Spatio-temporal modeling of infectious disease dynamics

A stochastic model, which is well suited to capture space–time dependence of an infectious disease, was employed in this study to describe the underlying spatial and temporal pattern of measles in Barisal Division, Bangladesh. The model has two components: an endemic component and an epidemic component; weights are used in the epidemic component for better accounting of the disease spread into different geographical regions. We illustrate our findings using a data set of monthly measles counts in the six districts of Barisal, from January 2000 to August 2009, collected from the Expanded Program on Immunization, Bangladesh. The negative binomial model with both the seasonal and autoregressive components was found to be suitable for capturing space–time dependence of measles in Barisal. Analyses were done using general optimization routines, which provided the maximum likelihood estimates with the corresponding standard errors.     

Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates

Abstract.  Previously, small area estimation under a nested error linear regression model was studied with area level covariates subject to measurement error. However, the information on observed covariates was not used in finding the Bayes predictor of a small area mean. In this paper, we first derive the fully efficient Bayes predictor by utilizing all the available data. We then estimate the regression and variance component parameters in the model to get an empirical Bayes (EB) predictor and show that the EB predictor is asymptotically optimal. In addition, we employ the jackknife method to obtain an estimator of mean squared prediction error (MSPE) of the EB predictor. Finally, we report the results of a simulation study on the performance of our EB predictor and associated jackknife MSPE estimators. Our results show that the proposed EB predictor can lead to significant gain in efficiency over the previously proposed EB predictor.     

Maximum likelihood and bayes prediction of current system lifetime

The Power Law Process is often used to analyse failure data of repairable systems undergoing development testing where the system failure intensity decreases as a result of repeated application of corrective actions. At the end of the development program, the system failure intensity is assumed to remain constant and the current system lifetime is assumed to be exponentially distributed. In this paper, prediction limits on the current system lifetime have been derived both in the maximum likelihood and Bayesian context. Exact values and a closed form approximation of percentage points of the pivotal quantity used in the classical approach are given in the case of failure truncated testing. For both failure and time truncated testing, the Bayesian approach is developed both when no prior knowledge is available and when information on the reliability growth rate can be given. A numerical example is also given.     

Classical and Bayesian estimation of P(Y < X) for Kumaraswamy's distribution

The aim of this paper is to study the estimation of the reliability R=P(Y<X) when X and Y are independent random variables that follow Kumaraswamy's distribution with different parameters. If we assume that the first shape parameter is common and known, the maximum-likelihood estimator (MLE), the exact confidence interval and the uniformly minimum variance unbiased estimator of R are obtained. Moreover, when the first parameter is common but unknown, MLEs, Bayes estimators, asymptotic distributions and confidence intervals for R are derived. Furthermore, Bayes and empirical Bayes estimators for R are obtained when the first parameter is common and known. Finally, when all four parameters are different and unknown, the MLE of R is obtained. Monte Carlo simulations are performed to compare the different proposed methods and conclusions on the findings are given.     

Bayes prediction of number of failures in weibull samples

This work is concerned with the Bayesian prediction problem of the number of components which will fail in a future time interval, when the failure times are Weibull distributed. Both the 1-sample and the 2-sample prediction problems are dealed with, and some choices of the prior densities on the distribution parameters are discussed which are relatively easy to work with and allow different degrees of knowledge on the failure mechanism to be incorporated in the predictive procedure. Useful relations between the predictive distribution on the number of future failures and the predictive distribution on the future failure times are derived. Numerical examples are also given.     

Bayes prediction of Poisson regression superpopulation mean with a non gamma prior

We consider Khamis' (1960) Laguerre expansion with gamma weight function as a class of “near-gamma” priors (K-prior) to obtain the Bayes predictor of a finite population mean under the Poisson regression superpopulation model using Zellner's balanced loss function (BLF). Kullback–Leibler (K-L) distance between gamma and some K-priors is tabulated to examine the quantitative prior robustness. Some numerical investigations are also conducted to illustrate the effects of a change in skewness and/or kurtosis on the Bayes predictor and the corresponding minimal Bayes predictive expected loss (MBPEL). Loss robustness with respect to the class of BLFs is also examined in terms of relative savings loss (RSL).     

Bayes and robust Bayes predictions in a subfamily of scale parameters under a precautionary loss function

ABSTRACT

This paper deals with Bayes, robust Bayes, and minimax predictions in a subfamily of scale parameters under an asymmetric precautionary loss function. In Bayesian statistical inference, the goal is to obtain optimal rules under a specified loss function and an explicit prior distribution over the parameter space. However, in practice, we are not able to specify the prior totally or when a problem must be solved by two statisticians, they may agree on the choice of the prior but not the values of the hyperparameters. A common approach to the prior uncertainty in Bayesian analysis is to choose a class of prior distributions and compute some functional quantity. This is known as Robust Bayesian analysis which provides a way to consider the prior knowledge in terms of a class of priors Γ for global prevention against bad choices of hyperparameters. Under a scale invariant precautionary loss function, we deal with robust Bayes predictions of Y based on X. We carried out a simulation study and a real data analysis to illustrate the practical utility of the prediction procedure.     

