Equation-solving estimator based on Metropolis-Hastings algorithm with delayed rejection

Numerous works have recently attempted to develop more efficient estimators for MCMC inference than classical ones. In this perspective and approximate nonstandard discrete distributions, Liang and Liu proposed the equation solving estimator as an alternative to the conventional frequency estimator. The specific MCMC method used is the Metropolis-Hastings (M-H) algorithm. In this work, we propose to adapt the equation-solving estimator to the context of simulation using the Metropolis-Hastings algorithm with delayed rejection (MHDR). Developed originally by Mira, this algorithm is considered an improved version of the standard M-H sampler which aims to reduce the variance of MCMC estimators. An application to a Bayesian hypothesis test problem shows the superiority of the equation-solving estimator, based on MHDR sampling, over the one introduced by Liang and Liu.     

On the performance of the Bayesian composite likelihood estimation of max-stable processes

The max-stable process is a natural approach for modelling extrenal dependence in spatial data. However, the estimation is difficult due to the intractability of the full likelihoods. One approach that can be used to estimate the posterior distribution of the parameters of the max-stable process is to employ composite likelihoods in the Markov chain Monte Carlo (MCMC) samplers, possibly with adjustment of the credible intervals. In this paper, we investigate the performance of the composite likelihood-based MCMC samplers under various settings of the Gaussian extreme value process and the Brown–Resnick process. Based on our findings, some suggestions are made to facilitate the application of this estimator in real data.     

Forward Simulation Markov Chain Monte Carlo with Applications to Stochastic Epidemic Models

For many stochastic models, it is difficult to make inference about the model parameters because it is impossible to write down a tractable likelihood given the observed data. A common solution is data augmentation in a Markov chain Monte Carlo (MCMC) framework. However, there are statistical problems where this approach has proved infeasible but where simulation from the model is straightforward leading to the popularity of the approximate Bayesian computation algorithm. We introduce a forward simulation MCMC (fsMCMC) algorithm, which is primarily based upon simulation from the model. The fsMCMC algorithm formulates the simulation of the process explicitly as a data augmentation problem. By exploiting non‐centred parameterizations, an efficient MCMC updating schema for the parameters and augmented data is introduced, whilst maintaining straightforward simulation from the model. The fsMCMC algorithm is successfully applied to two distinct epidemic models including a birth–death–mutation model that has only previously been analysed using approximate Bayesian computation methods.     

On estimating the current intensity of failure for the power-law process

We consider the problem of estimating the current failure intensity for the power-law (Weibull) process. Closed-form optimum estimators under the criteria of minimum risks as well as Pitman-closeness are derived for the failure truncated case. A unique Pitman-closest estimator which is also invariant of the choice of the loss function within a very wide class of loss functions is obtained. In the frequentist setup, no admissible estimator under these criteria are available for the time truncated scheme due to the lack of any pivotal quantity. We present a Bayesian approach, which circumvents this problem and provides a uniform solution. In the Bayesian framework, we provide an algorithm based on Markov Chain Monte Carlo (MCMC) technique, which facilitates the evaluation of the estimators. The theoretical findings are supplemented by substantial numerical investigation.     

Markov chain Monte Carlo estimation of a mixture item response theory model

Markov chain Monte Carlo (MCMC) algorithms have been shown to be useful for estimation of complex item response theory (IRT) models. Although an MCMC algorithm can be very useful, it also requires care in use and interpretation of results. In particular, MCMC algorithms generally make extensive use of priors on model parameters. In this paper, MCMC estimation is illustrated using a simple mixture IRT model, a mixture Rasch model (MRM), to demonstrate how the algorithm operates and how results may be affected by some commonly used priors. Priors on the probabilities of mixtures, label switching, model selection, metric anchoring, and implementation of the MCMC algorithm using WinBUGS are described, and their effects illustrated on parameter recovery in practical testing situations. In addition, an example is presented in which an MRM is fitted to a set of educational test data using the MCMC algorithm and a comparison is illustrated with results from three existing maximum likelihood estimation methods.     

