Basu–Dhar bivariate geometric distribution in the presence of covariates and censored data: a Bayesian approach

In this paper, we introduce classical and Bayesian approaches for the Basu–Dhar bivariate geometric distribution in the presence of covariates and censored data. This distribution is considered for the analysis of bivariate lifetime as an alternative to some existing bivariate lifetime distributions assuming continuous lifetimes as the Block and Basu or Marshall and Olkin bivariate distributions. Maximum likelihood and Bayesian estimators are presented. Two examples are considered to illustrate the proposed methodology: an example with simulated data and an example with medical bivariate lifetime data.     

An adaptive bayebsian solution to the control peoble1 for linear control systems

In this paper, a Bayesian procedure to solve the control problem in Linear Control Systems , when the precision Matrices of the obstsrvaiional and systems errors are unknown, is proposed. Prior information on those matrices is required, A numerical example illustrates the procedure applied to models Which contain bivariate observations and bivariate state vectors.     

Bayesian bivariate survival estimation

There is no easy extension of the Kaplan–Meier and Nelson–Aalen estimators to the bivariate case, and estimating bivariate survival distributions nonparametrically is associated with various nontrivial problems. The Dabrowska estimator will, for example, associate negative mass to some subsets. Bayesian methods hold some promise as they will avoid the negative mass problem, but are also prone to difficulties. We simplify and extend an example by Pruitt to show that the posterior distribution from a Dirichlet process prior is inconsistent. We construct a different nonparametric prior via Beta processes and provide an updating scheme that utilizes only the most relevant parts of the likelihood, and show that this leads to a consistent estimator.     

Bayesian inference for a flexible class of bivariate beta distributions

Several bivariate beta distributions have been proposed in the literature. In particular, Olkin and Liu [A bivariate beta distribution. Statist Probab Lett. 2003;62(4):407–412] proposed a 3 parameter bivariate beta model which Arnold and Ng [Flexible bivariate beta distributions. J Multivariate Anal. 2011;102(8):1194–1202] extend to 5 and 8 parameter models. The 3 parameter model allows for only positive correlation, while the latter models can accommodate both positive and negative correlation. However, these come at the expense of a density that is mathematically intractable. The focus of this research is on Bayesian estimation for the 5 and 8 parameter models. Since the likelihood does not exist in closed form, we apply approximate Bayesian computation, a likelihood free approach. Simulation studies have been carried out for the 5 and 8 parameter cases under various priors and tolerance levels. We apply the 5 parameter model to a real data set by allowing the model to serve as a prior to correlated proportions of a bivariate beta binomial model. Results and comparisons are then discussed.     

A Comparison of Frailty and Other Models for Bivariate Survival Data

Multivariate survival data arise when eachstudy subject may experience multiple events or when study subjectsare clustered into groups. Statistical analyses of such dataneed to account for the intra-cluster dependence through appropriatemodeling. Frailty models are the most popular for such failuretime data. However, there are other approaches which model thedependence structure directly. In this article, we compare thefrailty models for bivariate data with the models based on bivariateexponential and Weibull distributions. Bayesian methods providea convenient paradigm for comparing the two sets of models weconsider. Our techniques are illustrated using two examples.One simulated example demonstrates model choice methods developedin this paper and the other example, based on a practical dataset of onset of blindness among patients with diabetic Retinopathy,considers Bayesian inference using different models.     

Residual life estimation based on bivariate Wiener degradation process with time-scale transformations

The issue of residual life (RL) estimation plays an important role for products while they are in use, especially for expensive and reliability-critical products. For many products, they may have two or more performance characteristics (PCs). Here, an adaptive method of RL estimation based on bivariate Wiener degradation process with time-scale transformations is presented. It is assumed that a product has two PCs, and that each PC is governed by a Wiener process with a time-scale transformation. The dependency of PCs is characterized by the Frank copula function. Parameters are estimated by using the Bayesian Markov chain Monte Carlo method. Once new degradation information is available, the RL is re-estimated in an adaptive manner. A numerical example about fatigue cracks is given to demonstrate the usefulness and validity of the proposed method.     

