Objective Bayesian hypothesis testing in regression models with first-order autoregressive residuals

This article considers the objective Bayesian testing in the normal regression models with first-order autoregressive residuals. We propose some solutions based on a Bayesian model selection procedure to this problem where no subjective input is considered. We construct the proper priors for testing the autocorrelation coefficient based on measures of divergence between competing models, which is called the divergence-based (DB) priors and then propose the objective Bayesian decision-theoretic rule, which is called the Bayesian reference criterion (BRC). Finally, we derive the intrinsic test statistic for testing the autocorrelation coefficient. The behavior of the Bayes factor-based DB priors is examined by comparing with the BRC in a simulation study and an example.     

Bayesian LASSO-Regularized quantile regression for linear regression models with autoregressive errors

Quantile regression (QR) is a natural alternative for depicting the impact of covariates on the conditional distributions of a outcome variable instead of the mean. In this paper, we investigate Bayesian regularized QR for the linear models with autoregressive errors. LASSO-penalized type priors are forced on regression coefficients and autoregressive parameters of the model. Gibbs sampler algorithm is employed to draw the full posterior distributions of unknown parameters. Finally, the proposed procedures are illustrated by some simulation studies and applied to a real data analysis of the electricity consumption.     

Bayesian estimation of an AR(1) process with exponential white noise

The object of this paper is a Bayesian analysis of the autoregressive model X _t ?=?ρX _t?1?+?Y _t where 0?Y _t are independent random variables with an exponential distribution of parameter θ. Our study generalizes some results obtained by Turkmann (1990 Amaral Turkmann, M. A. (1990). Bayesian analysis of an autoregressive process with exponential white noise. Statistics, 4: 601–608.  [Google Scholar]). Our analysis is based on a more general non-informative prior which allows us to improve the estimators of ρ and θ.     

Bayesian inference for the Pareto lifetime model under progressive censoring with binomial removals

This paper considers the estimation and prediction problems when lifetimes are Pareto-distributed and are collected under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a Binomial distribution. The analysis is carried out within the Bayesian context.     

Bayes estimation for a bivariate survival model based on exponential distributions

Bayesian analysis of a bivariate survival model based on exponential distributions is discussed using both vague and conjugate prior distributions. Parameter and reliability estimators are given for the maximum likelihood technique and the Bayesian approach using both types of priors. A Monte Carlo study indicates the vague prior Bayes estimator of reliability performs better than its maximum likelihood counterpart.     

On the estimation of an instantaneous transformation for time series

We modify Ramsay's algorithm for estimating monotonic transformations in regression and extend it to autoregression, where strict monotonicity is an essential requirement. Compared with other methods, our method can capture some characteristics that are pertinent to the time series and is much easier to implement. An order selection method is introduced and developed. Some real data sets are analysed.     

Self-weighted least absolute deviation estimation for infinite variance autoregressive models

Summary.  How to undertake statistical inference for infinite variance autoregressive models has been a long-standing open problem. To solve this problem, we propose a self-weighted least absolute deviation estimator and show that this estimator is asymptotically normal if the density of errors and its derivative are uniformly bounded. Furthermore, a Wald test statistic is developed for the linear restriction on the parameters, and it is shown to have non-trivial local power. Simulation experiments are carried out to assess the performance of the theory and method in finite samples and a real data example is given. The results are entirely different from other published results and should provide new insights for future research on heavy-tailed time series.     

Bayesian temporal density estimation with autoregressive species sampling models

We propose a novel Bayesian nonparametric (BNP) model, which is built on a class of species sampling models, for estimating density functions of temporal data. In particular, we introduce species sampling mixture models with temporal dependence. To accommodate temporal dependence, we define dependent species sampling models by modeling random support points and weights through an autoregressive model, and then we construct the mixture models based on the collection of these dependent species sampling models. We propose an algorithm to generate posterior samples and present simulation studies to compare the performance of the proposed models with competitors that are based on Dirichlet process mixture models. We apply our method to the estimation of densities for the price of apartment in Seoul, the closing price in Korea Composite Stock Price Index (KOSPI), and climate variables (daily maximum temperature and precipitation) of around the Korean peninsula.     

A comparison of scientific and engineering criteria for Bayesian model selection

Given a set of possible models for variables X and a set of possible parameters for each model, the Bayesian estimate of the probability distribution for X given observed data is obtained by averaging over the possible models and their parameters. An often-used approximation for this estimate is obtained by selecting a single model and averaging over its parameters. The approximation is useful because it is computationally efficient, and because it provides a model that facilitates understanding of the domain. A common criterion for model selection is the posterior probability of the model. Another criterion for model selection, proposed by San Martini and Spezzafari (1984), is the predictive performance of a model for the next observation to be seen. From the standpoint of domain understanding, both criteria are useful, because one identifies the model that is most likely, whereas the other identifies the model that is the best predictor of the next observation. To highlight the difference, we refer to the posterior-probability and alternative criteria as the scientific criterion (SC) and engineering criterion (EC), respectively. When we are interested in predicting the next observation, the model-averaged estimate is at least as good as that produced by EC, which itself is at least as good as the estimate produced by SC. We show experimentally that, for Bayesian-network models containing discrete variables only, the predictive performance of the model average can be significantly better than those of single models selected by either criterion, and that differences between models selected by the two criterion can be substantial.     

