Empirical Bayes Estimators of Reliability for Lognormal Failure Model

Empirical Bayes (EB) procedures are considered for estimating the reliability R(t;?,?) = gaufc[(ln t -?)/?] for the lognormal failure model. EB estimators are obtained for the 2 cases: i)? is unknown and ? is known, and both ? and ? are unknown. The empirical Cdf of the maximum likelihood estimators of the parameters is used to obtain the EB estimators. ii) A smooth EB estimator of R(t;?,?) is developed when ? is unknown and ? is known. A modification of this estimator is proposed for both ? and ? unknown. In both cases, EB estimators are obtained for complete samples at each testing stage. Monte Carlo simulations are presented to compare the EB estimators and the maximum likelihood (ML) estimators of R(t;?,?). The simulations indicate that the smooth EB estimators have smaller mean squared errors than the other EB estimators or the ML estimators.     

A Bayesian Approach to Parameter and Reliability Estimation in the Poisson Distribution

For life testing procedures, a Bayesian analysis is developed with respect to a random intensity parameter in the Poisson distribution. Bayes estimators are derived for the Poisson parameter and the reliability function based on uniform and gamma prior distributions of that parameter. A Monte Carlo procedure is implemented to make possible an empirical mean-squared error comparison between Bayes and existing minimum variance unbiased, as well as maximum likelihood, estimators. As expected, the Bayes estimators have mean-squared errors that are appreciably smaller than those of the other two.     

On Bayes Estimation of Reliability for the Birnbaum-Saunders Fatigue Life Model

Estimation of reliability for the Birnbaum-Saunders fatigue life distribution is considered. The scale parameter is also the median lifetime, and assuming that the scale parameter is known, Bayes estimators of the reliability function are obtained for a family of proper conjugate priors as well as for Jeffreys' vague prior for the shape parameter. When both parameters are unknown, a modified Bayes estimator of reliability is proposed using a moment estimator of the scale parameter. In addition to being computationally simpler than the MLE of reliability, Monte Carlo simulations for small samples show that the modified Bayes estimator is better than the MME for all values of the shape parameter and as good as the MLE for small values of the shape parameter in the sense of root mean squared errors.     

NAR estimators of spatial covariance matrices for adaptive arraydetection

The theory of noise-alone-reference (NAR) power estimation is extended to the estimation of spatial covariance matrices. A NAR covariance estimator insensitive to signal presence is derived. The SNR (signal-to-noise ratio) loss incurred by using this estimator is independent of the input SNR and is less than that encountered with the maximum likelihood covariance estimator given that the same number of uncorrelated snapshots is available to both estimators. The analysis assumes first a deterministic signal. The results are extended and generalized to signals with unknown parameters or random signals. For the random signal case, generalized and quasi-NAR covariance estimators are presented     

Estimation of reliability in multi-component stress-strength model following a Burr distribution

This paper draws inferences about the reliability in a multi-component stress-strength system when both stress and strength are independently identically distributed (idd) Burr random variables. We consider both maximum likelihood and Bayes estimators of the system reliability. The two estimators are compared numerically by obtaining empirical efficiencies with respect to the maximum likelihood estimator (MLE) by generating 1000 random samples by a Monte Carlo simulation. It is found that the Bayes estimators are better than the corresponding MLEs for small samples (n_i ≤ 7; i = 1, 2). Moreover, the robustness of the Bayes estimators to the change of the prior parameters is also considered.     

Bayes estimation of reliability in stress-strength model of Weibull distribution with equal scale parameters

The paper provides a Bayesian approach to inference about the reliability in a multicomponent stress-strength system. We consider Bayes' estimator of the system reliability from data consisting of a random sample from the stress distribution and one from the strength distribution when the two distributions are Weibull with equal and known scale parameters. The estimator of λ, ratio of two shape parameters, is also considered. The proposed estimators can be compared with the maximum likelihood estimators (mles). However, the comparison is carried out for single component stress-strength system and the Monte Carlo efficiencies are obtained. It is found that the proposed estimators are better than the corresponding mles.     

Shrunken estimators for the scale parameter of classical Pareto distribution

The shrunken estimators for the scale parameter of classical Pareto distribution by shrinking the maximum likelihood estimator and the unbiased estimator towards the guess value are proposed. Comparisons with the usual estimators in terms of mean square error have been made. The proposed estimators are preferable in some regions of parametric space.     

An Empirical Bayes Approach for the Poisson Life Distribution

A smooth empirical Bayes estimator is derived for the intensity parameter (hazard rate) in the Poisson distribution as used in life testing. The reliability function is also estimated either by using the empirical Bayes estimate of the parameter, or by obtaining the expectation of the reliability function. The behavior of the empirical Bayes procedure is studied through Monte Carlo simulation in which estimates of mean-squared errors of the empirical Bayes estimators are compared with those of conventional estimators such as minimum variance unbiased or maximum likelihood. Results indicate a significant reduction in mean-squared error of the empirical Bayes estimators over the conventional variety.     

