Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation

We propose a two-stage algorithm for computing maximum likelihood estimates for a class of spatial models. The algorithm combines Markov chain Monte Carlo methods such as the Metropolis–Hastings–Green algorithm and the Gibbs sampler, and stochastic approximation methods such as the off-line average and adaptive search direction. A new criterion is built into the algorithm so stopping is automatic once the desired precision has been set. Simulation studies and applications to some real data sets have been conducted with three spatial models. We compared the algorithm proposed with a direct application of the classical Robbins–Monro algorithm using Wiebe's wheat data and found that our procedure is at least 15 times faster.     

Motor unit number estimation using reversible jump Markov chain Monte Carlo methods

Summary.  We present an application of reversible jump Markov chain Monte Carlo sampling from the field of neurophysiology where we seek to estimate the number of motor units within a single muscle. Such an estimate is needed for monitoring the progression of neuromuscular diseases such as amyotrophic lateral sclerosis. Our data consist of action potentials that were recorded from the surface of a muscle in response to stimuli of different intensities applied to the nerve supplying the muscle. During the gradual increase in intensity of the stimulus from the threshold to supramaximal, all motor units are progressively excited. However, at any given submaximal intensity of stimulus, the number of units that are excited is variable, because of random fluctuations in axonal excitability. Furthermore, the individual motor unit action potentials exhibit variability. To account for these biological properties, Ridall and co-workers developed a model of motor unit activation that is capable of describing the response where the number of motor units, N , is fixed. The purpose of this paper is to extend that model so that the possible number of motor units, N , is a stochastic variable. We illustrate the elements of our model, show that the results are reproducible and show that our model can measure the decline in motor unit numbers during the course of amyotrophic lateral sclerosis. Our method holds promise of being useful in the study of neurogenic diseases.     

A note on the estimation of rate parameters in stochastic compartmental models

Two approaches to estimating rate parameters in stochastic compartmental models are compared. It is shown that the methods are essentially equivalent both computationally and in terms of asymptotic efficiency. A direct comparison is made using data from a cancer treatment follow-up study. Keywords:Compartmental model, Markov process, Conditional mean and variance.     

Value at risk estimation under stochastic volatility models using adaptive PMCMC methods

In this paper, we propose a value-at-risk (VaR) estimation technique based on a new stochastic volatility model with leverage effect, nonconstant conditional mean and jump. In order to estimate the model parameters and latent state variables, we integrate the particle filter and adaptive Markov Chain Monte Carlo (MCMC) algorithms to develop a novel adaptive particle MCMC (A-PMCMC) algorithm. Comprehensive simulation experiments based on three stock indices and two foreign exchange time series show effectiveness of the proposed A-PMCMC algorithm and the VaR estimation technique.     

A semi-markov approach to stochastic compartmental models

In the past, various methods using either differential equations or differential-difference equations have been used to analyze stochastic compartmental models. In this paper a semi-Markov process approach is used to provide a framework for analyzing such models. The distribution function of the number of particles in each of the compartments is derived along with the stationary distributions. Various models found in the literature arising from biological and reliability applications are analyzed here using the semi-Markov process technique.     

Markov chain Monte Carlo methods for parameter estimation of the modified Weibull distribution

In this paper, the Markov chain Monte Carlo (MCMC) method is used to estimate the parameters of a modified Weibull distribution based on a complete sample. While maximum-likelihood estimation (MLE) is the most used method for parameter estimation, MCMC has recently emerged as a good alternative. When applied to parameter estimation, MCMC methods have been shown to be easy to implement computationally, the estimates always exist and are statistically consistent, and their probability intervals are convenient to construct. Details of applying MCMC to parameter estimation for the modified Weibull model are elaborated and a numerical example is presented to illustrate the methods of inference discussed in this paper. To compare MCMC with MLE, a simulation study is provided, and the differences between the estimates obtained by the two algorithms are examined.     

Quantifying uncertainty in transdimensional Markov chain Monte Carlo using discrete Markov models

Bayesian analysis often concerns an evaluation of models with different dimensionality as is necessary in, for example, model selection or mixture models. To facilitate this evaluation, transdimensional Markov chain Monte Carlo (MCMC) relies on sampling a discrete indexing variable to estimate the posterior model probabilities. However, little attention has been paid to the precision of these estimates. If only few switches occur between the models in the transdimensional MCMC output, precision may be low and assessment based on the assumption of independent samples misleading. Here, we propose a new method to estimate the precision based on the observed transition matrix of the model-indexing variable. Assuming a first-order Markov model, the method samples from the posterior of the stationary distribution. This allows assessment of the uncertainty in the estimated posterior model probabilities, model ranks, and Bayes factors. Moreover, the method provides an estimate for the effective sample size of the MCMC output. In two model selection examples, we show that the proposed approach provides a good assessment of the uncertainty associated with the estimated posterior model probabilities.

     

Parallel algorithms for Markov chain Monte Carlo methods in latent spatial Gaussian models

Markov chain Monte Carlo (MCMC) implementations of Bayesian inference for latent spatial Gaussian models are very computationally intensive, and restrictions on storage and computation time are limiting their application to large problems. Here we propose various parallel MCMC algorithms for such models. The algorithms' performance is discussed with respect to a simulation study, which demonstrates the increase in speed with which the algorithms explore the posterior distribution as a function of the number of processors. We also discuss how feasible problem size is increased by use of these algorithms.     

