isotonic) bootstrap method works, we derive some results that seem to be of independent interest.     

Bayesian analysis of hypothesis testing problems for general population: A Kullback–Leibler alternative

We consider a hypothesis problem with directional alternatives. We approach the problem from a Bayesian decision theoretic point of view and consider a situation when one side of the alternatives is more important or more probable than the other. We develop a general Bayesian framework by specifying a mixture prior structure and a loss function related to the Kullback–Leibler divergence. This Bayesian decision method is applied to Normal and Poisson populations. Simulations are performed to compare the performance of the proposed method with that of a method based on a classical z-test and a Bayesian method based on the “0–1” loss.     

Dynamic Bayesian analysis for irregularly and incompletely observed contingency tables

Pre-election surveys are usually conducted several times to forecast election results before the actual voting. It is common that each survey includes a substantial number of non-responses and that the successive survey results are seen as a stochastic multinomial time series evolving over time. We propose a dynamic Bayesian model to examine how multinomial time series evolve over time for the irregularly observed contingency tables and to determine how sensitively the dynamic structure reacts to an unexpected event, such as a candidate scandal. Further, we test whether non-responses are non-ignorable to determine if non-responses need to be imputed for better forecast. We also suggest a Bayesian method that overcomes the boundary solution problem and show that the proposed method outperforms the previous Bayesian methods. Our dynamic Bayesian model is applied to the two pre-election surveys for the 2007 Korea presidential candidate election and for the 1998 Ohio general election.     

A Bayesian approach for improved pavement performance prediction

We present a method for predicting future pavement distresses such as longitudinal cracking. These predicted distress values are used to plan road repairs. Large inherent variability in measured cracking and an extremely small number of observations are the nature of the pavement cracking data, which calls for a parametric Bayesian approach. We model theoretical pavement distress with a sigmoidal equation with coefficients based on prior engineering knowledge. We show that a Bayesian formulation akin to Kalman filtering gives sensible predictions and provides defendable uncertainty statements for predictions. The method is demonstrated on data collected by the Texas Transportation Institute at several sites in Texas. The predictions behave in a reasonable and statistically valid manner.     

A Bayesian approach to sequential analysis in post‐licensure vaccine safety surveillance

With rapid development of computing technology, Bayesian statistics have increasingly gained more attention in various areas of public health. However, the full potential of Bayesian sequential methods applied to vaccine safety surveillance has not yet been realized, despite acknowledged practical benefits and philosophical advantages of Bayesian statistics. In this paper, we describe how sequential analysis can be performed in a Bayesian paradigm in the field of vaccine safety. We compared the performance of the frequentist sequential method, specifically, Maximized Sequential Probability Ratio Test (MaxSPRT), and a Bayesian sequential method using simulations and a real world vaccine safety example. The performance is evaluated using three metrics: false positive rate, false negative rate, and average earliest time to signal. Depending on the background rate of adverse events, the Bayesian sequential method could significantly improve the false negative rate and decrease the earliest time to signal. We consider the proposed Bayesian sequential approach to be a promising alternative for vaccine safety surveillance.     

Bayesian Estimation of the Parameters of Bivariate Exponential Distributions

In this article, we develop an empirical Bayesian approach for the Bayesian estimation of parameters in four bivariate exponential (BVE) distributions. We have opted for gamma distribution as a prior for the parameters of the model in which the hyper parameters have been estimated based on the method of moments and maximum likelihood estimates (MLEs). A simulation study was conducted to compute empirical Bayesian estimates of the parameters and their standard errors. We use moment estimators or MLEs to estimate the hyper parameters of the prior distributions. Furthermore, we compare the posterior mode of parameters obtained by different prior distributions and the Bayesian estimates based on gamma priors are very close to the true values as compared to improper priors. We use MCMC method to obtain the posterior mean and compared the same using the improper priors and the classical estimates, MLEs.     

