Inference for Weibull distribution under generalized order statistics

Based on generalized order statistics from Weibull distribution the approach of Bayesian and non-Bayesian estimation are discussed. We present a simple and efficient simulational algorithm for generating a generalized order statistics sample from any continuous distribution. Specializations to Bayesian and non-Bayesian estimators, some lifetime parameters and confidence intervals of progressive II censoring and record values are obtained and compared with the existing results. Two examples are given to illustrate the proposed estimators and the simulation algorithm.     

A conjugate bayesian approach for calculating process capability indices

Process capability indices measure the ability of a production process to produce items within specification limits. The calculation of process capability indices has been focusing on using traditional frequency approach, which requires a large sample size for an accurate estimation. In order to eliminate this defect of traditional frequency approach on multi-batch and low volume production, Bayesian approach was used. The conjugate Bayesian approach is chosen to estimate the process distribution parameters. The algorithm with these conjugate Bayes estimators is proposed for measuring the process capability for multi-batch and low volume production. A case study is presented to demonstrate how the approach can be applied to actual data collected in practice.     

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.     

Averaging, maximum penalized likelihood and Bayesian estimation forimproving Gaussian mixture probability density estimates

We apply the idea of averaging ensembles of estimators to probability density estimation. In particular, we use Gaussian mixture models which are important components in many neural-network applications. We investigate the performance of averaging using three data sets. For comparison, we employ two traditional regularization approaches, i.e., a maximum penalized likelihood approach and a Bayesian approach. In the maximum penalized likelihood approach we use penalty functions derived from conjugate Bayesian priors such that an expectation maximization (EM) algorithm can be used for training. In all experiments, the maximum penalized likelihood approach and averaging improved performance considerably if compared to a maximum likelihood approach. In two of the experiments, the maximum penalized likelihood approach outperformed averaging. In one experiment averaging was clearly superior. Our conclusion is that maximum penalized likelihood gives good results if the penalty term in the cost function is appropriate for the particular problem. If this is not the case, averaging is superior since it shows greater robustness by not relying on any particular prior assumption. The Bayesian approach worked very well on a low-dimensional toy problem but failed to give good performance in higher dimensional problems.     

Hierarchical Bayes multivariate estimation of poverty rates based on increasing thresholds for small domains

A model-based small area method for calculating estimates of poverty rates based on different thresholds for subsets of the Italian population is proposed. The subsets are obtained by cross-classifying by household type and administrative region. The suggested estimators satisfy the following coherence properties: (i) within a given area, rates associated with increasing thresholds are monotonically increasing; (ii) interval estimators have lower and upper bounds within the interval (0, 1); (iii) when a large domain-specific sample is available the small area estimate is close to the one obtained using standard design-based methods; (iv) estimates of poverty rates should also be produced for domains for which there is no sample or when no poor households are included in the sample. A hierarchical Bayesian approach to estimation is adopted. Posterior distributions are approximated by means of MCMC computation methods. Empirical analysis is based on data from the 2005 wave of the EU-SILC survey.     

Parameter estimation in a model for misclassified Markov data — a Bayesian approach

We estimate parameters in the context of a discrete-time hidden Markov model with two latent states and two observed states through a Bayesian approach. We provide a Gibbs sampling algorithm for longitudinal data that ensures parameter identifiability. We examine two approaches to start the algorithm for estimation. The first approach generates the initial latent data from transition probability estimates under the false assumption of perfect classification. The second approach requires an initial guess of the classification probabilities and obtains bias-adjusted approximated estimators of the latent transition probabilities based on the observed data. These probabilities are then used to generate the initial latent data set based on the observed data set. Both approaches are illustrated on medical data and the performance of estimates is examined through simulation studies. The approach using bias-adjusted estimators is the best choice of the two options, since it generates a plausible initial latent data set. Our situation is particularly applicable to diagnostic testing, where specifying the range of plausible classification rates may be more feasible than specifying initial values for transition probabilities.     

Constrained Bayesian state estimation – A comparative study and a new particle filter based approach

This paper investigates constrained Bayesian state estimation problems by using a Particle Filter (PF) approach. Constrained systems with nonlinear model and non-Gaussian uncertainty are commonly encountered in practice. However, most of the existing Bayesian methods are unable to take constraints into account and require some simplifications. In this paper, a novel constrained PF algorithm based on acceptance/rejection and optimization strategies is proposed. The proposed method retains the ability of PF in nonlinear and non-Gaussian state estimation, while take advantage of optimization techniques in constraints handling. The performance of the proposed method is compared with other accepted Bayesian estimators. Extensive simulation results from three examples show the efficacy of the proposed method in constraints handling and its robustness against poor prior information.     

Random effects in promotion time cure rate models

In this paper, a survival model with long-term survivors and random effects, based on the promotion time cure rate model formulation for models with a surviving fraction is investigated. We present Bayesian and classical estimation approaches. The Bayesian approach is implemented using a Markov chain Monte Carlo (MCMC) based on the Metropolis-Hastings algorithms. For the second one, we use restricted maximum likelihood (REML) estimators. A simulation study is performed to evaluate the accuracy of the applied techniques for the estimates and their standard deviations. An example on an oropharynx cancer study is used to illustrate the model and the estimation approaches considered in the study.     