On parallelizable Markov chain Monte Carlo algorithms with waste-recycling

Parallelizable Markov chain Monte Carlo (MCMC) generates multiple proposals and parallelizes the evaluations of the likelihood function on different cores at each MCMC iteration. Inspired by Calderhead (Proc Natl Acad Sci 111(49):17408–17413, 2014), we introduce a general ‘waste-recycling’ framework for parallelizable MCMC, under which we show that using weighted samples from waste-recycling is preferable to resampling in terms of both statistical and computational efficiencies. We also provide a simple-to-use criteria, the generalized effective sample size, for evaluating efficiencies of parallelizable MCMC algorithms, which applies to both the waste-recycling and the vanilla versions. A moment estimator of the generalized effective sample size is provided and shown to be reasonably accurate by simulations.     

Zero variance Markov chain Monte Carlo for Bayesian estimators

Interest is in evaluating, by Markov chain Monte Carlo (MCMC) simulation, the expected value of a function with respect to a, possibly unnormalized, probability distribution. A general purpose variance reduction technique for the MCMC estimator, based on the zero-variance principle introduced in the physics literature, is proposed. Conditions for asymptotic unbiasedness of the zero-variance estimator are derived. A central limit theorem is also proved under regularity conditions. The potential of the idea is illustrated with real applications to probit, logit and GARCH Bayesian models. For all these models, a central limit theorem and unbiasedness for the zero-variance estimator are proved (see the supplementary material available on-line).     

Likelihood-based Estimation of the Regression Model with Scrambled Responses

A significant problem in the collection of responses to potentially sensitive questions, such as relating to illegal, immoral or embarrassing activities, is non-sampling error due to refusal to respond or false responses. Eichhorn & Hayre (1983) suggested the use of scrambled responses to reduce this form of bias. This paper considers a linear regression model in which the dependent variable is unobserved but for which the sum or product with a scrambling random variable of known distribution, is known. The performance of two likelihood-based estimators is investigated, namely of a Bayesian estimator achieved through a Markov chain Monte Carlo (MCMC) sampling scheme, and a classical maximum-likelihood estimator. These two estimators and an estimator suggested by Singh, Joarder & King (1996) are compared. Monte Carlo results show that the Bayesian estimator out-performs the classical estimators in all cases, and the relative performance of the Bayesian estimator improves as the responses become more scrambled.     

Coupling random inputs for parameter estimation in complex models

Complex stochastic models, such as individual-based models, are becoming increasingly popular. However this complexity can often mean that the likelihood is intractable. Performing parameter estimation on the model can then be difficult. One way of doing this when the complex model is relatively quick to simulate from is approximate Bayesian computation (ABC). Rejection-ABC algorithm is not always efficient so numerous other algorithms have been proposed. One such method is ABC with Markov chain Monte Carlo (ABC–MCMC). Unfortunately for some models this method does not perform well and some alternatives have been proposed including the fsMCMC algorithm (Neal and Huang, in: Scand J Stat 42:378–396, 2015) that explores the random inputs space as well unknown model parameters. In this paper we extend the fsMCMC algorithm and take advantage of the joint parameter and random input space in order to get better mixing of the Markov Chain. We also introduce a Gibbs step that conditions on the current accepted model and allows the parameters to move as well as the random inputs conditional on this accepted model. We show empirically that this improves the efficiency of the ABC–MCMC algorithm on a queuing model and an individual-based model of the group-living bird, the woodhoopoe.     

Monte Carlo integration with Markov chain

There are two conceptually distinct tasks in Markov chain Monte Carlo (MCMC): a sampler is designed for simulating a Markov chain and then an estimator is constructed on the Markov chain for computing integrals and expectations. In this article, we aim to address the second task by extending the likelihood approach of Kong et al. for Monte Carlo integration. We consider a general Markov chain scheme and use partial likelihood for estimation. Basically, the Markov chain scheme is treated as a random design and a stratified estimator is defined for the baseline measure. Further, we propose useful techniques including subsampling, regulation, and amplification for achieving overall computational efficiency. Finally, we introduce approximate variance estimators for the point estimators. The method can yield substantially improved accuracy compared with Chib's estimator and the crude Monte Carlo estimator, as illustrated with three examples.     