Exact Bayesian modeling for bivariate Poisson data and extensions

Bivariate count data arise in several different disciplines (epidemiology, marketing, sports statistics just to name a few) and the bivariate Poisson distribution being a generalization of the Poisson distribution plays an important role in modelling such data. In the present paper we present a Bayesian estimation approach for the parameters of the bivariate Poisson model and provide the posterior distributions in closed forms. It is shown that the joint posterior distributions are finite mixtures of conditionally independent gamma distributions for which their full form can be easily deduced by a recursively updating scheme. Thus, the need of applying computationally demanding MCMC schemes for Bayesian inference in such models will be removed, since direct sampling from the posterior will become available, even in cases where the posterior distribution of functions of the parameters is not available in closed form. In addition, we define a class of prior distributions that possess an interesting conjugacy property which extends the typical notion of conjugacy, in the sense that both prior and posteriors belong to the same family of finite mixture models but with different number of components. Extension to certain other models including multivariate models or models with other marginal distributions are discussed.     

Bayesian analysis of bivariate competing risks models with covariates

Bivariate exponential models have often been used for the analysis of competing risks data involving two correlated risk components. Competing risks data consist only of the time to failure and cause of failure. In situations where there is positive probability of simultaneous failure, possibly the most widely used model is the Marshall–Olkin (J. Amer. Statist. Assoc. 62 (1967) 30) bivariate lifetime model. This distribution is not absolutely continuous as it involves a singularity component. However, the likelihood function based on the competing risks data is then identifiable, and any inference, Bayesian or frequentist, can be carried out in a straightforward manner. For the analysis of absolutely continuous bivariate exponential models, standard approaches often run into difficulty due to the lack of a fully identifiable likelihood (Basu and Ghosh; Commun. Statist. Theory Methods 9 (1980) 1515). To overcome the nonidentifiability, the usual frequentist approach is based on an integrated likelihood. Such an approach is implicit in Wada et al. (Calcutta Statist. Assoc. Bull. 46 (1996) 197) who proved some related asymptotic results. We offer in this paper an alternative Bayesian approach. Since systematic prior elicitation is often difficult, the present study focuses on Bayesian analysis with noninformative priors. It turns out that with an appropriate reparameterization, standard noninformative priors such as Jeffreys’ prior and its variants can be applied directly even though the likelihood is not fully identifiable. Two noninformative priors are developed that consist of Laplace's prior for nonidentifiable parameters and Laplace's and Jeffreys's priors for identifiable parameters. The resulting Bayesian procedures possess some frequentist optimality properties as well. Finally, these Bayesian methods are illustrated with analyses of a data set originating out of a lung cancer clinical trial conducted by the Eastern Cooperative Oncology Group.     

Bayesian regression analysis of data with censored initiating and terminating times: applications to aids

Data with censored initiating and terminating times arises quite frequently in acquired immunodeficiency syndrome (AIDS) epidemiologic studies. Analysis of such data involves a complicated bivariate likelihood, which is difficult to deal with computationally. Bayesian analysis, op the other hand, presents added complexities that have yet to be resolved. By exploiting the simple form of a complete data likelihood and utilizing the power of a Markov Chain Monte Carlo (MCMC) algorithm, this paper presents a methodology for fitting Bayesian regression models to such data. The proposed methods extend the work of Sinha (1997), who considered non-parametric Bayesian analysis of this type of data. The methodology is illustiated with an application to a cohort of HIV infected hemophiliac patients.     

Toward an integrated Bayesian network approach to measurement error detection and correction

In this article, the quality of data produced by national statistical institutes and by governmental institutions is considered. In particular, the problem of measurement error is analyzed and an integrated Bayesian network decision support system based on non-parametric Bayesian networks is proposed for its detection and correction. Non-parametric Bayesian networks are graphical models expressing dependence structure via bivariate copulas associated to the edges of the graph. The network structure and the misreport probability are estimated using a validation sample. The Bayesian network model is proposed to decide: (i) which records have to be corrected; (ii) the kind and amount of correction to be adopted. The proposed correction procedure is applied to the Banca d’Italia Survey on Household Income and Wealth and, specifically, the bond amounts are analyzed. Finally, the sensitivity of the conditional distribution of the true value random variable given the observed one to different evidence configurations is studied.     

Reference priors for linear models with general covariance structures

We develop a new class of reference priors for linear models with general covariance structures. A general Markov chain Monte Carlo algorithm is also proposed for implementing the computation. We present several examples to demonstrate the results: Bayesian penalized spline smoothing, a Bayesian approach to bivariate smoothing for a spatial model, and prior specification for structural equation models.     

Estimation of Dynamic Bivariate Mixture Models

This note compares a Bayesian Markov chain Monte Carlo approach implemented by Watanabe with a maximum likelihood ML approach based on an efficient importance sampling procedure to estimate dynamic bivariate mixture models. In these models, stock price volatility and trading volume are jointly directed by the unobservable number of price-relevant information arrivals, which is specified as a serially correlated random variable. It is shown that the efficient importance sampling technique is extremely accurate and that it produces results that differ significantly from those reported by Watanabe.     