Prediction for future failures in Weibull distribution under hybrid censoring

In this paper, we consider the prediction of a future observation based on a type-I hybrid censored sample when the lifetime distribution of experimental units is assumed to be a Weibull random variable. Different classical and Bayesian point predictors are obtained. Bayesian predictors are obtained using squared error and linear-exponential loss functions. We also provide a simulation consistent method for computing Bayesian prediction intervals. Monte Carlo simulations are performed to compare the performances of the different methods, and one data analysis has been presented for illustrative purposes.     

Methods for Constructing Uncertainty Intervals for Queries of Bayesian Nets

In this paper the issue of finding uncertainty intervals for queries in a Bayesian Network is reconsidered. The investigation focuses on Bayesian Nets with discrete nodes and finite populations. An earlier asymptotic approach is compared with a simulation‐based approach, together with further alternatives, one based on a single sample of the Bayesian Net of a particular finite population size, and another which uses expected population sizes together with exact probabilities. We conclude that a query of a Bayesian Net should be expressed as a probability embedded in an uncertainty interval. Based on an investigation of two Bayesian Net structures, the preferred method is the simulation method. However, both the single sample method and the expected sample size methods may be useful and are simpler to compute. Any method at all is more useful than none, when assessing a Bayesian Net under development, or when drawing conclusions from an ‘expert’ system.     

Bayesian reliability when system and subsystem failure data are obtained in the same time period

Previously, Bayesian anomaly was reported for estimating reliability when subsystem failure data and system failure data were obtained from the same time period. As a result, a practical method for mitigating Bayesian anomaly was developed. In the first part of this paper, however, we show that the Bayesian anomaly can be avoided as long as the same failure information is incorporated in the model. In the second part of this paper, we consider a problem of estimating the Bayesian reliability when the failure count data on subsystems and systems are obtained from the same time period. We show that Bayesian anomaly does not exist when using the multinomial distribution with the Dirichlet prior distribution. A numerical example is given to compare the proposed method with the previous methods.     

Bayesian meta-analysis of correlation coefficients through power prior

This article proposes a Bayesian approach for meta-analysis of correlation coefficients through power prior. The primary purpose of this method is to allow meta-analytic researchers to evaluate the contribution and influence of each individual study to the estimated overall effect size though power prior. We use the relationship between high-performance work systems and financial performance as an example to illustrate how to apply this method. We also introduce free online software that can be used to conduct Bayesian meta-analysis proposed in this study. Implications and future directions are also discussed in this article.     

Bayesian quantile regression for hierarchical linear models

The paper proposes a Bayesian quantile regression method for hierarchical linear models. Existing approaches of hierarchical linear quantile regression models are scarce and most of them were not from the perspective of Bayesian thoughts, which is important for hierarchical models. In this paper, based on Bayesian theories and Markov Chain Monte Carlo methods, we introduce Asymmetric Laplace distributed errors to simulate joint posterior distributions of population parameters and across-unit parameters and then derive their posterior quantile inferences. We run a simulation as the proposed method to examine the effects on parameters induced by units and quantile levels; the method is also applied to study the relationship between Chinese rural residents' family annual income and their cultivated areas. Both the simulation and real data analysis indicate that the method is effective and accurate.     

Statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data

In this article, the statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data is discussed. The estimation of the parameters for Gompertz distribution is obtained using maximum likelihood method (MLE) and Bayesian method under three different loss functions. We also proved the existence and uniqueness of the MLE. The one-sample Bayesian prediction intervals are obtained. The work is done for different values of the parameters. We apply the Monto Carlo simulation to compare the proposed methods, also an example is discussed to construct the Prediction intervals.     

Bayesian analysis of seasonal variation when the sample size and the amplitude are small

We propose what appears to be the first Bayesian procedure for the analysis of seasonal variation when the sample size and the amplitude are small. Such data occur often in the medical sciences, where seasonality analyses and environmental considerations can help clarify disease etiologies. The method is explained in terms of a simple physico-geometric setting. We present the Bayesian version of a frequentist test that performs well. Two examples of real data illustrate the procedure's application.     