Marginal maximum a posteriori estimation using Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) methods, while facilitating the solution of many complex problems in Bayesian inference, are not currently well adapted to the problem of marginal maximum a posteriori (MMAP) estimation, especially when the number of parameters is large. We present here a simple and novel MCMC strategy, called State-Augmentation for Marginal Estimation (SAME), which leads to MMAP estimates for Bayesian models. We illustrate the simplicity and utility of the approach for missing data interpolation in autoregressive time series and blind deconvolution of impulsive processes.     

Likelihood estimation for longitudinal zero-inflated power series regression models

In this paper, a zero-inflated power series regression model for longitudinal count data with excess zeros is presented. We demonstrate how to calculate the likelihood for such data when it is assumed that the increment in the cumulative total follows a discrete distribution with a location parameter that depends on a linear function of explanatory variables. Simulation studies indicate that this method can provide improvements in obtaining standard errors of the estimates. We also calculate the dispersion index for this model. The influence of a small perturbation of the dispersion index of the zero-inflated model on likelihood displacement is also studied. The zero-inflated negative binomial regression model is illustrated on data regarding joint damage in psoriatic arthritis.     

Markov chain Monte Carlo methods for family trees using a parallel processor

A 1024 CPU parallel computer is used to obtain simulated genotypes in the Tristan da Cunha pedigree using random local updating methods. A four-colour theorem is invoked to justify simultaneous updating. Multiple copies of the program are run simultaneously. These results are used to infer the source of the B allele of the ABO blood group that is present in the population.     

Empirical transform estimation for indexed stochastic models

We present a method for estimating the parameters in indexed stochastic models via a least squares approach based on empirical transforms. Asymptotic approximations are derived for the distribution of the resulting estimators. An explicit expression for the mean-squared error provides a natural way of selecting the transform variable, and a numerical example illustrates the performance of the resulting method. A common finding, which we term 'diagonal optimization', occurs when multiparameter models are fitted by using transforms. Diagonal optimization arises when optimal performance results from equating the elements of the transform vector, and we provide a heuristic explanation of why this occurs.     

Estimation for aggregate models: The aggregate Markov chain

An argument in favour of projecting the score function for models involving incomplete data is presented. Projection is then applied to the aggregate Markov-chain model resulting in weighted least-squares estimators. The limit theory and efficiency of these estimators are studied using martingale limit theory.     

Markov chain models for multivariate repeated binary data analysis

Repeated categorical outcomes frequently occur in clinical trials. Muenz and Rubinstein (1985) presented Markov chain models to analyze binary repeated data in a breast cancer study. We extend their method to the setting when more than one repeated outcome variable is of interest. In a randomized clinical trial of breast cancer, we investigate the dependency of toxicities on predictor variables and the relationship among multiple toxic effects.     

Steady-state Markov chain models for certain q-confluent hypergeometric distributions

This paper draws attention to those members of the q-confluent hypergeometric family of discrete distributions that either (i) have special properties or (ii) arise as steady-state distributions from interesting Markov chains. They include (i) the Exton and O/U distributions and (ii) the q-hyper-Poisson I, Morse, confluent Bailey–Daum, and confluent q-Chu–Vandermonde distributions.     

Penalized maximum likelihood estimation for Gaussian hidden Markov models

ABSTRACT

The likelihood function of a Gaussian hidden Markov model is unbounded, which is why the maximum likelihood estimator (MLE) is not consistent. A penalized MLE is introduced along with a rigorous consistency proof.     

On the coefficient of variation for residence time distributions of some stochastic compartmental models

The study of residence time distributions is motivated by the desire to develop new practical tools for the statistical analysis of compartmental systems. In particular, Gibaldi and Perrier (1982) describe three alternative models for a two-compartment system, which were noted to be "indistinguishable based solely on plasma or urinary excretion data," defining a residence time distribution. In this paper, properties of the coefficient of variation of the residence time distributions are developed for these three models. In addition to the standard Markovian model with exponential retention times, properties are also derived for a non-Markovian model with gamma retention times. A coefficient of variation may be estimated from commonly available elimination data, and may be used in principle to discriminate between these three models.     

Analysis of generalized linear mixed models via a stochastic approximation algorithm with Markov chain Monte-Carlo method

In recent years much effort has been devoted to maximum likelihood estimation of generalized linear mixed models. Most of the existing methods use the EM algorithm, with various techniques in handling the intractable E-step. In this paper, a new implementation of a stochastic approximation algorithm with Markov chain Monte Carlo method is investigated. The proposed algorithm is computationally straightforward and its convergence is guaranteed. A simulation and three real data sets, including the challenging salamander data, are used to illustrate the procedure and to compare it with some existing methods. The results indicate that the proposed algorithm is an attractive alternative for problems with a large number of random effects or with high dimensional intractable integrals in the likelihood function.     

On a convergent stochastic estimation algorithm for frailty models

A maximum likelihood estimation procedure is presented for the frailty model. The procedure is based on a stochastic Expectation Maximization algorithm which converges quickly to the maximum likelihood estimate. The usual expectation step is replaced by a stochastic approximation of the complete log-likelihood using simulated values of unobserved frailties whereas the maximization step follows the same lines as those of the Expectation Maximization algorithm. The procedure allows to obtain at the same time estimations of the marginal likelihood and of the observed Fisher information matrix. Moreover, this stochastic Expectation Maximization algorithm requires less computation time. A wide variety of multivariate frailty models without any assumption on the covariance structure can be studied. To illustrate this procedure, a Gaussian frailty model with two frailty terms is introduced. The numerical results based on simulated data and on real bladder cancer data are more accurate than those obtained by using the Expectation Maximization Laplace algorithm and the Monte-Carlo Expectation Maximization one. Finally, since frailty models are used in many fields such as ecology, biology, economy, …, the proposed algorithm has a wide spectrum of applications.