Inferences on Weibull parameters with conventional type-I censoring

In this article we consider the statistical inferences of the unknown parameters of a Weibull distribution when the data are Type-I censored. It is well known that the maximum likelihood estimators do not always exist, and even when they exist, they do not have explicit expressions. We propose a simple fixed point type algorithm to compute the maximum likelihood estimators, when they exist. We also propose approximate maximum likelihood estimators of the unknown parameters, which have explicit forms. We construct the confidence intervals of the unknown parameters using asymptotic distribution and also by using the bootstrapping technique. Bayes estimates and the corresponding highest posterior density credible intervals of the unknown parameters are also obtained under fairly general priors on the unknown parameters. The Bayes estimates cannot be obtained explicitly. We propose to use the Gibbs sampling technique to compute the Bayes estimates and also to construct the highest posterior density credible intervals. Different methods have been compared by Monte Carlo simulations. One real data set has been analyzed for illustrative purposes.     

Fuzzy Bayesian system reliability assessment based on prior two‐parameter exponential distribution under different loss functions

The fuzzy Bayesian system reliability assessment based on prior two‐parameter exponential distribution under squared error symmetric loss function and precautionary asymmetric loss function is proposed in this paper. In order to apply the Bayesian approach, the fuzzy parameters are assumed as fuzzy random variables with fuzzy prior distributions. Because the goal of the paper is to obtain fuzzy Bayes point estimators of system reliability assessment, prior distributions of location‐scale family has been changed to scale family with change variable. On the other hand, also the computational procedures to evaluate the membership degree of any given Bayes point estimate of system reliability have been provided. In order to achieve this purpose, we transform the original problem into a non‐linear programming problem. This non‐linear programming problem is then divided into four sub‐problems for the purpose of simplifying computation. Finally, the sub‐problems can be solved by using any commercial optimizers, e.g. GAMS or LINGO. Copyright © 2010 John Wiley & Sons, Ltd.     

Generalized exponential distribution: Bayesian estimations

Recently two-parameter generalized exponential distribution has been introduced by the authors. In this paper we consider the Bayes estimators of the unknown parameters under the assumptions of gamma priors on both the shape and scale parameters. The Bayes estimators cannot be obtained in explicit forms. Approximate Bayes estimators are computed using the idea of Lindley. We also propose Gibbs sampling procedure to generate samples from the posterior distributions and in turn computing the Bayes estimators. The approximate Bayes estimators obtained under the assumptions of non-informative priors, are compared with the maximum likelihood estimators using Monte Carlo simulations. One real data set has been analyzed for illustrative purposes.     

两单元并联可修系统的参数估计

为了给出两单元并联可修系统未知参数的Bayes估计,假设单元的失效时间和修理时间服从参数不同的指数分布,选取LINEX损失函数,给出了参数估计的计算公式,并进行了计算机仿真。与平方损失下的Bayes估计进行比较的结果表明,选用LINEX损失下得到的估计有较小的均方误差。     

Exact likelihood inference for two exponential populations under joint Type-II censoring

Comparative lifetime experiments are of paramount importance when the object of a study is to ascertain the relative merits of two competing products in regard to the duration of their service life. In this paper, we discuss exact inference for two exponential populations when Type-II censoring is implemented on the two samples in a combined manner. We obtain the conditional maximum likelihood estimators (MLEs) of the two exponential mean parameters. We then derive the moment generating functions and the exact distributions of these MLEs along with exact confidence intervals and simultaneous confidence regions. Moreover, simultaneous approximate confidence regions based on the asymptotic normality of the MLEs and simultaneous credible confidence regions from a Bayesian viewpoint are also discussed. A comparison of the exact, approximate, bootstrap and Bayesian intervals is also made in terms of coverage probabilities. Finally, an example is presented in order to illustrate all the methods of inference discussed here.     

Suboptimal behavior of Bayes and MDL in classification under misspecification

We show that forms of Bayesian and MDL inference that are often applied to classification problems can be inconsistent. This means that there exists a learning problem such that for all amounts of data the generalization errors of the MDL classifier and the Bayes classifier relative to the Bayesian posterior both remain bounded away from the smallest achievable generalization error. From a Bayesian point of view, the result can be reinterpreted as saying that Bayesian inference can be inconsistent under misspecification, even for countably infinite models. We extensively discuss the result from both a Bayesian and an MDL perspective.     

Comparison of estimates using record statistics from Weibull model: Bayesian and non-Bayesian approaches

This paper develops a Bayesian analysis in the context of record statistics values from the two-parameter Weibull distribution. The ML and the Bayes estimates based on record values are derived for the two unknown parameters and some survival time parameters e.g. reliability and hazard functions. The Bayes estimates are obtained based on a conjugate prior for the scale parameter and a discrete prior for the shape parameter of this model. This is done with respect to both symmetric loss function (squared error loss), and asymmetric loss function (linear-exponential (LINEX)) loss function. The maximum likelihood and the different Bayes estimates are compared via a Monte Carlo simulation study. A practical example consisting of real record values using the data from an accelerated test on insulating fluid reported by Nelson was used for illustration and comparison. Finally, Bayesian predictive density function, which is necessary to obtain bounds for predictive interval of future record is derived and discussed using a numerical example. The results may be of interest in a situation where only record values are stored.     

