Weakly Informative Reparameterizations for Location-Scale Mixtures

While mixtures of Gaussian distributions have been studied for more than a century, the construction of a reference Bayesian analysis of those models remains unsolved, with a general prohibition of improper priors due to the ill-posed nature of such statistical objects. This difficulty is usually bypassed by an empirical Bayes resolution. By creating a new parameterization centered on the mean and possibly the variance of the mixture distribution itself, we manage to develop here a weakly informative prior for a wide class of mixtures with an arbitrary number of components. We demonstrate that some posterior distributions associated with this prior and a minimal sample size are proper. We provide Markov chain Monte Carlo (MCMC) implementations that exhibit the expected exchangeability. We only study here the univariate case, the extension to multivariate location-scale mixtures being currently under study. An R package called Ultimixt is associated with this article. Supplementary material for this article is available online.     

Implementing random scan Gibbs samplers

Summary  The Gibbs sampler, being a popular routine amongst Markov chain Monte Carlo sampling methodologies, has revolutionized the application of Monte Carlo methods in statistical computing practice. The performance of the Gibbs sampler relies heavily on the choice of sweep strategy, that is, the means by which the components or blocks of the random vector X of interest are visited and updated. We develop an automated, adaptive algorithm for implementing the optimal sweep strategy as the Gibbs sampler traverses the sample space. The decision rules through which this strategy is chosen are based on convergence properties of the induced chain and precision of statistical inferences drawn from the generated Monte Carlo samples. As part of the development, we analytically derive closed form expressions for the decision criteria of interest and present computationally feasible implementations of the adaptive random scan Gibbs sampler via a Gaussian approximation to the target distribution. We illustrate the results and algorithms presented by using the adaptive random scan Gibbs sampler developed to sample multivariate Gaussian target distributions, and screening test and image data. Research by RL and ZY supported in part by a US National Science Foundation FRG grant 0139948 and a grant from Lawrence Livermore National Laboratory, Livermore, California, USA.     

Gaussian cubature: A practitioner’s guide

Accurate modeling of management and economic processes often requires that researchers accurately approximate the expectations of functions of random variables. While commonly employed, Monte Carlo simulation techniques generally require large sample sizes to insure accuracy. For functions that are computationally burdensome, the Monte Carlo approach may be impractical. We propose a method to generate samples from multivariate distributions that contain far fewer points than reliable Monte Carlo samples, yet retain much of the original distributions’ information. Our method, Gaussian cubatures generated via linear programming, is designed to be feasible for joint, but independent distributions. While heuristic for joint, dependent distributions, this method appears to be very reliable and to accurately approximate expectations of an important class of functions.     

Efficient Bayesian Inference for Multivariate Probit Models With Sparse Inverse Correlation Matrices

We propose a Bayesian approach for inference in the multivariate probit model, taking into account the association structure between binary observations. We model the association through the correlation matrix of the latent Gaussian variables. Conditional independence is imposed by setting some off-diagonal elements of the inverse correlation matrix to zero and this sparsity structure is modeled using a decomposable graphical model. We propose an efficient Markov chain Monte Carlo algorithm relying on a parameter expansion scheme to sample from the resulting posterior distribution. This algorithm updates the correlation matrix within a simple Gibbs sampling framework and allows us to infer the correlation structure from the data, generalizing methods used for inference in decomposable Gaussian graphical models to multivariate binary observations. We demonstrate the performance of this model and of the Markov chain Monte Carlo algorithm on simulated and real datasets. This article has online supplementary materials.     

Geometric Ergodicity of Metropolis-Hastings Algorithms for Conditional Simulation in Generalized Linear Mixed Models

Conditional simulation is useful in connection with inference and prediction for a generalized linear mixed model. We consider random walk Metropolis and Langevin-Hastings algorithms for simulating the random effects given the observed data, when the joint distribution of the unobserved random effects is multivariate Gaussian. In particular we study the desirable property of geometric ergodicity, which ensures the validity of central limit theorems for Monte Carlo estimates.     

MCMC Estimation of Restricted Covariance Matrices

This article is motivated by the difficulty of applying standard simulation techniques when identification constraints or theoretical considerations induce covariance restrictions in multivariate models. To deal with this difficulty, we build upon a decomposition of positive definite matrices and show that it leads to straightforward Markov chain Monte Carlo samplers for restricted covariance matrices. We introduce the approach by reviewing results for multivariate Gaussian models without restrictions, where standard conjugate priors on the elements of the decomposition induce the usual Wishart distribution on the precision matrix and vice versa. The unrestricted case provides guidance for constructing efficient Metropolis–Hastings and accept-reject Metropolis–Hastings samplers in more complex settings, and we describe in detail how simulation can be performed under several important constraints. The proposed approach is illustrated in a simulation study and two applications in economics. Supplemental materials for this article (appendixes, data, and computer code) are available online.     