Exponentially smoothed control charts for the inverse-gaussian process

Exponential smoothing technique is used to obtain centrality and dispersion control charts for an Inverse-Gaussian(IG)process. The centrality and dispersion control charts use linear combinations of IG and chi-square random variables, respectively. Simulation results are used to verify the approximate distribution of the linear combinations. A method for choosing the “best” value for the weighting constant is presented. An example illustrates the computations that are involved in finding the control limits.     

Shared frailty model based on reversed hazard rate for left censored data

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in the individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin or family data), the shared frailty models were suggested. In this article, we introduce the shared gamma frailty models with the reversed hazard rate. We develop the Bayesian estimation procedure using the Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. We apply the model to a real life bivariate survival dataset.     

A new look at the inverse Gaussian distribution with applications to insurance and economic data

Insurance and economic data are often positive, and we need to take into account this peculiarity in choosing a statistical model for their distribution. An example is the inverse Gaussian (IG), which is one of the most famous and considered distributions with positive support. With the aim of increasing the use of the IG distribution on insurance and economic data, we propose a convenient mode-based parameterization yielding the reparametrized IG (rIG) distribution; it allows/simplifies the use of the IG distribution in various branches of statistics, and we give some examples. In nonparametric statistics, we define a smoother based on rIG kernels. By construction, the estimator is well-defined and does not allocate probability mass to unrealistic negative values. We adopt likelihood cross-validation to select the smoothing parameter. In robust statistics, we propose the contaminated IG distribution, a heavy-tailed generalization of the rIG distribution to accommodate mild outliers. Finally, for model-based clustering and semiparametric density estimation, we present finite mixtures of rIG distributions. We use the EM algorithm to obtain maximum likelihood estimates of the parameters of the mixture and contaminated models. We use insurance data about bodily injury claims, and economic data about incomes of Italian households, to illustrate the models.     

Correlated gamma frailty models for bivariate survival data

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin or family data) the shared frailty models were suggested. Shared frailty models are used despite their limitations. To overcome their disadvantages correlated frailty models may be used. In this article, we introduce the gamma correlated frailty models with two different baseline distributions namely, the generalized log logistic, and the generalized Weibull. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. Also we apply these models to a real life bivariate survival dataset related to the kidney infection data and a better model is suggested for the data.     

Using simulation methods for bayesian econometric models: inference, development,and communication

This paper surveys the fundamental principles of subjective Bayesian inference in econometrics and the implementation of those principles using posterior simulation methods. The emphasis is on the combination of models and the development of predictive distributions. Moving beyond conditioning on a fixed number of completely specified models, the paper introduces subjective Bayesian tools for formal comparison of these models with as yet incompletely specified models. The paper then shows how posterior simulators can facilitate communication between investigators (for example, econometricians) on the one hand and remote clients (for example, decision makers) on the other, enabling clients to vary the prior distributions and functions of interest employed by investigators. A theme of the paper is the practicality of subjective Bayesian methods. To this end, the paper describes publicly available software for Bayesian inference, model development, and communication and provides illustrations using two simple econometric models.     

Comparisons of some bivariate regression models

The bivariate negative binomial regression (BNBR) and the bivariate Poisson log-normal regression (BPLR) models have been used to describe count data that are over-dispersed. In this paper, a new bivariate generalized Poisson regression (BGPR) model is defined. An advantage of the new regression model over the BNBR and BPLR models is that the BGPR can be used to model bivariate count data with either over-dispersion or under-dispersion. In this paper, we carry out a simulation study to compare the three regression models when the true data-generating process exhibits over-dispersion. In the simulation experiment, we observe that the bivariate generalized Poisson regression model performs better than the bivariate negative binomial regression model and the BPLR model.     

Approximation of the Bivariate Renewal Function

Recently, Kambo and his co-researchers (2012) proposed a method of approximation for evaluating the one-dimensional renewal function based on the first three moments. Their method is simple and elegant, which gives exact values for well-known distributions. In this article, we propose an analogous method for the evaluation of bivariate renewal function based on the first two moments of the variables and their joint moment. The proposed method yields exact results for certain widely used bivariate distributions like bivariate exponential distribution, bivariate Weibull distributions, and bivariate Pareto distributions. An illustrative example in the form of a two-dimensional warranty problem is considered and comparisons of our method are made with the results of other models.