Bayesian analysis for a stress-strength system under noninformative priors

Noninformative priors are used for estimating the reliability of a stress-strength system. Several reference priors (cf. Berger and Bernardo 1989, 1992) are derived. A class of priors is found by matching the coverage probabilities of one-sided Bayesian credible intervals with the corresponding frequentist coverage probabilities. It turns out that none of the reference priors is a matching prior. Sufficient conditions for propriety of posteriors under reference priors and matching priors are provided. A simple matching prior is compared with three reference priors when sample sizes are small. The study shows that the matching prior performs better than Jeffreys's prior and reference priors in meeting the target coverage probabilities.     

Bayesian analysis of agricultural field experiments

The paper describes Bayesian analysis for agricultural field experiments, a topic that has received very little previous attention, despite a vast frequentist literature. Adoption of the Bayesian paradigm simplifies the interpretation of the results, especially in ranking and selection. Also, complex formulations can be analysed with comparative ease, by using Markov chain Monte Carlo methods. A key ingredient in the approach is the need for spatial representations of the unobserved fertility patterns. This is discussed in detail. Problems caused by outliers and by jumps in fertility are tackled via hierarchical t formulations that may find use in other contexts. The paper includes three analyses of variety trials for yield and one example involving binary data; none is entirely straightforward. Some comparisons with frequentist analyses are made.     

Bayesian correlated factor analysis of socio-demographic indicators

Recent changes in European family dynamics are often linked to common latent trends of economic and ideational change. Using Bayesian factor analysis, we extract three latent variables from eight socio-demographic indicators related to family formation, dissolution, and gender system and collected on 19 European countries within four periods (1970, 1980, 1990, 1998). The flexibility of the Bayesian approach allows us to introduce an innovative temporal factor model, adding the temporal dimension to the traditional factorial analysis. The underlying structure of the Bayesian factor model proposed reflects our idea of an autoregressive pattern in the latent variables relative to adjacent time periods. The results we obtain are consistent with current interpretations in European demographic trends.     

Bayesian robustness in meta‐analysis for studies with zero responses

Statistical meta‐analysis is mostly carried out with the help of the random effect normal model, including the case of discrete random variables. We argue that the normal approximation is not always able to adequately capture the underlying uncertainty of the original discrete data. Furthermore, when we examine the influence of the prior distributions considered, in the presence of rare events, the results from this approximation can be very poor. In order to assess the robustness of the quantities of interest in meta‐analysis with respect to the choice of priors, this paper proposes an alternative Bayesian model for binomial random variables with several zero responses. Particular attention is paid to the coherence between the prior distributions of the study model parameters and the meta‐parameter. Thus, our method introduces a simple way to examine the sensitivity of these quantities to the structure dependence selected for study. For illustrative purposes, an example with real data is analysed, using the proposed Bayesian meta‐analysis model for binomial sparse data. Copyright © 2016 John Wiley & Sons, Ltd.     

Bayesian statistical inference of the loglogistic model with interval-censored lifetime data

Interval-censored data arise when a failure time say, T cannot be observed directly but can only be determined to lie in an interval obtained from a series of inspection times. The frequentist approach for analysing interval-censored data has been developed for some time now. It is very common due to unavailability of software in the field of biological, medical and reliability studies to simplify the interval censoring structure of the data into that of a more standard right censoring situation by imputing the midpoints of the censoring intervals. In this research paper, we apply the Bayesian approach by employing Lindley's 1980, and Tierney and Kadane 1986 numerical approximation procedures when the survival data under consideration are interval-censored. The Bayesian approach to interval-censored data has barely been discussed in literature. The essence of this study is to explore and promote the Bayesian methods when the survival data been analysed are is interval-censored. We have considered only a parametric approach by assuming that the survival data follow a loglogistic distribution model. We illustrate the proposed methods with two real data sets. A simulation study is also carried out to compare the performances of the methods.     

Bayesian estimation of the generalized lognormal distribution using objective priors

The generalized lognormal distribution plays an important role in analysing data from different life testing experiments. In this paper, we consider Bayesian analysis of this distribution using various objective priors for the model parameters. Specifically, we derive expressions for the Jeffreys-type priors, the reference priors with different group orderings of the parameters, and the first-order matching priors. We also study the properties of the posterior distributions of the parameters under these improper priors. It is shown that only two of them result in proper posterior distributions. Numerical simulation studies are conducted to compare the performances of the Bayesian estimators under the considered priors and the maximum likelihood estimates. Finally, a real-data application is also provided for illustrative purposes.     