Bayesian bridge-randomized penalized quantile regression estimation for linear regression model with AP(q) perturbation

Bridge penalized regression has many desirable statistical properties such as unbiasedness, sparseness as well as ‘oracle’. In Bayesian framework, bridge regularized penalty can be implemented based on generalized Gaussian distribution (GGD) prior. In this paper, we incorporate Bayesian bridge-randomized penalty and its adaptive version into the quantile regression (QR) models with autoregressive perturbations to conduct Bayesian penalization estimation. Employing the working likelihood of the asymmetric Laplace distribution (ALD) perturbations, the Bayesian joint hierarchical models are established. Based on the mixture representations of the ALD and generalized Gaussian distribution (GGD) priors of coefficients, the hybrid algorithms based on Gibbs sampler and Metropolis-Hasting sampler are provided to conduct fully Bayesian posterior estimation. Finally, the proposed Bayesian procedures are illustrated by some simulation examples and applied to a real data application of the electricity consumption.     

Bayesian population estimation for small sample capture-recapture data using noninformative priors

One critical issue in the Bayesian approach is choosing the priors when there is not enough prior information to specify hyperparameters. Several improper noninformative priors for capture-recapture models were proposed in the literature. It is known that the Bayesian estimate can be sensitive to the choice of priors, especially when sample size is small to moderate. Yet, how to choose a noninformative prior for a given model remains a question. In this paper, as the first step, we consider the problem of estimating the population size for _Mt$M_{\mathrm{t}}$ model using noninformative priors. The _Mt$M_{\mathrm{t}}$ model has prodigious application in wildlife management, ecology, software liability, epidemiological study, census under-count, and other research areas. Four commonly used noninformative priors are considered. We find that the choice of noninformative priors depends on the number of sampling occasions only. The guidelines on the choice of noninformative priors are provided based on the simulation results. Propriety of applying improper noninformative prior is discussed. Simulation studies are developed to inspect the frequentist performance of Bayesian point and interval estimates with different noninformative priors under various population sizes, capture probabilities, and the number of sampling occasions. The simulation results show that the Bayesian approach can provide more accurate estimates of the population size than the MLE for small samples. Two real-data examples are given to illustrate the method.     

A Bayesian analysis of a change in the parameters of autoregressive time series

In this article, we consider a Bayesian analysis of a possible change in the parameters of autoregressive time series of known order p, AR(p). An unconditional Bayesian test based on highest posterior density (HPD) credible sets is determined. The test is useful to detect a change in any one of the parameters separately. Using the Gibbs sampler algorithm, we approximate the posterior densities of the change point and other parameters to calculate the p-values that define our test.     

Bayesian autoregressive filter and squared gain estimation

This article describes a Bayesian small sample approach to making inferences for the operator (or filter) and squared gain of a p — th order Gaussian univariate autoregressive process. Simultaneous pos¬terior probability bands are developed for the real and the imaginary parts of the frequency-response function of an autoregressive operator as well as for the squared gain of an autoregressive process.     

Nonparametric Bayesian analysis for multi-site hidden Markov model

The hidden Markov model (HMM) provides an attractive framework for modeling long-term persistence in a variety of applications including pattern recognition. Unlike typical mixture models, hidden Markov states can represent the heterogeneity in data and it can be extended to a multivariate case using a hierarchical Bayesian approach. This article provides a nonparametric Bayesian modeling approach to the multi-site HMM by considering stick-breaking priors for each row of an infinite state transition matrix. This extension has many advantages over a parametric HMM. For example, it can provide more flexible information for identifying the structure of the HMM than parametric HMM analysis, such as the number of states in HMM. We exploit a simulation example and a real dataset to evaluate the proposed approach.     

Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors

Summary.  We develop an efficient way to select the best subset autoregressive model with exogenous variables and generalized autoregressive conditional heteroscedasticity errors. One main feature of our method is to select important autoregressive and exogenous variables, and at the same time to estimate the unknown parameters. The method proposed uses the stochastic search idea. By adopting Markov chain Monte Carlo techniques, we can identify the best subset model from a large of number of possible choices. A simulation experiment shows that the method is very effective. Misspecification in the mean equation can also be detected by our model selection method. In the application to the stock-market data of seven countries, the lagged 1 US return is found to have a strong influence on the other stock-market returns.     

贝叶斯统计推断及其主要进展

贝叶斯统计推断作为现代统计分析方法的重要内容,对于统计学理论的发展具有里程碑的作用。深入总结其研究的主要进展,具有重要的现实意义。在查阅国内外重要学术研究资料的基础上,从贝叶斯统计推断的思想、与古典统计的研究思路比较和贝叶斯统计推断研究的主要进展三个方面作了综述与介绍,力图达到认识贝叶斯统计推断及其研究现状的目的。     

Exact and Monte Carlo calculations of integrated likelihoods for the latent class model

The latent class model or multivariate multinomial mixture is a powerful approach for clustering categorical data. It uses a conditional independence assumption given the latent class to which a statistical unit is belonging. In this paper, we exploit the fact that a fully Bayesian analysis with Jeffreys non-informative prior distributions does not involve technical difficulty to propose an exact expression of the integrated complete-data likelihood, which is known as being a meaningful model selection criterion in a clustering perspective. Similarly, a Monte Carlo approximation of the integrated observed-data likelihood can be obtained in two steps: an exact integration over the parameters is followed by an approximation of the sum over all possible partitions through an importance sampling strategy. Then, the exact and the approximate criteria experimentally compete, respectively, with their standard asymptotic BIC approximations for choosing the number of mixture components. Numerical experiments on simulated data and a biological example highlight that asymptotic criteria are usually dramatically more conservative than the non-asymptotic presented criteria, not only for moderate sample sizes as expected but also for quite large sample sizes. This research highlights that asymptotic standard criteria could often fail to select some interesting structures present in the data.