Bayesian estimation of the intraclass correlation coefficients in the mixed linear model

The method of determining Bayesian estimators for the special ratios of variance components called the intraclass correlation coefficients is presented. The exact posterior distribution for these ratios of variance components is obtained. The approximate posterior mean of this distribution is also derived. All computations are non-iterative and avoid numerical integration.     

A Markov modulated Poisson model for software reliability

In this paper, we consider a latent Markov process governing the intensity rate of a Poisson process model for software failures. The latent process enables us to infer performance of the debugging operations over time and allows us to deal with the imperfect debugging scenario. We develop the Bayesian inference for the model and also introduce a method to infer the unknown dimension of the Markov process. We illustrate the implementation of our model and the Bayesian approach by using actual software failure data.     

Bayesian semiparametric customer base segmentation of mobile phone users based on longitudinal traffic data

Customer segmentation is one of the most important purposes of customer base analysis for telecommunication companies. Because companies accumulate very large amounts of data on customer behavior, segmentation is typically achieved by profiling and clustering traffic behavior jointly with demographic data and contracts characteristics. Unfortunately, most algorithms and models used for segmentation do not take into account the longitudinal characteristics of data. In particular, in telecommunication traffic analysis, the importance of decreasing patterns of traffic in customers' lives is well known, and it is relevant to aggregate all clients with such a pattern, while other unknown clusters may be of interest for the marketing manager. Our approach to address this problem is based on specifying the distribution of functions as a mixture of a parametric hierarchical model describing the decreasing pattern segment and a nonparametric contamination that allows unanticipated curve shapes in subjects' traffic. The parametric component is chosen based on prior knowledge, while the contamination is characterized as a functional Dirichlet process. Copyright © 2014 John Wiley & Sons, Ltd.     

A semiparametric Bayesian approach to the analysis of financial time series with applications to value at risk estimation

GARCH models are commonly used for describing, estimating and predicting the dynamics of financial returns. Here, we relax the usual parametric distributional assumptions of GARCH models and develop a Bayesian semiparametric approach based on modeling the innovations using the class of scale mixtures of Gaussian distributions with a Dirichlet process prior on the mixing distribution. The proposed specification allows for greater flexibility in capturing the usual patterns observed in financial returns. It is also shown how to undertake Bayesian prediction of the Value at Risk (VaR). The performance of the proposed semiparametric method is illustrated using simulated and real data from the Hang Seng Index (HSI) and Bombay Stock Exchange index (BSE30).     

A discrete time model for software reliability with application to a flight control software

This work is motivated by a particular software reliability problem in a unit of flight control software developed by the Indian Space Research Organization (ISRO), in which the testing of the software is carried out in multiple batches, each consisting of several runs. As the errors are found during the runs within a batch, they are noted, but not debugged immediately; they are debugged only at the end of that particular batch of runs. In this work, we introduce a discrete time model suitable for this type of periodic debugging schedule and describe maximum likelihood estimation for the model parameters. This model is used to estimate the reliability of the software. We also develop a method to determine the additional number of error‐free test runs required for the estimated reliability to achieve a specific target with some high probability. We analyze the test data on the flight control software of ISRO. Copyright © 2011 John Wiley & Sons, Ltd.     

A semiparametric additive rates model for clustered recurrent event data

Recurrent event data often arises in biomedical studies, and individuals within a cluster might not be independent. We propose a semiparametric additive rates model for clustered recurrent event data, wherein the covariates are assumed to add to the unspecified baseline rate. For the inference on the model parameters, estimating equation approaches are developed, and both large and finite sample properties of the proposed estimators are established.     

A prior near-ignorance Gaussian process model for nonparametric regression

This paper proposes a prior near-ignorance model for regression based on a set of Gaussian Processes (GP). GPs are natural prior distributions for Bayesian regression. They offer a great modeling flexibility and have found widespread application in many regression problems. However, a GP requires the prior elicitation of its mean function, which represents our prior belief about the shape of the regression function, and of the covariance between any two function values.In the absence of prior information, it may be difficult to fully specify these infinite dimensional parameters. In this work, by modeling the prior mean of the GP as a linear combination of a set of basis functions and assuming as prior for the combination coefficients a set of conjugate distributions obtained as limits of truncate exponential priors, we have been able to model prior ignorance about the mean of the GP. The resulting model satisfies translation invariance, learning and, under some constraints, convergence, which are desirable properties for a prior near-ignorance model. Moreover, it is shown in this paper how this model can be extended to allow for a weaker specification of the GP covariance between function values, by letting each basis function to vary in a set of functions.Application to hypothesis testing has shown how the use of this model induces the capability of automatically detecting when a reliable decision cannot be made based on the available data.     