Bayesian analysis for incomplete multi-way contingency tables with nonignorable nonresponse

We propose Bayesian methods with five types of priors to estimate cell probabilities in an incomplete multi-way contingency table under nonignorable nonresponse. In this situation, the maximum likelihood (ML) estimates often fall in the boundary solution, causing the ML estimates to become unstable. To deal with such a multi-way table, we present an EM algorithm which generalizes the previous algorithm used for incomplete one-way tables. Three of the five types of priors were previously introduced while the other two are newly proposed to reflect different response patterns between respondents and nonrespondents. Data analysis and simulation studies show that Bayesian estimates based on the old three priors can be worse than the ML regardless of occurrence of boundary solution, contrary to previous studies. The Bayesian estimates from the two new priors are most preferable when a boundary solution occurs. We provide an illustrating example using data for a study of the relationship between a mother's smoking and her newborn's weight.     

Bayesian emulation of complex multi-output and dynamic computer models

Computer models are widely used in scientific research to study and predict the behaviour of complex systems. The run times of computer-intensive simulators are often such that it is impractical to make the thousands of model runs that are conventionally required for sensitivity analysis, uncertainty analysis or calibration. In response to this problem, highly efficient techniques have recently been developed based on a statistical meta-model (the emulator) that is built to approximate the computer model. The approach, however, is less straightforward for dynamic simulators, designed to represent time-evolving systems. Generalisations of the established methodology to allow for dynamic emulation are here proposed and contrasted. Advantages and difficulties are discussed and illustrated with an application to the Sheffield Dynamic Global Vegetation Model, developed within the UK Centre for Terrestrial Carbon Dynamics.     

Non parametric Bayesian analysis of the two-sample problem with censoring

Testing for differences between two groups is a fundamental problem in statistics, and due to developments in Bayesian non parametrics and semiparametrics there has been renewed interest in approaches to this problem. Here we describe a new approach to developing such tests and introduce a class of such tests that take advantage of developments in Bayesian non parametric computing. This class of tests uses the connection between the Dirichlet process (DP) prior and the Wilcoxon rank sum test but extends this idea to the DP mixture prior. Here tests are developed that have appropriate frequentist sampling procedures for large samples but have the potential to outperform the usual frequentist tests. Extensions to interval and right censoring are considered and an application to a high-dimensional data set obtained from an RNA-Seq investigation demonstrates the practical utility of the method.     

Bayesian interim analysis of lifetime data

In hypothesis testing involving censored lifetime data that are independently distributed according to an accelerated-failure-time model, it is often of interest to predict whether continuation of the experiment will significantly alter the inferences drawn at an interim point. Approaching the problem from a Bayesian viewpoint, we suggest a possible solution based on Laplace approximations to the posterior distribution of the parameters of interest and on Markov-chain Monte Carlo. We apply our results to Weibull data from a carcinogenesis experiment on mice.     

The design of experiments for correlated error structures: Layout and robustness

Observations with correlated error structures are sometimes unavoidable. Appropriate designs and analyses are reviewed for such situations. Serious problems can occur if conventional designs and analyses are used when correlated errors and layout are ignored or when the error structure is not known. Robust designs are discussed which guard against these problems.     

Bayesian analysis of a multiple-recapture model

In the Bayesian analysis of a multiple-recapture census, different diffuse prior distributions can lead to markedly different inferences about the population size N. Through consideration of the Fisher information matrix it is shown that the number of captures in each sample typically provides little information about N. This suggests that if there is no prior information about capture probabilities, then knowledge of just the sample sizes and not the number of recaptures should leave the distribution of Nunchanged. A prior model that has this property is identified and the posterior distribution is examined. In particular, asymptotic estimates of the posterior mean and variance are derived. Differences between Bayesian and classical point and interval estimators are illustrated through examples.     

Bayesian analysis of reliability in multicomponent stress-strength models

This paper provides a Bayesian treatment of the problem of inference about the reliability of a multicomponent stress-strength system which functions if s or more of k identical components simultaneously operate. All stresses and strengths are assumed to be independent, exponentially distributed random variables. Exact and approximate asymptotic posterior distributions for the reliability are derived, and the results are illustrated by a numerical example.     

Bayesian analysis of correlated mixed categorical data by incorporating historical prior information

In this article, we develop statistical models for analysis of correlated mixed categorical (binary and ordinal) response data arising in medical and epidemi-ologic studies. There is evidence in the literature to suggest that models including correlation structure can lead to substantial improvement in precision of estimation or are more appropriate (accurate). We use a very rich class of scale mixture of multivariate normal (SMMVN) iink functions to accommodate heavy tailed distributions. In order to incorporate available historical information, we propose a unified prior elicitation scheme based on SMMVN-link models. Further, simulation-based techniques are developed to assess model adequacy. Finally, a real data example from prostate cancer studies is used to illustrate the proposed methodologies.     

