Profile Hellinger distance estimation

The successful application of the Hellinger distance approach to fully parametric models is well known. The corresponding optimal estimators, known as minimum Hellinger distance (MHD) estimators, are efficient and have excellent robustness properties [Beran R. Minimum Hellinger distance estimators for parametric models. Ann Statist. 1977;5:445–463]. This combination of efficiency and robustness makes MHD estimators appealing in practice. However, their application to semiparametric statistical models, which have a nuisance parameter (typically of infinite dimension), has not been fully studied. In this paper, we investigate a methodology to extend the MHD approach to general semiparametric models. We introduce the profile Hellinger distance and use it to construct a minimum profile Hellinger distance estimator of the finite-dimensional parameter of interest. This approach is analogous in some sense to the profile likelihood approach. We investigate the asymptotic properties such as the asymptotic normality, efficiency, and adaptivity of the proposed estimator. We also investigate its robustness properties. We present its small-sample properties using a Monte Carlo study.     

Analyses of infectious disease data from household outbreaks by Markov chain Monte Carlo methods

The analysis of infectious disease data presents challenges arising from the dependence in the data and the fact that only part of the transmission process is observable. These difficulties are usually overcome by making simplifying assumptions. The paper explores the use of Markov chain Monte Carlo (MCMC) methods for the analysis of infectious disease data, with the hope that they will permit analyses to be made under more realistic assumptions. Two important kinds of data sets are considered, containing temporal and non-temporal information, from outbreaks of measles and influenza. Stochastic epidemic models are used to describe the processes that generate the data. MCMC methods are then employed to perform inference in a Bayesian context for the model parameters. The MCMC methods used include standard algorithms, such as the Metropolis–Hastings algorithm and the Gibbs sampler, as well as a new method that involves likelihood approximation. It is found that standard algorithms perform well in some situations but can exhibit serious convergence difficulties in others. The inferences that we obtain are in broad agreement with estimates obtained by other methods where they are available. However, we can also provide inferences for parameters which have not been reported in previous analyses.     

Approximation and information topologies

In this paper approximation properties of finite dimensional parametric models are described in terms of an information metric: the Hellinger distance. Under conditions on the parametric family given solely in terms of a comparison of the Hellinger distance with the parameter metric, optimal rates of convergence are described. It is also shown how to use these conditions on the parametric family to determine whether consistent estimation is possible. We give applications of the theorems to regular and non-regular parametric families, and to nonlinear regression.     

Minimum hellinger distance estimation for normal models

A robust estimator introduced by Beran (1977a, 1977b), which is based on the minimum Hellinger distance between a projection model density and a nonparametric sample density, is studied empirically. An extensive simulation provides an estimate of the small sample distribution and supplies empirical evidence of the estimator performance for a normal location-scale model. While the performance of the minimum Hellinger distance estimator is seen to be competitive with the maximum likelihood estimator at the true model, its robustness to deviations from normality is shown to be competitive in this setting with that obtained from the M-estimator and the Cramér-von Mises minimum distance estimator. Beran also introduced a goodness-of-fit statisticH ₂, based on the minimized Hellinger distance between a member of a parametric family of densities and a nonparametric density estimate. We investigate the statistic H (the square root of H ₂) as a test for normality when both location and scale are unspecified. Empirically derived critical values are given which do not require extensive tables. The power of the statistic H compares favorably with four other widely used tests for normality.     

Convergence in the Wasserstein Metric for Markov Chain Monte Carlo Algorithms with Applications to Image Restoration

Abstract

In this paper, we show how the time for convergence to stationarity of a Markov chain can be assessed using the Wasserstein metric, rather than the usual choice of total variation distance. The Wasserstein metric may be more easily applied in some applications, particularly those on continuous state spaces. Bounds on convergence time are established by considering the number of iterations required to approximately couple two realizations of the Markov chain to within ε tolerance. The particular application considered is the use of the Gibbs sampler in the Bayesian restoration of a degraded image, with pixels that are a continuous grey-scale and with pixels that can only take two colours. On finite state spaces, a bound in the Wasserstein metric can be used to find a bound in total variation distance. We use this relationship to get a precise O(N log N) bound on the convergence time of the stochastic Ising model that holds for appropriate values of its parameter as well as other binary image models. Our method employing convergence in the Wasserstein metric can also be applied to perfect sampling algorithms involving coupling from the past to obtain estimates of their running times.     