Bayesian joint quantile regression for mixed effects models with censoring and errors in covariates

In this paper, we discuss Bayesian joint quantile regression of mixed effects models with censored responses and errors in covariates simultaneously using Markov Chain Monte Carlo method. Under the assumption of asymmetric Laplace error distribution, we establish a Bayesian hierarchical model and derive the posterior distributions of all unknown parameters based on Gibbs sampling algorithm. Three cases including multivariate normal distribution and other two heavy-tailed distributions are considered for fitting random effects of the mixed effects models. Finally, some Monte Carlo simulations are performed and the proposed procedure is illustrated by analyzing a group of AIDS clinical data set.     

Closed-form valuations of basket options using a multivariate normal inverse Gaussian model

This paper uses a multivariate normal inverse Gaussian model to develop closed-form pricing formulas for both geometric and arithmetic basket options. For geometric basket options, an exact analytical solution is possible; for arithmetic basket options, the formula is an approximation. The model is based on a jump-driven financial process, which is known empirically to be more realistic than a geometric Brownian motion. By comparing our results to Monte Carlo experiments, we confirm the internal consistency of our formulas. The “Greeks” can be derived from the closed-form formulas in a straightforward manner.     

Rejoinder

Royston proposed a normal probability plot to detect nonnormality of univariate data. The normal probability plot was provided with normalized acceptance regions to enhance its interpretability. By using the theory of spherical distributions and the idea of principal component analysis, we propose an approach to extending Royston’s normal plot to detecting nonmultivariate normality in analyzing high-dimensional data. The performance of the proposed multivariate normal plot is demonstrated by Monte Carlo studies and illustrated by two real datasets.

Datasets, computer code and documentation of the code are available in the online supplements.     

Additive Positive Stable Frailty Models

In this article, we describe an additive stable frailty model for multivariate times to events data using a flexible baseline hazard, and assuming that the frailty component for each individual is described by additive functions of independent positive stable random variables with possibly different stability indices. Dependence properties of this frailty model are investigated. To carry out inference, the likelihood function is derived by replacing high-dimensional integration by Monte Carlo simulation. Markov chain Monte Carlo algorithms enable estimation and model checking in the Bayesian framework.      

Goodness-of-fit tests based on the empirical characteristic function

The paper is devoted to the supremum-type multivariate goodness-of-fit tests based on the empirical characteristic function. Particular attention is devoted to the composite hypothesis of normality and Gaussian distribution mixture model. An analytical way to approximate the null asymptotic distributions of the considered test statistics is discussed applying the theory of large excursions of differentiable Gaussian random fields. The produced comparative Monte Carlo power study shows that the considered tests are powerful competitors to the existing classical criteria, clearly dominating in verification of the goodness-of-fit hypotheses against the specific types of alternatives.     

Product and process yield estimation with Gaussian quadrature (GQ) reduction: Improvements over the GQ full factorial approach

Manufacturing or multivariate yield, the fraction of unscreened products which conforms to all product specification limits, is an important and commonly used metric for assessing and improving the quality of a production process. Current procedures for multivariate yield evaluation, such as Monte Carlo simulation, require substantial computing effort, making the iterative adjustment of design parameters often impractical. This paper introduces a new approach to multivariate yield evaluation based on a numerical integration procedure called Gaussian quadrature reduction (GQR). The advantage of this approach is a large reduction in the computational burden associated with multivariate yield evaluation with virtually no loss in accuracy of the estimates. The proposed procedure can be generalized to evaluate many other multivariate criteria such as expected costs and the desirability index. The method is demonstrated for three yield evaluation test problems, and comparisons to Monte Carlo-based evaluations are presented.     

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??Kolmogorov-Smirnov (KS), Cramer-von Mises (CM) and Anderson-Darling (AD) test, which are based on empirical distribution function (EDF), are well-known statistics in testing univariate normality. In this paper, we focus on the high dimensional case and propose a family of generalized EDF based statistics to test the high-dimensional normal distribution by reducing the dimension of the variable. Not only can we approximate the corresponding critical values of three statistics by Monte Carlo method, we also can investigate the approximate distributions of proposed statistics based on approximate formulas in univariate case under null hypothesis. The Monte Carlo simulation is carried out to demonstrate that the performance of proposed statistics is more competitive than existing methods under some alternative hypotheses. Finally, the proposed tests are applied to real data to illustrate their utility.     