Reference analysis for Birnbaum-Saunders distribution

In this paper we consider the Bayesian estimators for the unknown parameters of the Birnbaum-Saunders distribution under the reference prior. The Bayesian estimators cannot be obtained in closed forms. An approximate Bayesian approach is proposed using the idea of Lindley and Gibbs sampling procedure is also used to obtain the Bayesian estimators. These results are compared using Monte Carlo simulations with the maximum likelihood method and another approximate Bayesian approach Laplace’s approximation. Two real data sets are analyzed for illustrative purposes.     

Variational Bayesian Blind Image Deconvolution: A review

In this paper we provide a review of the recent literature on Bayesian Blind Image Deconvolution (BID) methods. We believe that two events have marked the recent history of BID: the predominance of Variational Bayes (VB) inference as a tool to solve BID problems and the increasing interest of the computer vision community in solving BID problems. VB inference in combination with recent image models like the ones based on Super Gaussian (SG) and Scale Mixture of Gaussians (SMG) representations have led to the use of very general and powerful tools to provide clear images from blurry observations. In the provided review emphasis is paid on VB inference and the use of SG and SMG models with coverage of recent advances in sampling methods. We also provide examples of current state of the art BID methods and discuss problems that very likely will mark the near future of BID.     

Codebook-Based Bayesian Speech Enhancement for Nonstationary Environments

In this paper, we propose a Bayesian minimum mean squared error approach for the joint estimation of the short-term predictor parameters of speech and noise, from the noisy observation. We use trained codebooks of speech and noise linear predictive coefficients to model the a priori information required by the Bayesian scheme. In contrast to current Bayesian estimation approaches that consider the excitation variances as part of the a priori information, in the proposed method they are computed online for each short-time segment, based on the observation at hand. Consequently, the method performs well in nonstationary noise conditions. The resulting estimates of the speech and noise spectra can be used in a Wiener filter or any state-of-the-art speech enhancement system. We develop both memoryless (using information from the current frame alone) and memory-based (using information from the current and previous frames) estimators. Estimation of functions of the short-term predictor parameters is also addressed, in particular one that leads to the minimum mean squared error estimate of the clean speech signal. Experiments indicate that the scheme proposed in this paper performs significantly better than competing methods     

Optimal mean-square-error calibration of classifier error estimators under Bayesian models

A recently proposed Bayesian modeling framework for classification facilitates both the analysis and optimization of error estimation performance. The Bayesian error estimator is then defined to have optimal mean-square error performance, but in many situations closed-form representations are unavailable and approximations may not be feasible. To address this, we present a method to optimally calibrate arbitrary error estimators for minimum mean-square error performance within a supposed Bayesian framework. Assuming a fixed sample size, classification rule and error estimation rule, as well as a fixed Bayesian model, the calibration is done by first computing a calibration function that maps error estimates to their optimally calibrated values off-line. Once found, this calibration function may be easily applied to error estimates on the fly whenever the assumptions apply. We demonstrate that calibrated error estimators offer significant improvement in performance relative to classical error estimators under Bayesian models with both linear and non-linear classification rules.     

Bayesian computation for geometric process in maintenance problems

Geometric process modeling is a useful tool to study repairable deteriorating systems in maintenance problems. This model has been used in a variety of situations such as the determination of the optimal replacement policy and the optimal inspection-repair-replacement policy for standby systems, and the analysis of data with trend. In this article, Bayesian inference for the geometric process with several popular life distributions, for instance, the exponential distribution and the lognormal distribution, are studied. The Gibbs sampler and the Metropolis algorithm are used to compute the Bayes estimators of the parameters in the geometric process. Simulation results are presented to illustrate the use of our procedures.     

Improving biosignal processing through modeling uncertainty: Bayes vs. non-bayes in sleep staging

In this paper we report about an investigation of Bayesian inference applied to neural networks multilayer perceptrons (MLP), in particular in the task of automatic sleep staging based on electroencephalogram (EEG) and electrooculogram (EOG) signals. The main focus was on evaluating the use of so-called "doubt-levels" and "confidence intervals" ("error bars") in improving the results by rejecting uncertain cases and patterns not well represented by the training set. Bayesian inference is used to arrive at distributions of network weights based on training data. We compare the results of the full-blown Bayesian method with results obtained from a k-nearest neighbor classifier. The results show that the Bayesian technique significantly outperforms the k-nearest-neighbor classifier. At the same time, we show that Bayesian inference, for which we have developed an extension for the calculation of error bars in the latent space of hidden units, can indeed be used for improving results by rejecting cases below a doubt-level threshold of probability, as well as for the rejection of artifacts. The performance of the Bayesian solution, however, is not significantly better than alternative techniques such as doubt levels applied to a maximum posterior approach, or the use of density estimation for outlier rejection. We conclude that Bayesian inference is a valid and valuable technique for model estimation but in the given application does not lead to improved results over simpler techniques.     