Minimum Hellinger distance estimation for randomized play the winner design

Response-adaptive designs in clinical trials incorporate information from prior patient responses in order to assign better performing treatments to the future patients of a clinical study. An example of a response adaptive design that has received much attention in recent years is the randomized play the winner design (RPWD). Beran [1977. Minimum Hellinger distance estimates for parametric models. Ann. Statist. 5, 445–463] investigated the problem of minimum Hellinger distance procedure (MHDP) for continuous data and showed that minimum Hellinger distance estimator (MHDE) of a finite dimensional parameter is as efficient as the MLE (maximum likelihood estimator) under a true model assumption. This paper develops minimum Hellinger distance methodology for data generated using RPWD. A new algorithm using the Monte Carlo approximation to the estimating equation is proposed. Consistency and asymptotic normality of the estimators are established and the robustness and small sample performance of the estimators are illustrated using simulations. The methodology when applied to the clinical trial data conducted by Eli-Lilly and Company, brings out the treatment effect in one of the strata using the frequentist techniques compared to the Bayesian argument of Tamura et al [1994. A case study of an adaptive clinical trialin the treatment of out-patients with depressive disorder. J. Amer. Statist. Assoc. 89, 768–776].     

On minimum Hellinger distance estimation

Efficiency and robustness are two fundamental concepts in parametric estimation problems. It was long thought that there was an inherent contradiction between the aims of achieving robustness and efficiency; that is, a robust estimator could not be efficient and vice versa. It is now known that the minimum Hellinger distance approached introduced by Beran [R. Beran, Annals of Statistics 1977;5:445–463] is one way of reconciling the conflicting concepts of efficiency and robustness. For parametric models, it has been shown that minimum Hellinger estimators achieve efficiency at the model density and simultaneously have excellent robustness properties. In this article, we examine the application of this approach in two semiparametric models. In particular, we consider a two‐component mixture model and a two‐sample semiparametric model. In each case, we investigate minimum Hellinger distance estimators of finite‐dimensional Euclidean parameters of particular interest and study their basic asymptotic properties. Small sample properties of the proposed estimators are examined using a Monte Carlo study. The results can be extended to semiparametric models of general form as well. The Canadian Journal of Statistics 37: 514–533; 2009 © 2009 Statistical Society of Canada     

Bayesian analysis of hierarchical linear mixed modeling using the multivariate t distribution

This article presents a fully Bayesian approach to modeling incomplete longitudinal data using the t linear mixed model with AR(p) dependence. Markov chain Monte Carlo (MCMC) techniques are implemented for computing posterior distributions of parameters. To facilitate the computation, two types of auxiliary indicator matrices are incorporated into the model. Meanwhile, the constraints on the parameter space arising from the stationarity conditions for the autoregressive parameters are handled by a reparametrization scheme. Bayesian predictive inferences for the future vector are also investigated. An application is illustrated through a real example from a multiple sclerosis clinical trial.     

Markov Chain Monte Carlo in small worlds

As the number of applications for Markov Chain Monte Carlo (MCMC) grows, the power of these methods as well as their shortcomings become more apparent. While MCMC yields an almost automatic way to sample a space according to some distribution, its implementations often fall short of this task as they may lead to chains which converge too slowly or get trapped within one mode of a multi-modal space. Moreover, it may be difficult to determine if a chain is only sampling a certain area of the space or if it has indeed reached stationarity. In this paper, we show how a simple modification of the proposal mechanism results in faster convergence of the chain and helps to circumvent the problems described above. This mechanism, which is based on an idea from the field of “small-world” networks, amounts to adding occasional “wild” proposals to any local proposal scheme. We demonstrate through both theory and extensive simulations, that these new proposal distributions can greatly outperform the traditional local proposals when it comes to exploring complex heterogenous spaces and multi-modal distributions. Our method can easily be applied to most, if not all, problems involving MCMC and unlike many other remedies which improve the performance of MCMC it preserves the simplicity of the underlying algorithm.     