Optimal network design for Bayesian spatial prediction of multivariate non-Gaussian environmental data

This paper deals with the problem of increasing air pollution monitoring stations in Tehran city for efficient spatial prediction. As the data are multivariate and skewed, we introduce two multivariate skew models through developing the univariate skew Gaussian random field proposed by Zareifard and Jafari Khaledi [21 H. Zareifard and M. Jafari Khaledi, Non-Gaussian modeling of spatial data using scale mixing of a unified skew Gaussian process, J. Multivariate Anal. 114 (2013), pp. 16–28. doi: 10.1016/j.jmva.2012.07.003[Crossref], [Web of Science ®] , [Google Scholar]]. These models provide extensions of the linear model of coregionalization for non-Gaussian data. In the Bayesian framework, the optimal network design is found based on the maximum entropy criterion. A Markov chain Monte Carlo algorithm is developed to implement posterior inference. Finally, the applicability of two proposed models is demonstrated by analyzing an air pollution data set.     

General linear estimators under the prediction error sum of squares criterion in a linear regression model

In this paper, the notion of the general linear estimator and its modified version are introduced using the singular value decomposition theorem in the linear regression model y=X β+e to improve some classical linear estimators. The optimal selections of the biasing parameters involved are theoretically given under the prediction error sum of squares criterion. A numerical example and a simulation study are finally conducted to illustrate the superiority of the proposed estimators.     

Bayes and Empirical Bayes Inference in Changepoint Problems

ABSTRACT

In the case of the random design nonparametric regression, the double smoothing technique is applied to estimate the multivariate regression function. The proposed estimator has desirable properties in both the finite sample and the asymptotic cases. In the finite sample case, it has bounded conditional (and unconditional) bias and variance. On the other hand, in the asymptotic case, it has the same mean square error as the local linear estimator in Fan (Design-Adaptive Nonparametric Regression. Journal of the American Statistical Association 1992, 87, 998–1004; Local Linear Regression Smoothers and Their Minimax Efficiencies. Annals of Statistics 1993, 21, 196–216). Simulation studies demonstrate that the proposed estimator is better than the local linear estimator, because it has a smaller sample mean integrated square error and gives smoother estimates.     

Convergence rates for empirical bayes estimators of parameters in linear regressin models

In this paper, the convergence rates of the EB estimators of the regression coefficients and the error variance in a linear model are obtained. The rates can approximate to O(n¹) arbitrarily. The convergency of the EB estimators of the regression coefiicients and the variance components in a variance component model is also investigated. The investigation makes use of the results concerning the convergence rates of the EB estimators of the parameters in multi-parameter exponential families.     

Predictive Inference for Big,Spatial, Non‐Gaussian Data: MODIS Cloud Data and its Change‐of‐Support

Remote sensing of the earth with satellites yields datasets that can be massive in size, nonstationary in space, and non‐Gaussian in distribution. To overcome computational challenges, we use the reduced‐rank spatial random effects (SRE) model in a statistical analysis of cloud‐mask data from NASA's Moderate Resolution Imaging Spectroradiometer (MODIS) instrument on board NASA's Terra satellite. Parameterisations of cloud processes are the biggest source of uncertainty and sensitivity in different climate models’ future projections of Earth's climate. An accurate quantification of the spatial distribution of clouds, as well as a rigorously estimated pixel‐scale clear‐sky‐probability process, is needed to establish reliable estimates of cloud‐distributional changes and trends caused by climate change. Here we give a hierarchical spatial‐statistical modelling approach for a very large spatial dataset of 2.75 million pixels, corresponding to a granule of MODIS cloud‐mask data, and we use spatial change‐of‐Support relationships to estimate cloud fraction at coarser resolutions. Our model is non‐Gaussian; it postulates a hidden process for the clear‐sky probability that makes use of the SRE model, EM‐estimation, and optimal (empirical Bayes) spatial prediction of the clear‐sky‐probability process. Measures of prediction uncertainty are also given.     

Properties and Applications of the Generalized Likelihood as a Summary Function for Prediction Problems

The generalized likelihood plays an important role in parametric inference for prediction and empirical Bayesian models. This paper emphasizes the utility of the generalized likelihood as a summarization procedure in general prediction models. Properties of the generalized likelihood when used in this setting, and examples of its use as a data analytic tool are given in a series of numerical examples.     

Bayesian accrual modeling and prediction in multicenter clinical trials with varying center activation times

Investigators who manage multicenter clinical trials need to pay careful attention to patterns of subject accrual, and the prediction of activation time for pending centers is potentially crucial for subject accrual prediction. We propose a Bayesian hierarchical model to predict subject accrual for multicenter clinical trials in which center activation times vary. We define center activation time as the time at which a center can begin enrolling patients in the trial. The difference in activation times between centers is assumed to follow an exponential distribution, and the model of subject accrual integrates prior information for the study with actual enrollment progress. We apply our proposed Bayesian multicenter accrual model to two multicenter clinical studies. The first is the PAIN‐CONTRoLS study, a multicenter clinical trial with a goal of activating 40 centers and enrolling 400 patients within 104 weeks. The second is the HOBIT trial, a multicenter clinical trial with a goal of activating 14 centers and enrolling 200 subjects within 36 months. In summary, the Bayesian multicenter accrual model provides a prediction of subject accrual while accounting for both center‐ and individual patient‐level variation.     