Efficient and accurate approximate Bayesian inference with an application to insurance data

Efficient and accurate Bayesian Markov chain Monte Carlo methodology is proposed for the estimation of event rates under an overdispersed Poisson distribution. An approximate Gibbs sampling method and an exact independence-type Metropolis-Hastings algorithm are derived, based on a log-normal/gamma mixture density that closely approximates the conditional distribution of the Poisson parameters. This involves a moment matching process, with the exact conditional moments obtained employing an entropy distance minimisation (Kullback-Liebler divergence) criterion. A simulation study is conducted and demonstrates good Bayes risk properties and robust performance for the proposed estimators, as compared with other estimating approaches under various loss functions. Actuarial data on insurance claims are used to illustrate the methodology. The approximate analysis displays superior Markov chain Monte Carlo mixing efficiency, whilst providing almost identical inferences to those obtained with exact methods.     

Incorporating historical control information into quantal bioassay with Bayesian approach

A Bayesian approach with an iterative reweighted least squares is used to incorporate historical control information into quantal bioassays to estimate the dose-response relationship, where the logit of the historical control responses are assumed to have a normal distribution. The parameters from this normal distribution are estimated from both empirical and full Bayesian approaches with a marginal likelihood function being approximated by Laplace’s Method. A comparison is made using real data between estimates that include the historical control information and those that do not. It was found that the inclusion of the historical control information improves the efficiency of the estimators. In addition, this logit-normal formulation is compared with the traditional beta-binomial for its improvement in parameter estimates. Consequently the estimated dose-response relationship is used to formulate the point estimator and confidence bands for ED(100p) for various values of risk rate p and the potency for any dose level.     

Weighted average least squares estimation with nonspherical disturbances and an application to the Hong Kong housing market

The recently proposed ‘weighted average least squares’ (WALS) estimator is a Bayesian combination of frequentist estimators. It has been shown that the WALS estimator possesses major advantages over standard Bayesian model averaging (BMA) estimators: the WALS estimator has bounded risk, allows a coherent treatment of ignorance and its computational effort is negligible. However, the sampling properties of the WALS estimator as compared to BMA estimators are heretofore unexamined. The WALS theory is further extended to allow for nonspherical disturbances, and the estimator is illustrated with data from the Hong Kong real estate market. Monte Carlo evidence shows that the WALS estimator performs significantly better than standard BMA and pretest alternatives.     

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.     

EM algorithm for one-shot device testing under the exponential distribution

The EM algorithm is a powerful technique for determining the maximum likelihood estimates (MLEs) in the presence of binary data since the maximum likelihood estimators of the parameters cannot be expressed in a closed-form. In this paper, we consider one-shot devices that can be used only once and are destroyed after use, and so the actual observation is on the conditions rather than on the real lifetimes of the devices under test. Here, we develop the EM algorithm for such data under the exponential distribution for the lifetimes. Due to the advances in manufacturing design and technology, products have become highly reliable with long lifetimes. For this reason, accelerated life tests are performed to collect useful information on the parameters of the lifetime distribution. For such a test, the Bayesian approach with normal prior was proposed recently by Fan et al. (2009). Here, through a simulation study, we show that the EM algorithm and the mentioned Bayesian approach are both useful techniques for analyzing such binary data arising from one-shot device testing and then make a comparative study of their performance and show that, while the Bayesian approach is good for highly reliable products, the EM algorithm method is good for moderate and low reliability situations.     

Bayesian invariant measurements of generalization

The problem of evaluating different learning rules and other statistical estimators is analysed. A new general theory of statistical inference is developed by combining Bayesian decision theory with information geometry. It is coherent and invariant. For each sample a unique ideal estimate exists and is given by an average over the posterior. An optimal estimate within a model is given by a projection of the ideal estimate. The ideal estimate is a sufficient statistic of the posterior, so practical learning rules are functions of the ideal estimator. If the sole purpose of learning is to extract information from the data, the learning rule must also approximate the ideal estimator. This framework is applicable to both Bayesian and non-Bayesian methods, with arbitrary statistical models, and to supervised, unsupervised and reinforcement learning schemes.     

Bayesian analysis of Birnbaum-Saunders distribution with partial information

In Bayesian analysis with objective priors, it should be justified that the posterior distribution is proper. In this paper, we show that the reference prior (or independent Jeffreys prior) of a two-parameter Birnbaum-Saunders distribution will result in an improper posterior distribution. However, the posterior distributions are proper based on the reference priors with partial information (RPPI). Based on censored samples, slice sampling is utilized to obtain the Bayesian estimators based on RPPI. Monte Carlo simulations are used to compare the efficiencies of different RPPIs, to assess the sensitivity of the choice of the priors, and to compare the Bayesian estimators with the maximum likelihood estimators, for various scales of sample size and degree of censoring. A real data set is analyzed for illustrative purpose.     

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.     

Fault detection of actuator faults in unmanned underwater vehicles

A fault-diagnostic system for unmanned underwater vehicles has been designed and tested in real operating conditions. Actuator faults have been considered, relying on approximate models of the vehicles’ dynamics. Fault detection and diagnosis is accomplished by evaluating any significant change in the behaviour of the vehicle. This task is performed by a bank of estimators: a filter is implemented for each actuator fault type, including the no-fault case. The estimators used are extended Kalman filters (EKF), due to the presence of nonlinearities in the dynamic models. Experimental results are reported, to demonstrate the effectiveness of the proposed approach.     

瑞利分布的可靠性分析

基于逐步增加Ⅱ型截尾样本,研究了瑞利分布可靠性指标的贝叶斯估计及其容许性。在不同的损失函数下给出了分布参数、可靠度函数、失效率函数的贝叶斯估计及参数的最短可信区间估计,并证明了贝叶斯估计具有容许性。最后运用蒙特卡洛方法对各结果的均方误差进行了比较。