Bayesian analysis of mortality data

Congdon argued that the use of parametric modelling of mortality data is necessary in many practical demographical problems. In this paper, we focus on a form of model introduced by Heligman and Pollard in 1980, and we adopt a Bayesian analysis, using Markov chain Monte Carlo simulation, to produce the posterior summaries required. This opens the way to richer, more flexible inference summaries and avoids the numerical problems that are encountered with classical methods. Particular methodologies to cope with incomplete life-tables and a derivation of joint lifetimes, median times to death and related quantities of interest are also presented.     

Bayesian analysis of spatial data using different variance and neighbourhood structures

In disease mapping, the overall goal is to study the incidence or mortality risk caused by a specific disease in a number of geographical regions. It is common to assume that the response variable follows a Poisson distribution, whose average rate can be explained by a group of covariates and a random effect. For this random effect, it is considered conditional autoregressive (CAR) models, which carry information about the neighbourhood relationship between the regions. The focus of this paper was to explore and compare some CAR models proposed in the literature. An application with epidemiological data was conducted to model the risk of death due to Crohn's Disease and Ulcerative Colitis in the State of São Paulo – Brazil. Finally, a simulation study was done to strengthen the results and assess the performance of the models in the presence of various levels of spatial dependence.     

Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information

In the life test, predicting higher failure times than the largest failure time of the observed is an important issue. Although the Rayleigh distribution is a suitable model for analyzing the lifetime of components that age rapidly over time because its failure rate function is an increasing linear function of time, the inference for a two-parameter Rayleigh distribution based on upper record values has not been addressed from the Bayesian perspective. This paper provides Bayesian analysis methods by proposing a noninformative prior distribution to analyze survival data, using a two-parameter Rayleigh distribution based on record values. In addition, we provide a pivotal quantity and an algorithm based on the pivotal quantity to predict the behavior of future survival records. We show that the proposed method is superior to the frequentist counterpart in terms of the mean-squared error and bias through Monte carlo simulations. For illustrative purposes, survival data on lung cancer patients are analyzed, and it is proved that the proposed model can be a good alternative when prior information is not given.     

Bayesian Analysis of a Queueing System with a Long-Tailed Arrival Process

Internet traffic data is characterized by some unusual statistical properties, in particular, the presence of heavy-tailed variables. A typical model for heavy-tailed distributions is the Pareto distribution although this is not adequate in many cases. In this article, we consider a mixture of two-parameter Pareto distributions as a model for heavy-tailed data and use a Bayesian approach based on the birth-death Markov chain Monte Carlo algorithm to fit this model. We estimate some measures of interest related to the queueing system k-Par/M/1 where k-Par denotes a mixture of k Pareto distributions. Heavy-tailed variables are difficult to model in such queueing systems because of the lack of a simple expression for the Laplace Transform (LT). We use a procedure based on recent LT approximating results for the Pareto/M/1 system. We illustrate our approach with both simulated and real data.     

Bayesian analysis of the amended Davidson model for paired comparison using noninformative and informative priors

In recent years, numerous statisticians have focused their attention on the Bayesian analysis of different paired comparison models. While studying paired comparison techniques, the Davidson model is considered to be one of the famous paired comparison models in the available literature. In this article, we have introduced an amendment in the Davidson model which has been commenced to accommodate the option of not distinguishing the effects of two treatments when they are compared pairwise. Having made this amendment, the Bayesian analysis of the Amended Davidson model is performed using the noninformative (uniform and Jeffreys’) and informative (Dirichlet–gamma–gamma) priors. To study the model and to perform the Bayesian analysis with the help of an example, we have obtained the joint and marginal posterior distributions of the parameters, their posterior estimates, graphical presentations of the marginal densities, preference and predictive probabilities and the posterior probabilities to compare the treatment parameters.     

A gamma-mixture class of distributions with Bayesian application

In this article, a subjective Bayesian approach is followed to derive estimators for the parameters of the normal model by assuming a gamma-mixture class of prior distributions, which includes the gamma and the noncentral gamma as special cases. An innovative approach is proposed to find the analytical expression of the posterior density function when a complicated prior structure is ensued. The simulation studies and a real dataset illustrate the modeling advantages of this proposed prior and support some of the findings.