Finite-sample investigation of likelihood and Bayes inference for the symmetric von Mises-Fisher distribution

We consider likelihood and Bayes analyses for the symmetric matrix von Mises-Fisher (matrix Fisher) distribution, which is a common model for three-dimensional orientations (represented by 3×3 orthogonal matrices with a positive determinant). One important characteristic of this model is a 3×3 rotation matrix representing the modal rotation, and an important challenge is to establish accurate confidence regions for it with an interpretable geometry for practical implementation. While we provide some extensions of one-sample likelihood theory (e.g., Euler angle parametrizations of modal rotation), our main contribution is the development of MCMC-based Bayes inference through non-informative priors. In one-sample problems, the Bayes methods allow the construction of inference regions with transparent geometry and accurate frequentist coverages in a way that standard likelihood inference cannot. Simulation is used to evaluate the performance of Bayes and likelihood inference regions. Furthermore, we illustrate how the Bayes framework extends inference from one-sample problems to more complicated one-way random effects models based on the symmetric matrix Fisher model in a computationally straightforward manner. The inference methods are then applied to a human kinematics example for illustration.     

Efficient Bayesian spatial prediction with mobile sensor networks using Gaussian Markov random fields

In this paper, we consider the problem of predicting a large scale spatial field using successive noisy measurements obtained by mobile sensing agents. The physical spatial field of interest is discretized and modeled by a Gaussian Markov random field (GMRF) with uncertain hyperparameters. From a Bayesian perspective, we design a sequential prediction algorithm to exactly compute the predictive inference of the random field. The main advantages of the proposed algorithm are: (1) the computational efficiency due to the sparse structure of the precision matrix, and (2) the scalability as the number of measurements increases. Thus, the prediction algorithm correctly takes into account the uncertainty in hyperparameters in a Bayesian way and is also scalable to be usable for mobile sensor networks with limited resources. We also present a distributed version of the prediction algorithm for a special case. An adaptive sampling strategy is presented for mobile sensing agents to find the most informative locations in taking future measurements in order to minimize the prediction error and the uncertainty in hyperparameters simultaneously. The effectiveness of the proposed algorithms is illustrated by numerical experiments.     

Model selection by bootstrap penalization for classification

We consider the binary classification problem. Given an i.i.d. sample drawn from the distribution of an χ×{0,1}?valued random pair, we propose to estimate the so-called Bayes classifier by minimizing the sum of the empirical classification error and a penalty term based on Efron’s or i.i.d. weighted bootstrap samples of the data. We obtain exponential inequalities for such bootstrap type penalties, which allow us to derive non-asymptotic properties for the corresponding estimators. In particular, we prove that these estimators achieve the global minimax risk over sets of functions built from Vapnik-Chervonenkis classes. The obtained results generalize Koltchinskii (2001) and Bartlett et al.’s (2002) ones for Rademacher penalties that can thus be seen as special examples of bootstrap type penalties. To illustrate this, we carry out an experimental study in which we compare the different methods for an intervals model selection problem.     

Improving the performance of kurtosis estimator

In this communication, sample measures of kurtosis adapted by various software packages are compared for data from normal and non-normal populations. Further, two improved estimators of population kurtosis are proposed and their performance is compared with the currently used measures. The suggested estimators have considerably lower mean squared error (MSE) for various sampling designs in our simulation study. Two empirical examples are given to illustrate the usefulness of suggested estimators in practice.     

An improved mean estimator for judgment post-stratification

We prove that the standard nonparametric mean estimator for judgment post-stratification is inadmissible under squared error loss within a certain class of linear estimators. We derive alternate estimators that are admissible in this class, and we show that one of them is always better than the standard estimator. The reduction in mean squared error from using this alternate estimator can be as large as 10% for small set sizes and small sample sizes.     

Inference for a simple step-stress model with competing risks for failure from the exponential distribution under time constraint

In reliability analysis, accelerated life-testing allows for gradual increment of stress levels on test units during an experiment. In a special class of accelerated life tests known as step-stress tests, the stress levels increase discretely at pre-fixed time points, and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. In this article, we consider the simple step-stress model under time constraint when the lifetime distributions of the different risk factors are independently exponentially distributed. Under this setup, we derive the maximum likelihood estimators (MLEs) of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. Since it is found that the MLEs do not exist when there is no failure by any particular risk factor within the specified time frame, the exact sampling distributions of the MLEs are derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions, the parametric bootstrap method, and the Bayesian posterior distribution, we discuss the construction of confidence intervals and credible intervals for the parameters. Their performance is assessed through Monte Carlo simulations and finally, we illustrate the methods of inference discussed here with an example.