Determination of sample size for selecting the smallest of k poisson population means

A procedure for selecting a Poisson population with smallest mean is considered using an indifference zone approach. The objective is to determine the smallest sample size n required from k ≥ 2 populations in order to attain the desired probability of correct selection. Since the means procedure is not consistent with respect to the difference or ratio alone, two distance measures are used simultaneously to overcome the difficulty in obtaining the smallest probability of correct selection that is greater than some specified limit. The constants required to determine n are computed and tabulated. The asymptotic results are derived using a normal approximation. A comparison with the exact results indicates that the proposed approximation works well. Only in the extreme cases small increases in n are observed. An example of industrial accident data is used to illustrate this procedure.     

On convergence of properly weighted samples to the target distribution

We consider importance sampling as well as other properly weighted samples with respect to a target distribution π$\pi$ from a different point of view. By considering the associated weights as sojourn times until the next jump, we define appropriate jump processes. When the original sample sequence forms an ergodic Markov chain, the associated jump process is an ergodic semi-Markov process with stationary distribution π$\pi$. In this respect, properly weighted samples behave very similarly to standard Markov chain Monte Carlo (MCMC) schemes in that they exhibit convergence to the target distribution as well. Indeed, some standard MCMC procedures like the Metropolis–Hastings algorithm are included in this context. Moreover, when the samples are independent and the mean weight is bounded above, we describe a slight modification in order to achieve exact (weighted) samples from the target distribution.     

Bayesian inference for pairwise interacting point processes

Pairwise interacting point processes are commonly used to model spatial point patterns. To perform inference, the established frequentist methods can produce good point estimates when the interaction in the data is moderate, but some methods may produce severely biased estimates when the interaction in strong. Furthermore, because the sampling distributions of the estimates are unclear, interval estimates are typically obtained by parametric bootstrap methods. In the current setting however, the behavior of such estimates is not well understood. In this article we propose Bayesian methods for obtaining inferences in pairwise interacting point processes. The requisite application of Markov chain Monte Carlo (MCMC) techniques is complicated by an intractable function of the parameters in the likelihood. The acceptance probability in a Metropolis-Hastings algorithm involves the ratio of two likelihoods evaluated at differing parameter values. The intractable functions do not cancel, and hence an intractable ratio r must be estimated within each iteration of a Metropolis-Hastings sampler. We propose the use of importance sampling techniques within MCMC to address this problem. While r may be estimated by other methods, these, in general, are not readily applied in a Bayesian setting. We demonstrate the validity of our importance sampling approach with a small simulation study. Finally, we analyze the Swedish pine sapling dataset (Strand 1972) and contrast the results with those in the literature.     

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.     

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.     

Bayesian comparison of diagnostic tests with largely non-informative missing data

This work was motivated by a real problem of comparing binary diagnostic tests based upon a gold standard, where the collected data showed that the large majority of classifications were incomplete and the feedback received from the medical doctors allowed us to consider the missingness as non-informative. Taking into account the degree of data incompleteness, we used a Bayesian approach via MCMC methods for drawing inferences of interest on accuracy measures. Its direct implementation by well-known software demonstrated serious problems of chain convergence. The difficulties were overcome by the proposal of a simple, efficient and easily adaptable data augmentation algorithm, performed through an ad hoc computer program.     