Robust Gaussian graphical modeling

A new Gaussian graphical modeling that is robustified against possible outliers is proposed. The likelihood function is weighted according to how the observation is deviated, where the deviation of the observation is measured based on its likelihood. Test statistics associated with the robustified estimators are developed. These include statistics for goodness of fit of a model. An outlying score, similar to but more robust than the Mahalanobis distance, is also proposed. The new scores make it easier to identify outlying observations. A Monte Carlo simulation and an analysis of a real data set show that the proposed method works better than ordinary Gaussian graphical modeling and some other robustified multivariate estimators.     

Automated Factor Slice Sampling

Markov chain Monte Carlo (MCMC) algorithms offer a very general approach for sampling from arbitrary distributions. However, designing and tuning MCMC algorithms for each new distribution can be challenging and time consuming. It is particularly difficult to create an efficient sampler when there is strong dependence among the variables in a multivariate distribution. We describe a two-pronged approach for constructing efficient, automated MCMC algorithms: (1) we propose the “factor slice sampler,” a generalization of the univariate slice sampler where we treat the selection of a coordinate basis (factors) as an additional tuning parameter, and (2) we develop an approach for automatically selecting tuning parameters to construct an efficient factor slice sampler. In addition to automating the factor slice sampler, our tuning approach also applies to the standard univariate slice samplers. We demonstrate the efficiency and general applicability of our automated MCMC algorithm with a number of illustrative examples. This article has online supplementary materials.     

Conditionally heteroscedastic factorial HMMs for time series in finance

In this article, we develop a new approach within the framework of asset pricing models that incorporates two key features of the latent volatility: co‐movement among conditionally heteroscedastic financial returns and switching between different unobservable regimes. By combining latent factor models with hidden Markov chain models we derive a dynamical local model for segmentation and prediction of multivariate conditionally heteroscedastic financial time series. We concentrate more precisely on situations where the factor variances are modelled by univariate generalized quadratic autoregressive conditionally heteroscedastic processes. The expectation maximization algorithm that we have developed for the maximum likelihood estimation is based on a quasi‐optimal switching Kalman filter approach combined with a generalized pseudo‐Bayesian approximation, which yield inferences about the unobservable path of the common factors, their variances and the latent variable of the state process. Extensive Monte Carlo simulations and preliminary experiments obtained with daily foreign exchange rate returns of eight currencies show promising results. Copyright © 2007 John Wiley & Sons, Ltd.     

Propriétés asymptotiques d'un estimateur dans un modèle non linéaire à effets mixtes

We propose an estimator of nonlinear mixed effects model's parameters, obtained by maximization of simulated pseudo likelihood. This simulated criterion is constructed from the likelihood of a Gaussian model whose means and variances are given by Monte Carlo approximations of means and variances of the true model. If the number of experimental units and the sample size of Monte Carlo simulations are respectively-denoted by N and K, we obtain the strong consistency and asymptotic normality of the estimator when N tends to infinity and the ratio √N/K tends to zero.     

Gauss色噪声激励下含黏弹力摩擦系统的随机响应分析

研究了Gauss色噪声激励下含黏弹力、弱非线性阻尼的摩擦振子的随机响应.将适用于光滑系统的随机平均法推广到了非光滑摩擦系统,进而得到系统振幅、位移及速度的稳态概率密度函数.同时结合材料的黏弹性,研究了摩擦力和Gauss色噪声对系统响应的影响.研究表明,摩擦力、黏弹力及噪声项的相关参数均可引起随机P-分岔,并且在一定范围内系统响应对摩擦力极为敏感.此外,理论结果与Monte Carlo 模拟结果吻合较好,验证了方法的有效性.     

Some necessary uniform tests for spherical symmetry

While spherical distributions have been used in many statistical models for high-dimensional data analysis, there are few easily implemented statistics for testing spherical symmetry for the underlying distribution of high-dimensional data. Many existing statistics for this purpose were constructed by the theory of empirical processes and turn out to converge slowly to their limiting distributions. Some existing statistics for the same purpose were given in the form of high-dimensional integrals that are not easily evaluated in numerical computation. In this paper, we develop some necessary tests for spherical symmetry based on both univariate and multivariate uniform statistics. These statistics are easily evaluated numerically and have simple limiting distributions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and power.     

Bayesian and Maximum Likelihood Estimation for Gaussian Processes on an Incomplete Lattice

This article proposes a new approach for Bayesian and maximum likelihood parameter estimation for stationary Gaussian processes observed on a large lattice with missing values. We propose a Markov chain Monte Carlo approach for Bayesian inference, and a Monte Carlo expectation-maximization algorithm for maximum likelihood inference. Our approach uses data augmentation and circulant embedding of the covariance matrix, and provides likelihood-based inference for the parameters and the missing data. Using simulated data and an application to satellite sea surface temperatures in the Pacific Ocean, we show that our method provides accurate inference on lattices of sizes up to 512 × 512, and is competitive with two popular methods: composite likelihood and spectral approximations.