A Bayesian sequential design with adaptive randomization for 2‐sided hypothesis test

Bayesian sequential and adaptive randomization designs are gaining popularity in clinical trials thanks to their potentials to reduce the number of required participants and save resources. We propose a Bayesian sequential design with adaptive randomization rates so as to more efficiently attribute newly recruited patients to different treatment arms. In this paper, we consider 2‐arm clinical trials. Patients are allocated to the 2 arms with a randomization rate to achieve minimum variance for the test statistic. Algorithms are presented to calculate the optimal randomization rate, critical values, and power for the proposed design. Sensitivity analysis is implemented to check the influence on design by changing the prior distributions. Simulation studies are applied to compare the proposed method and traditional methods in terms of power and actual sample sizes. Simulations show that, when total sample size is fixed, the proposed design can obtain greater power and/or cost smaller actual sample size than the traditional Bayesian sequential design. Finally, we apply the proposed method to a real data set and compare the results with the Bayesian sequential design without adaptive randomization in terms of sample sizes. The proposed method can further reduce required sample size.     

A comparison of estimators for regression models with change points

We consider two problems concerning locating change points in a linear regression model. One involves jump discontinuities (change-point) in a regression model and the other involves regression lines connected at unknown points. We compare four methods for estimating single or multiple change points in a regression model, when both the error variance and regression coefficients change simultaneously at the unknown point(s): Bayesian, Julious, grid search, and the segmented methods. The proposed methods are evaluated via a simulation study and compared via some standard measures of estimation bias and precision. Finally, the methods are illustrated and compared using three real data sets. The simulation and empirical results overall favor both the segmented and Bayesian methods of estimation, which simultaneously estimate the change point and the other model parameters, though only the Bayesian method is able to handle both continuous and dis-continuous change point problems successfully. If it is known that regression lines are continuous then the segmented method ranked first among methods.     

Adaptive Bayesian compound designs for dose finding studies

We consider the problem of how to efficiently and safely design dose finding studies. Both current and novel utility functions are explored using Bayesian adaptive design methodology for the estimation of a maximum tolerated dose (MTD). In particular, we explore widely adopted approaches such as the continual reassessment method and minimizing the variance of the estimate of an MTD. New utility functions are constructed in the Bayesian framework and are evaluated against current approaches. To reduce computing time, importance sampling is implemented to re-weight posterior samples thus avoiding the need to draw samples using Markov chain Monte Carlo techniques. Further, as such studies are generally first-in-man, the safety of patients is paramount. We therefore explore methods for the incorporation of safety considerations into utility functions to ensure that only safe and well-predicted doses are administered. The amalgamation of Bayesian methodology, adaptive design and compound utility functions is termed adaptive Bayesian compound design (ABCD). The performance of this amalgamation of methodology is investigated via the simulation of dose finding studies. The paper concludes with a discussion of results and extensions that could be included into our approach.     

A Bayesian kriged Kalman model for short-term forecasting of air pollution levels

Summary.  Short-term forecasts of air pollution levels in big cities are now reported in news-papers and other media outlets. Studies indicate that even short-term exposure to high levels of an air pollutant called atmospheric particulate matter can lead to long-term health effects. Data are typically observed at fixed monitoring stations throughout a study region of interest at different time points. Statistical spatiotemporal models are appropriate for modelling these data. We consider short-term forecasting of these spatiotemporal processes by using a Bayesian kriged Kalman filtering model. The spatial prediction surface of the model is built by using the well-known method of kriging for optimum spatial prediction and the temporal effects are analysed by using the models underlying the Kalman filtering method. The full Bayesian model is implemented by using Markov chain Monte Carlo techniques which enable us to obtain the optimal Bayesian forecasts in time and space. A new cross-validation method based on the Mahalanobis distance between the forecasts and observed data is also developed to assess the forecasting performance of the model implemented.