半参数回归模型独立情形的分离法估计

对半参数回归模型,用L2最佳逼近加矩估计的方法,推出其非参数部分的依L2与强相合联合收敛意义下的估计,及参数部分的强相合与相合渐近正态估计,并设计实行了一个模拟实验.     

隐马尔可夫因子分析模型的半参数贝叶斯分析

因子模型在刻画潜在因素(因子)与观测变量间的影响关系并进而解释多元观测指标(变量)间的相关性方面具有重要作用.在实际应用中,观测数据往往呈现出时序变异多峰,偏态等特性.将经典的因子分析延伸到带有时齐隐马尔可夫模型的动力因子模型,并建立了半参数贝叶斯分析程序.分块GIBBS抽样器用以后验抽样.经验结果展示所建立的统计程序是有效的.     

误差为线性过程时半参数模型的估计

对半参数模型Ynt=β·tni＋g（xni）＋εni,（1≤i≤n）,利用一般权函数并综合最小二乘法,定义了β,g的估计量βn,gn．在误差为线性过程时,获得了βn和gn的r阶矩相合性及gn的渐进正态性．     

Asymptotic properties for the semiparametric regression model with randomly censored data

Suppose that the patients’ survival times.Y, are random variables following the semiparametric regression modelY = Xβ +g(T) + ε, where (X,T) is a radom vector taking values inR×[0,1],βis an unknown parameter,g (*) is an unknown smooth regression function andE is the random error with zero mean and variance σ². It is assumed that (X,T) is independent of E. The estimators andg _n (*) of P andg(*) are defined, respectively, when the observations are randomly censored on the right and the censoring distribution is unknown. Moreover, it is shown that is asymptotically normal andg _n (*) is weak consistence with rateO _p(n-1/3). Project supported by China Postdoctoral Science Foundation and the National Natural Science Foundation of China.     

Posterior mode estimation for the generalized linear model

Posterior mode estimators are proposed, which arise from simply expressed prior opinion about expected outcomes, roughly as follows: a conjugate family of prior distributions is determined by a given variance function. Using a conjugate prior, a posterior mode estimator and its estimated (co-)variances are obtained through conventional maximum likelihood computations, by means of small alterations to the observed outcomes and/or to the modelled variance function. Within the conjugate family, for purposes of inference about the regression vector, a reference prior is proposed for a given choice of linear design of the canonical link. The resulting approximate reference inferences approximate the Bayesian inferences which arise from a minimally informative reference prior. A set of subjective prior upper and lower percentage points for the expected outcomes can be used to determine a conjugate family member. Alternatively, a set of subjective prior means and standard deviations determines a member. The subfamily of priors determinable by percentage points either includes or approximates the proposed reference prior.The research of the first-named author was funded in part by a Natural Sciences and Engineering Research Council grant.The second named author gratefully acknowledges the support of the National Science Foundation, grant #DMS-8901494 and of the Kansas Geological Survey where he visited during the term of the majority of this research.     

Bayesian sequential reliability for Weibull and related distributions

Assume that the probability density function for the lifetime of a newly designed product has the form: [H(t)/Q()] exp{–H(t)/Q()}. The Exponential(), Rayleigh, WeibullW(, ) and Pareto pdf's are special cases.Q() will be assumed to have an inverse Gamma prior. Assume thatm independent products are to be tested with replacement. A Bayesian Sequential Reliability Demonstration Testing plan is used to eigher accept the product and start formal production, or reject the product for reengineering. The test criterion is the intersection of two goals, a minimal goal to begin production and a mature product goal. The exact values of various risks and the distribution of total number of failures are evaluated. Based on a result about a Poisson process, the expected stopping time for the exponential failure time is also found. Included in these risks and expected stopping times are frequentist versions, thereof, so that the results also provide frequentist answers for a class of interesting stopping rules.This research was supported by NSF grants DMS-8703620 and DMS-8923071, and forms part of the Ph.D. Thesis of the first author, the development of which was supported in part by a David Ross grant at Purdue University. The authors thank the editors and a referee for insightful comments and suggestions.     

Asymptotics of the goodness-of-fit test for a partial linear model with randomly censored data

In this paper, we discuss the problem of testing the hypothesis that the underlying regression is a partial linear model. A test statistic, which is based on the quadratic form of a cusum process of residuals, is proposed. The asymptotic distributions of the test statistic under null hypothesis and the local alternative hypothesis are given. The number simulation shows that the test is available.     