The precision of regression-type estimator for incremental cost–effectiveness ratio

The estimation of incremental cost–effectiveness ratio (ICER) has received increasing attention recently. It is expressed in terms of the ratio of the change in costs of a therapeutic intervention to the change in the effects of the intervention. Despite the intuitive interpretation of ICER as an additional cost per additional benefit unit, it is a challenge to estimate the distribution of a ratio of two stochastically dependent distributions. A vast literature regarding the statistical methods of ICER has developed in the past two decades, but none of these methods provide an unbiased estimator. Here, to obtain the unbiased estimator of the cost–effectiveness ratio (CER), the zero intercept of the bivariate normal regression is assumed. In equal sample sizes, the Iman–Conover algorithm is applied to construct the desired variance–covariance matrix of two random bivariate samples, and the estimation then follows the same approach as CER to obtain the unbiased estimator of ICER. The bootstrapping method with the Iman–Conover algorithm is employed for unequal sample sizes. Simulation experiments are conducted to evaluate the proposed method. The regression-type estimator performs overwhelmingly better than the sample mean estimator in terms of mean squared error in all cases.     

Bayesian analysis of circular distributions based on non-negative trigonometric sums

ABSTRACT

Fernández-Durán [Circular distributions based on nonnegative trigonometric sums. Biometrics. 2004;60:499–503] developed a new family of circular distributions based on non-negative trigonometric sums that is suitable for modelling data sets that present skewness and/or multimodality. In this paper, a Bayesian approach to deriving estimates of the unknown parameters of this family of distributions is presented. Because the parameter space is the surface of a hypersphere and the dimension of the hypersphere is an unknown parameter of the distribution, the Bayesian inference must be based on transdimensional Markov Chain Monte Carlo (MCMC) algorithms to obtain samples from the high-dimensional posterior distribution. The MCMC algorithm explores the parameter space by moving along great circles on the surface of the hypersphere. The methodology is illustrated with real and simulated data sets.     

Estimation Procedure for a Multiple Time Series Model

Given a multiple time series sharing common autoregressive patterns, we estimate an additive model. The autoregressive component and the individual random effects are estimated by integrating maximum likelihood estimation and best linear unbiased predictions in a backfitting algorithm. The simulation study illustrated that the estimation procedure provides an alternative to the Arellano–Bond generalized method of moments (GMM) estimator of the panel model when T > N and the Arellano–Bond generally diverges. The estimator has high predictive ability. In cases where T ≤ N, the backfitting estimator is at least comparable to Arellano–Bond estimator.     

Marginal likelihood estimation from the Metropolis output: tips and tricks for efficient implementation in generalized linear latent variable models

The marginal likelihood can be notoriously difficult to compute, and particularly so in high-dimensional problems. Chib and Jeliazkov employed the local reversibility of the Metropolis–Hastings algorithm to construct an estimator in models where full conditional densities are not available analytically. The estimator is free of distributional assumptions and is directly linked to the simulation algorithm. However, it generally requires a sequence of reduced Markov chain Monte Carlo runs which makes the method computationally demanding especially in cases when the parameter space is large. In this article, we study the implementation of this estimator on latent variable models which embed independence of the responses to the observables given the latent variables (conditional or local independence). This property is employed in the construction of a multi-block Metropolis-within-Gibbs algorithm that allows to compute the estimator in a single run, regardless of the dimensionality of the parameter space. The counterpart one-block algorithm is also considered here, by pointing out the difference between the two approaches. The paper closes with the illustration of the estimator in simulated and real-life data sets.     

Bayesian clustering for continuous-time hidden Markov models

We develop clustering procedures for longitudinal trajectories based on a continuous-time hidden Markov model (CTHMM) and a generalized linear observation model. Specifically, in this article we carry out finite and infinite mixture model-based clustering for a CTHMM and achieve inference using Markov chain Monte Carlo (MCMC). For a finite mixture model with a prior on the number of components, we implement reversible-jump MCMC to facilitate the trans-dimensional move between models with different numbers of clusters. For a Dirichlet process mixture model, we utilize restricted Gibbs sampling split–merge proposals to improve the performance of the MCMC algorithm. We apply our proposed algorithms to simulated data as well as a real-data example, and the results demonstrate the desired performance of the new sampler.     