Sup–Hellinger consistency for local density regression

Abstract

This article studies the consistency of a local density regression model under a supremum Hellinger distance. Such model applies a piecewise structure where a mixture of Dirichlet process model (MDP) is assigned as the fixed density on each piece. The piecewise construction is a straightforward way to establish sup–Hellinger consistency in a regression settings. A specific piecewise density example is presented in a simulation study.     

Bayesian variable selection in a finite mixture of linear mixed-effects models

Mixture of linear mixed-effects models has received considerable attention in longitudinal studies, including medical research, social science and economics. The inferential question of interest is often the identification of critical factors that affect the responses. We consider a Bayesian approach to select the important fixed and random effects in the finite mixture of linear mixed-effects models. To accomplish our goal, latent variables are introduced to facilitate the identification of influential fixed and random components and to classify the membership of observations in the longitudinal data. A spike-and-slab prior for the regression coefficients is adopted to sidestep the potential complications of highly collinear covariates and to handle large p and small n issues in the variable selection problems. Here we employ Markov chain Monte Carlo (MCMC) sampling techniques for posterior inferences and explore the performance of the proposed method in simulation studies, followed by an actual psychiatric data analysis concerning depressive disorder.     

Bayes and maximum likelihood for $$L^1$$-Wasserstein deconvolution of Laplace mixtures

We consider the problem of recovering a distribution function on the real line from observations additively contaminated with errors following the standard Laplace distribution. Assuming that the latent distribution is completely unknown leads to a nonparametric deconvolution problem. We begin by studying the rates of convergence relative to the \(L^2\)-norm and the Hellinger metric for the direct problem of estimating the sampling density, which is a mixture of Laplace densities with a possibly unbounded set of locations: the rate of convergence for the Bayes’ density estimator corresponding to a Dirichlet process prior over the space of all mixing distributions on the real line matches, up to a logarithmic factor, with the \(n^{-3/8}\log ^{1/8}n\) rate for the maximum likelihood estimator. Then, appealing to an inversion inequality translating the \(L^2\)-norm and the Hellinger distance between general kernel mixtures, with a kernel density having polynomially decaying Fourier transform, into any \(L^p\)-Wasserstein distance, \(p\ge 1\), between the corresponding mixing distributions, provided their Laplace transforms are finite in some neighborhood of zero, we derive the rates of convergence in the \(L^1\)-Wasserstein metric for the Bayes’ and maximum likelihood estimators of the mixing distribution. Merging in the \(L^1\)-Wasserstein distance between Bayes and maximum likelihood follows as a by-product, along with an assessment on the stochastic order of the discrepancy between the two estimation procedures.     

Adaptive independence samplers

Markov chain Monte Carlo (MCMC) is an important computational technique for generating samples from non-standard probability distributions. A major challenge in the design of practical MCMC samplers is to achieve efficient convergence and mixing properties. One way to accelerate convergence and mixing is to adapt the proposal distribution in light of previously sampled points, thus increasing the probability of acceptance. In this paper, we propose two new adaptive MCMC algorithms based on the Independent Metropolis–Hastings algorithm. In the first, we adjust the proposal to minimize an estimate of the cross-entropy between the target and proposal distributions, using the experience of pre-runs. This approach provides a general technique for deriving natural adaptive formulae. The second approach uses multiple parallel chains, and involves updating chains individually, then updating a proposal density by fitting a Bayesian model to the population. An important feature of this approach is that adapting the proposal does not change the limiting distributions of the chains. Consequently, the adaptive phase of the sampler can be continued indefinitely. We include results of numerical experiments indicating that the new algorithms compete well with traditional Metropolis–Hastings algorithms. We also demonstrate the method for a realistic problem arising in Comparative Genomics.     

R package rjmcmc: reversible jump MCMC using post‐processing

The rjmcmc package for R implements the post‐processing reversible jump Markov chain Monte Carlo (MCMC) algorithm of Barker & Link. MCMC output from each of the models is used to estimate posterior model probabilities and Bayes factors. Automatic differentiation is used to simplify implementation. The package is demonstrated on two examples.