Nonlinear general path models for degradation data with dynamic covariates

Degradation data have been widely used to estimate product reliability. Because of technology advancement, time‐varying usage and environmental variables, which are called dynamic covariates, can be easily recorded nowadays, in addition to the traditional degradation measurements. The use of dynamic covariates is appealing because they have the potential to explain more variability in degradation paths. We propose a class of general path models to incorporate dynamic covariates for modeling of degradation paths. Physically motivated nonlinear functions are used to describe the degradation paths, and random effects are used to describe unit‐to‐unit variability. The covariate effects are modeled by shape‐restricted splines. The estimation of unknown model parameters is challenging because of the involvement of nonlinear relationships, random effects, and shaped‐restricted splines. We develop an efficient procedure for parameter estimations. The performance of the proposed method is evaluated by simulations. An outdoor coating weathering dataset is used to illustrate the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotics for a censored generalized linear model with unknown link function

For censored response variable against projected co-variable, a generalized linear model with an unknown link function can cover almost all existing models under censorship. Its special cases include the accelerated failure time model with censored data. Such a model in the uncensored case is called the single-index model in econometrics. In this paper, we systematically study the asymptotic properties. We derive the central limit theorem and the law of the iterated logarithm for an estimator of the direction parameter. We also obtain the optimal convergence rate of an estimator of the unknown link function in the model.      

A Bayesian approach for analyzing case 2 interval-censored data under the semiparametric proportional odds model

Analyzing interval-censored data is difficult due to its complex data structure containing left-, interval-, and right-censored observations. An easy-to-implement Bayesian approach is proposed under the proportional odds (PO) model for analyzing such data. The nondecreasing baseline log odds function is modeled with a linear combination of monotone splines. Two efficient Gibbs samplers are developed based on two different data augmentations using the relationship between the PO model and the logistic distribution. In the first data augmentation, the logistic distribution is achieved by the scaled normal mixture with the scale parameter related to the Kolmogorov-Smirnove distribution. In the second data augmentation, the logistic distribution is approximated by a Student’s t distribution up to a scale constant. The proposed methods are evaluated by simulation studies and illustrated with an application of an HIV data set.     

Bayesian multiperiod forecasts for ARX models

Bayestian muliperiod forecasts for AR models with random independent exogenous variables under normal-gamma and normal-inverted Wishart prior assumptions are investigated. By suitably arranging the integration order of the model's parameters, at-density mixture approximation is analytically derived to provide an estimator of the posterior predictive density for any future observation. In particular, a suitablet-density is proposed by a convenient closed form. The precision of the discussed methods is examined by using some simulated data and one set of real data up to lead-six-ahead forecasts. It is found that the numerical results of the discussed methods are rather close. In particular, when sample sizes are sufficiently large, it is encouraging to apply a convenientt-density in practical usage. In fact, thist-density estimator asymptotically converges to the true density.This research was supported by the National Science Council, Republic of China under contract #NSC82-0208-M-008-086.     

A mixed integer linear programming model for reliability optimisation in the component deployment problem

Component deployment is a combinatorial optimisation problem in software engineering that aims at finding the best allocation of software components to hardware resources in order to optimise quality attributes, such as reliability. The problem is often constrained because of the limited hardware resources, and the communication network, which may connect only certain resources. Owing to the non-linear nature of the reliability function, current optimisation methods have focused mainly on heuristic or metaheuristic algorithms. These are approximate methods, which find near-optimal solutions in a reasonable amount of time. In this paper, we present a mixed integer linear programming (MILP) formulation of the component deployment problem. We design a set of experiments where we compare the MILP solver to methods previously used to solve this problem. Results show that the MILP solver is efficient in finding feasible solutions even where other methods fail, or prove infeasibility where feasible solutions do not exist.     

Bayesian networks with a logistic regression model for the conditional probabilities

Logistic regression techniques can be used to restrict the conditional probabilities of a Bayesian network for discrete variables. More specifically, each variable of the network can be modeled through a logistic regression model, in which the parents of the variable define the covariates. When all main effects and interactions between the parent variables are incorporated as covariates, the conditional probabilities are estimated without restrictions, as in a traditional Bayesian network. By incorporating interaction terms up to a specific order only, the number of parameters can be drastically reduced. Furthermore, ordered logistic regression can be used when the categories of a variable are ordered, resulting in even more parsimonious models. Parameters are estimated by a modified junction tree algorithm. The approach is illustrated with the Alarm network.