Bayesian inference and prediction of the Pareto distribution based on ordered ranked set sampling

In this paper, order statistics from independent and non identically distributed random variables is used to obtain ordered ranked set sampling (ORSS). Bayesian inference of unknown parameters under a squared error loss function of the Pareto distribution is determined. We compute the minimum posterior expected loss (the posterior risk) of the derived estimates and compare them with those based on the corresponding simple random sample (SRS) to assess the efficiency of the obtained estimates. Two-sample Bayesian prediction for future observations is introduced by using SRS and ORSS for one- and m-cycle. A simulation study and real data are applied to show the proposed results.     

Theoretical evaluation of prediction error in linear regression with a bivariate response variable containing missing data

Methods for linear regression with multivariate response variables are well described in statistical literature. In this study we conduct a theoretical evaluation of the expected squared prediction error in bivariate linear regression where one of the response variables contains missing data. We make the assumption of known covariance structure for the error terms. On this basis, we evaluate three well-known estimators: standard ordinary least squares, generalized least squares, and a James–Stein inspired estimator. Theoretical risk functions are worked out for all three estimators to evaluate under which circumstances it is advantageous to take the error covariance structure into account.     

Testing for structural breaks in the presence of data perturbations: impacts and wavelet-based improvements

This paper investigates how classical measurement error and additive outliers (AO) influence tests for structural change based on F-statistics. We derive theoretically the impact of general additive disturbances in the regressors on the asymptotic distribution of these tests for structural change. The small sample properties in the case of classical measurement error and AO are investigated via Monte Carlo simulations, revealing that sizes are biased upwards and that powers are reduced. Two-wavelet-based denoising methods are used to reduce these distortions. We show that these two methods can significantly improve the performance of structural break tests.     

Shrinkage and modification techniques in estimation of variance and the related problems: A review

One of the surprising decision-theoretic results Charles Stein discovered is the inadmissibility of the uniformly minimum variance unbiased estirnator(UMVUE) of the variance of a normal distribution with an unknown mean. Some methods for deriving estimators better than the UMVUE were given by Stein. Brown, Brewster and Zidek. Recently Kubokawa established a novel approach, called the IERD method, by use of which one gets a unified class of improved estimators including their previous procedures. This paper gives a review for a series of these decision-theoretical developments as well as surveys the study of the variance-estimation problem from various aspects. Related to this issue, the paper enumerates several topics with the situations where the usual plain estimators are required to be shrunken or modified, and gives reasonable procedures improving the usual ones through the IERD method.     

Empirical Bayes estimates of finite mixture of negative binomial regression models and its application to highway safety

The empirical Bayes (EB) method is commonly used by transportation safety analysts for conducting different types of safety analyses, such as before–after studies and hotspot analyses. To date, most implementations of the EB method have been applied using a negative binomial (NB) model, as it can easily accommodate the overdispersion commonly observed in crash data. Recent studies have shown that a generalized finite mixture of NB models with K mixture components (GFMNB-K) can also be used to model crash data subjected to overdispersion and generally offers better statistical performance than the traditional NB model. So far, nobody has developed how the EB method could be used with finite mixtures of NB models. The main objective of this study is therefore to use a GFMNB-K model in the calculation of EB estimates. Specifically, GFMNB-K models with varying weight parameters are developed to analyze crash data from Indiana and Texas. The main finding shows that the rankings produced by the NB and GFMNB-2 models for hotspot identification are often quite different, and this was especially noticeable with the Texas dataset. Finally, a simulation study designed to examine which model formulation can better identify the hotspot is recommended as our future research.     

Reml and best linear unbiased prediction in state space models

This article takes a hierarchical model approach to the estimation of state space models with diffuse initial conditions. An initial state is said to be diffuse when it cannot be assigned a proper prior distribution. In state space models this occurs either when fixed effects are present or when modelling nonstationarity in the state transition equation. Whereas much of the literature views diffuse states as an initialization problem, we follow the approach of Sallas and Harville (1981,1988) and incorporate diffuse initial conditions via noninformative prior distributions into hierarchical linear models. We apply existing results to derive the restricted loglike-lihood and appropriate modifications to the standard Kalman filter and smoother. Our approach results in a better understanding of De Jong's (1991) contributions. This article also shows how to adjust the standard Kalman filter, the fixed inter- val smoother and the state space model forecasting recursions, together with their mean square errors, for he presence of diffuse components. Using a hierarchical model approach it is shown that the estimates obtained are Best Linear Unbiased Predictors (BLUP).