Computation of Gaussian orthant probabilities in high dimension

We study the computation of Gaussian orthant probabilities, i.e. the probability that a Gaussian variable falls inside a quadrant. The Geweke–Hajivassiliou–Keane (GHK) algorithm (Geweke, Comput Sci Stat 23:571–578 1991, Keane, Simulation estimation for panel data models with limited dependent variables, 1993, Hajivassiliou, J Econom 72:85–134, 1996, Genz, J Comput Graph Stat 1:141–149, 1992) is currently used for integrals of dimension greater than 10. In this paper, we show that for Markovian covariances GHK can be interpreted as the estimator of the normalizing constant of a state-space model using sequential importance sampling. We show for an AR(1) the variance of the GHK, properly normalized, diverges exponentially fast with the dimension. As an improvement we propose using a particle filter. We then generalize this idea to arbitrary covariance matrices using Sequential Monte Carlo with properly tailored MCMC moves. We show empirically that this can lead to drastic improvements on currently used algorithms. We also extend the framework to orthants of mixture of Gaussians (Student, Cauchy, etc.), and to the simulation of truncated Gaussians.     

Marginal Likelihood Estimation with the Cross-Entropy Method

We consider an adaptive importance sampling approach to estimating the marginal likelihood, a quantity that is fundamental in Bayesian model comparison and Bayesian model averaging. This approach is motivated by the difficulty of obtaining an accurate estimate through existing algorithms that use Markov chain Monte Carlo (MCMC) draws, where the draws are typically costly to obtain and highly correlated in high-dimensional settings. In contrast, we use the cross-entropy (CE) method, a versatile adaptive Monte Carlo algorithm originally developed for rare-event simulation. The main advantage of the importance sampling approach is that random samples can be obtained from some convenient density with little additional costs. As we are generating independent draws instead of correlated MCMC draws, the increase in simulation effort is much smaller should one wish to reduce the numerical standard error of the estimator. Moreover, the importance density derived via the CE method is grounded in information theory, and therefore, is in a well-defined sense optimal. We demonstrate the utility of the proposed approach by two empirical applications involving women's labor market participation and U.S. macroeconomic time series. In both applications, the proposed CE method compares favorably to existing estimators.     

A computational procedure for estimation of the mixing time of the random-scan Metropolis algorithm

Many situations, especially in Bayesian statistical inference, call for the use of a Markov chain Monte Carlo (MCMC) method as a way to draw approximate samples from an intractable probability distribution. With the use of any MCMC algorithm comes the question of how long the algorithm must run before it can be used to draw an approximate sample from the target distribution. A common method of answering this question involves verifying that the Markov chain satisfies a drift condition and an associated minorization condition (Rosenthal, J Am Stat Assoc 90:558–566, 1995; Jones and Hobert, Stat Sci 16:312–334, 2001). This is often difficult to do analytically, so as an alternative, it is typical to rely on output-based methods of assessing convergence. The work presented here gives a computational method of approximately verifying a drift condition and a minorization condition specifically for the symmetric random-scan Metropolis algorithm. Two examples of the use of the method described in this article are provided, and output-based methods of convergence assessment are presented in each example for comparison with the upper bound on the convergence rate obtained via the simulation-based approach.     

Classical and Bayesian estimation of stress-strength reliability in type II censored Pareto distributions

ABSTRACT

In this paper, the stress-strength reliability, R, is estimated in type II censored samples from Pareto distributions. The classical inference includes obtaining the maximum likelihood estimator, an exact confidence interval, and the confidence intervals based on Wald and signed log-likelihood ratio statistics. Bayesian inference includes obtaining Bayes estimator, equi-tailed credible interval, and highest posterior density (HPD) interval given both informative and non-informative prior distributions. Bayes estimator of R is obtained using four methods: Lindley's approximation, Tierney-Kadane method, Monte Carlo integration, and MCMC. Also, we compare the proposed methods by simulation study and provide a real example to illustrate them.