Alternative Bayesian Estimators for Vector-Autoregressive Models

This paper compares frequentist risks of several Bayesian estimators of the VAR lag parameters and covariance matrix under alternative priors. With the constant prior on the VAR lag parameters, the asymmetric LINEX estimator for the lag parameters does better overall than the posterior mean. The posterior mean of covariance matrix performs well in most cases. The choice of prior has more significant effects on the estimates than the form of estimators. The shrinkage prior on the VAR lag parameters dominates the constant prior, while Yang and Berger's reference prior on the covariance matrix dominates the Jeffreys prior. Estimation of a VAR using the U.S. macroeconomic data reveals significant differences between estimates under the shrinkage and constant priors

Estimation of multivariate normal covariance and precision matrices in a star-shape model with missing data

AbstractIn this paper, we study the problem of estimating the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in a star-shape model with missing data. By considering a type of Cholesky decomposition of the precision matrix Ω=Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements, we get the MLEs of the covariance matrix and precision matrix and prove that both of them are biased. Based on the MLEs, unbiased estimators of the covariance matrix and precision matrix are obtained. A special group G, which is a subgroup of the group consisting all lower triangular matrices, is introduced. By choosing the left invariant Haar measure on G as a prior, we obtain the closed forms of the best equivariant estimates of Ω under any of the Stein loss, the entropy loss, and the symmetric loss. Consequently, the MLE of the precision matrix (covariance matrix) is inadmissible under any of the above three loss functions. Some simulation results are given for illustration

Thermal Insulation Coatings in Energy Saving

The surface temperature of object rises due to the accumulated heat when it absorbs solar energy, the excessive temperature caused by solar radiation will result in many inconveniences and even troubles in industrial production and daily life; in order to maintain the proper temperature of the object, a large amount of energy is consumed. The development of effective and economic thermal insulation materials is the key to meet the urgent needs for energy saving and emission reduction. In the face of variety of choices of thermal insulation materials, thermal insulation coating become more and more popular due to its good thermal insulation performance, economic, easy to use, and adaptability for a wide range of substrates. With the thermal insulation functional fillers (briefly called fillers in the following text) in coating system, the films can show a certain thermal insulation effect by reflecting, radiating, or isolating heat. As a result, when covered by thermal insulation coatings, the surface temperature of object would be greatly decreased. In this case, a large amount of energy consumed for cooling down the objects exposed to sunlight could be saved, which means the energy consumption can be reduced effectively by just covering with a thermal insulation coating on the surface of object

Noninformative Priors and Frequentist Risks of Bayesian Estimators of Vector-Autoregressive Models

In this study, we examine posterior properties and frequentist risks of Bayesian estimators based on several non-informative priors in Vector Autoregressive (VAR) models. We prove existence of the posterior distributions and posterior moments under a general class of priors. Using a variety of priors in this class we conduct numerical simulations of posteriors. We find that in most examples Bayesian estimators with a shrinkage prior on the VAR coefficients and the reference prior of Yang and Berger (1994) on the VAR covariance matrix dominate MLE, Bayesian estimators with the diffuse prior, and Bayesian estimators with the prior used in RATS. We also examine the informative Minnesota prior and find that its performance depends on the nature of the data sample and on the tightness of the Minnesota prior. A tightly set Minnesota prior is better when the data generating processes are similar to random walks, but the shrinkage prior or constant prior can be better otherwise

Bayesian Estimator of Vector-Autoregressive Model Under the Entropy Loss

The present study makes two contributions to the Bayesian Vector-Autoregression (VAR) literature. The first contribution is derivation of the Bayesian VAR estimator under the intrinsic entropy loss. The Bayesian estimator, which is distinctly different from the posterior mean, involves the frequentist expectation of a function of VAR variables. We find that the condition that allows for a closed-form expression of the frequentist expectation is violated even when the VAR is stationary, making it difficult to compute the Bayesian estimates via standard Markov Chain Monte Carlo (MCMC) procedures. The second contribution of the paper concerns MCMC simulation of the Bayesian estimator without using the closed-form expression of the frequentist expectation. A novelty of our MCMC algorithms is that they jointly simulate the posteriors of frequentist moments of VAR variables as well as the posteriors of VAR parameters. Numerical simulations show that the algorithms are surprisingly efficient

Bayesian analysis for the Lomax model using noninformative priors

The Lomax distribution is an important member in the distribution family. In this paper, we systematically develop an objective Bayesian analysis of data from a Lomax distribution. Noninformative priors, including probability matching priors, the maximal data information (MDI) prior, Jeffreys prior and reference priors, are derived. The propriety of the posterior under each prior is subsequently validated. It is revealed that the MDI prior and one of the reference priors yield improper posteriors, and the other reference prior is a second-order probability matching prior. A simulation study is conducted to assess the frequentist performance of the proposed Bayesian approach. Finally, this approach along with the bootstrap method is applied to a real data set

Posterior propriety of an objective prior for generalized hierarchical normal linear models

Bayesian Hierarchical models has been widely used in modern statistical application. To deal with the data having complex structures, we propose a generalized hierarchical normal linear (GHNL) model which accommodates arbitrarily many levels, usual design matrices and ‘vanilla’ covariance matrices. Objective hyperpriors can be employed for the GHNL model to express ignorance or match frequentist properties, yet the common objective Bayesian approaches are infeasible or fraught with danger in hierarchical modelling. To tackle this issue, [Berger, J., Sun, D., & Song, C. (2020b). An objective prior for hyperparameters in normal hierarchical models. Journal of Multivariate Analysis, 178, 104606. https://doi.org/10.1016/j.jmva.2020.104606] proposed a particular objective prior and investigated its properties comprehensively. Posterior propriety is important for the choice of priors to guarantee the convergence of MCMC samplers. James Berger conjectured that the resulting posterior is proper for a hierarchical normal model with arbitrarily many levels, a rigorous proof of which was not given, however. In this paper, we complete this story and provide an user-friendly guidance. One main contribution of this paper is to propose a new technique for deriving an elaborate upper bound on the integrated likelihood, but also one unified approach to checking the posterior propriety for linear models. An efficient Gibbs sampling method is also introduced and outperforms other sampling approaches considerably