Bayesian sequential inference for nonlinear multivariate diffusions

In this paper, we adapt recently developed simulation-based sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m−1 latent data points between every pair of observations. Sequential MCMC methods are then used to sample the posterior distribution of the latent data and the model parameters on-line. The method is applied to the estimation of parameters in a simple stochastic volatility model (SV) of the U.S. short-term interest rate. We also provide a simulation study to validate our method, using synthetic data generated by the SV model with parameters calibrated to match weekly observations of the U.S. short-term interest rate.     

Diagnostic tools for approximate Bayesian computation using the coverage property

Approximate Bayesian computation (ABC) is an approach to sampling from an approximate posterior distribution in the presence of a computationally intractable likelihood function. A common implementation is based on simulating model, parameter and dataset triples from the prior, and then accepting as samples from the approximate posterior, those model and parameter pairs for which the corresponding dataset, or a summary of that dataset, is ‘close’ to the observed data. Closeness is typically determined though a distance measure and a kernel scale parameter. Appropriate choice of that parameter is important in producing a good quality approximation. This paper proposes diagnostic tools for the choice of the kernel scale parameter based on assessing the coverage property, which asserts that credible intervals have the correct coverage levels in appropriately designed simulation settings. We provide theoretical results on coverage for both model and parameter inference, and adapt these into diagnostics for the ABC context. We re‐analyse a study on human demographic history to determine whether the adopted posterior approximation was appropriate. Code implementing the proposed methodology is freely available in the R package abctools .     

Bayesian inference for partially observed stochastic epidemics

The analysis of infectious disease data is usually complicated by the fact that real life epidemics are only partially observed. In particular, data concerning the process of infection are seldom available. Consequently, standard statistical techniques can become too complicated to implement effectively. In this paper Markov chain Monte Carlo methods are used to make inferences about the missing data as well as the unknown parameters of interest in a Bayesian framework. The methods are applied to real life data from disease outbreaks.     

Bayesian inference for rare errors in populations with unequal unit sizes

We describe a Bayesian model for a scenario in which the population of errors contains many 0s and there is a known covariate. This kind of structure typically occurs in auditing, and we use auditing as the driving application of the method. Our model is based on a categorization of the error population together with a Bayesian nonparametric method of modelling errors within some of the categories. Inference is through simulation. We conclude with an example based on a data set provided by the UK's National Audit Office.     

Bayesian inference for the mean and standard deviation of a normal population when only the sample size,mean and range are observed

Consider a random sample X₁, X₂,…, X_n, from a normal population with unknown mean and standard deviation. Only the sample size, mean and range are recorded and it is necessary to estimate the unknown population mean and standard deviation. In this paper the estimation of the mean and standard deviation is made from a Bayesian perspective by using a Markov Chain Monte Carlo (MCMC) algorithm to simulate samples from the intractable joint posterior distribution of the mean and standard deviation. The proposed methodology is applied to simulated and real data. The real data refers to the sugar content (^oBRIX level) of orange juice produced in different countries.     

Bayesian inference for epidemics with two levels of mixing

Abstract.  Methodology for Bayesian inference is considered for a stochastic epidemic model which permits mixing on both local and global scales. Interest focuses on estimation of the within- and between-group transmission rates given data on the final outcome. The model is sufficiently complex that the likelihood of the data is numerically intractable. To overcome this difficulty, an appropriate latent variable is introduced, about which asymptotic information is known as the population size tends to infinity. This yields a method for approximate inference for the true model. The methods are applied to real data, tested with simulated data, and also applied to a simple epidemic model for which exact results are available for comparison.     

Improved inference for MCP-Mod approach using time-to-event endpoints with small sample sizes

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.     

Parameter recovery for a skew-normal IRT model under a Bayesian approach: hierarchical framework,prior and kernel sensitivity and sample size

Item response theory (IRT) comprises a set of statistical models which are useful in many fields, especially when there is an interest in studying latent variables (or latent traits). Usually such latent traits are assumed to be random variables and a convenient distribution is assigned to them. A very common choice for such a distribution has been the standard normal. Recently, Azevedo et al. [Bayesian inference for a skew-normal IRT model under the centred parameterization, Comput. Stat. Data Anal. 55 (2011), pp. 353–365] proposed a skew-normal distribution under the centred parameterization (SNCP) as had been studied in [R.B. Arellano-Valle and A. Azzalini, The centred parametrization for the multivariate skew-normal distribution, J. Multivariate Anal. 99(7) (2008), pp. 1362–1382], to model the latent trait distribution. This approach allows one to represent any asymmetric behaviour concerning the latent trait distribution. Also, they developed a Metropolis–Hastings within the Gibbs sampling (MHWGS) algorithm based on the density of the SNCP. They showed that the algorithm recovers all parameters properly. Their results indicated that, in the presence of asymmetry, the proposed model and the estimation algorithm perform better than the usual model and estimation methods. Our main goal in this paper is to propose another type of MHWGS algorithm based on a stochastic representation (hierarchical structure) of the SNCP studied in [N. Henze, A probabilistic representation of the skew-normal distribution, Scand. J. Statist. 13 (1986), pp. 271–275]. Our algorithm has only one Metropolis–Hastings step, in opposition to the algorithm developed by Azevedo et al., which has two such steps. This not only makes the implementation easier but also reduces the number of proposal densities to be used, which can be a problem in the implementation of MHWGS algorithms, as can be seen in [R.J. Patz and B.W. Junker, A straightforward approach to Markov Chain Monte Carlo methods for item response models, J. Educ. Behav. Stat. 24(2) (1999), pp. 146–178; R.J. Patz and B.W. Junker, The applications and extensions of MCMC in IRT: Multiple item types, missing data, and rated responses, J. Educ. Behav. Stat. 24(4) (1999), pp. 342–366; A. Gelman, G.O. Roberts, and W.R. Gilks, Efficient Metropolis jumping rules, Bayesian Stat. 5 (1996), pp. 599–607]. Moreover, we consider a modified beta prior (which generalizes the one considered in [3 Azevedo, C. L.N., Bolfarine, H. and Andrade, D. F. 2011. Bayesian inference for a skew-normal IRT model under the centred parameterization. Comput. Stat. Data Anal., 55: 353–365. [Crossref], [Web of Science ®] , [Google Scholar]]) and a Jeffreys prior for the asymmetry parameter. Furthermore, we study the sensitivity of such priors as well as the use of different kernel densities for this parameter. Finally, we assess the impact of the number of examinees, number of items and the asymmetry level on the parameter recovery. Results of the simulation study indicated that our approach performed equally as well as that in [3 Azevedo, C. L.N., Bolfarine, H. and Andrade, D. F. 2011. Bayesian inference for a skew-normal IRT model under the centred parameterization. Comput. Stat. Data Anal., 55: 353–365. [Crossref], [Web of Science ®] , [Google Scholar]], in terms of parameter recovery, mainly using the Jeffreys prior. Also, they indicated that the asymmetry level has the highest impact on parameter recovery, even though it is relatively small. A real data analysis is considered jointly with the development of model fitting assessment tools. The results are compared with the ones obtained by Azevedo et al. The results indicate that using the hierarchical approach allows us to implement MCMC algorithms more easily, it facilitates diagnosis of the convergence and also it can be very useful to fit more complex skew IRT models.     

Likelihood‐based and marginal inference methods for recurrent event data with covariate measurement error

Recurrent event data arise commonly in medical and public health studies. The analysis of such data has received extensive research attention and various methods have been developed in the literature. Depending on the focus of scientific interest, the methods may be broadly classified as intensity‐based counting process methods, mean function‐based estimating equation methods, and the analysis of times to events or times between events. These methods and models cover a wide variety of practical applications. However, there is a critical assumption underlying those methods–variables need to be correctly measured. Unfortunately, this assumption is frequently violated in practice. It is quite common that some covariates are subject to measurement error. It is well known that covariate measurement error can substantially distort inference results if it is not properly taken into account. In the literature, there has been extensive research concerning measurement error problems in various settings. However, with recurrent events, there is little discussion on this topic. It is the objective of this paper to address this important issue. In this paper, we develop inferential methods which account for measurement error in covariates for models with multiplicative intensity functions or rate functions. Both likelihood‐based inference and robust inference based on estimating equations are discussed. The Canadian Journal of Statistics 40: 530–549; 2012 © 2012 Statistical Society of Canada     

Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process

We consider a one-dimensional diffusion process X , with ergodic property, with drift b ( x , θ) and diffusion coefficient a ( x , θ) depending on an unknown parameter θ that may be multidimensional. We are interested in the estimation of θ and dispose, for that purpose, of a discretized trajectory, observed at n equidistant times t_i = iΔ , i = 0, ..., n . We study a particular class of estimating functions of the form ∑ f (θ, X _{t i −1}) which, under the assumption that the integral of f with respect to the invariant measure is null, provide us with a consistent and asymptotically normal estimator. We determine the choice of f that yields the estimator with minimum asymptotic variance within the class and indicate how to construct explicit estimating functions based on the generator of the diffusion. Finally the theoretical study is completed with simulations.     

A Bayesian model for measurement and misclassification errors alongside missing data,with an application to higher education participation in Australia

In this paper we consider the impact of both missing data and measurement errors on a longitudinal analysis of participation in higher education in Australia. We develop a general method for handling both discrete and continuous measurement errors that also allows for the incorporation of missing values and random effects in both binary and continuous response multilevel models. Measurement errors are allowed to be mutually dependent and their distribution may depend on further covariates. We show that our methodology works via two simple simulation studies. We then consider the impact of our measurement error assumptions on the analysis of the real data set.     

Stochastic differential equation systems for an SIS epidemic model with vaccination and immigration

Two Itô stochastic differential equation (SDE) systems are constructed for a Susceptible-Infected-Susceptible epidemic model with temporary vaccination. A constant number of new members enter the population and total size of the population is variable. Some conditions for disease extinction in the stochastic models are established and compared with conditions in deterministic one. It is shown that the two stochastic models are equivalent in the sense that their solutions come from same distribution. In addition, the SDE models are simulated and the equivalence of the two stochastic models is confirmed by numerical examples. The probability distribution for extinction is also obtained numerically, provided there exists a probability for disease persistence whereas the expected duration of epidemic is acquired when extinction occurs with probability 1.