Modified `practical Bayes-estimators' [reliability theory]

This paper presents a new formulation of `practical Bayes-estimators' (PBE) for the 2-parameter Weibull model when both parameters are unknown. Overcoming some limitations of the first formulation gave rise to this work, but the results are beyond this intent. These estimators are a tool to improve technical knowledge by using a few experimental data. In this case, the controversy about whether to use Bayes or classical methods is surmounted since estimators, like maximum likelihood, give estimates that often appear unlikely on the basis of technical knowledge of the engineers. A Monte Carlo study supports the following conclusions: if the shape parameter is greater than one, modified PBE maintain the good properties of practical Bayes estimators; otherwise the modified PBE are much better and do not suffer from the past limitation regarding the formulation of the prior interval on the shape parameter itself; and when there are very few data the modified PBE work as a filter that always improves (on average) the prior information if it is poor, or substantially confirms it if it is good. From this viewpoint, Bayes theorem allows statistics to help engineering and not vice versa     

Computationally efficient angle estimation for signals with knownwaveforms

This paper presents a large sample decoupled maximum likelihood (DEML) angle estimator for uncorrelated narrowband plane waves with known waveforms and unknown amplitudes arriving at a sensor array in the presence of unknown and arbitrary spatially colored noise. The DEML estimator decouples the multidimensional problem of the exact ML estimator to a set of 1-D problems and, hence, is computationally efficient. We shall derive the asymptotic statistical performance of the DEML estimator and compare the performance with its Cramer-Rao bound (CRB), i.e., the best possible performance for the class of asymptotically unbiased estimators. We will show that the DEML estimator is asymptotically statistically efficient for uncorrelated signals with known waveforms. We will also show that for moderately correlated signals with known waveforms, the DEML estimator is no longer a large sample maximum likelihood (ML) estimator, but the DEML estimator may still be used for angle estimation, and the performance degradation relative to the CRB is small. We shall show that the DEML estimator can also be used to estimate the arrival angles of desired signals with known waveforms in the presence of interfering or jamming signals by modeling the interfering or jamming signals as random processes with an unknown spatial covariance matrix. Finally, several numerical examples showing the performance of the DEML estimator are presented in this paper     

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Kundu  D. Gupta  R.D. 《Reliability, IEEE Transactions on》2006,55(2):270-280

An Empirical Bayes Approach to Reliability

A Bayesian reliability estimation technique known as the ``empirical Bayes approach' is developed which uses previous experience nce to get a Bayesian point estimator. The techniques require no knowledge of the form of the unknown prior distribution and are robust to assumptions about its form. Empirical Bayes techniques are applicable to situations in which prior, independent observations of the random variable X from the random couple (?, X) are available where ? is the observed parameter of interest distributed in accordance with the unknown prior distribution. Performance comparisons of the empirical Bayes and other well established techniques are developed by examples for the binomial, exponential, Normal, and Poisson situations which often occur in reliability problems. In all cases the empirical Bayes estimator performed better than the classical estimator in minimizing the average squared error.     

On maximum likelihood estimation for the two parameter Weibull distribution

This paper examines recent results presented on maximum likelihood estimation for the two parameter Weibull distribution. In particular, we seek to explain some recently reported values for estimator bias when the data for analysis contains both times to failure and censored times in operation; our discussion centres on the generation of sample data sets. We conclude that, under appropriate conditions, estimators are asymptotically unbiased, with relatively low bias in small to moderate samples. We then present the results of some further experiments which suggest that the previously reported values for estimator bias can be attributed to the method of generating sample data sets in simulation experiments.     

Estimation of the Equivalent Number of Looks in Polarimetric Synthetic Aperture Radar Imagery

This paper addresses estimation of the equivalent number of looks (ENL), an important parameter in statistical modeling of multilook synthetic aperture radar (SAR) images. Two new ENL estimators are discovered by looking at certain moments of the multilook polarimetric covariance matrix, which is commonly used to represent multilook polarimetric SAR (PolSAR) data, and assuming that the covariance matrix is complex Wishart distributed. First, a second-order trace moment provides a polarimetric extension of the ENL definition and also a matrix-variate version of the conventional ENL estimator. The second estimator is obtained from the log-determinant matrix moment and is also shown to be the maximum likelihood estimator under the Wishart model. It proves to have much lower variance than any other known ENL estimator, whether applied to single-polarization or PolSAR data. Moreover, this estimator is less affected by texture and thus provides more accurate results than other estimators should the assumption of Gaussian statistics for the complex scattering coefficients be violated. These are the first known estimators to use the full covariance matrix as input, rather than individual intensity channels, and therefore to utilize all the statistical information available. We finally demonstrate how an ENL estimate can be computed automatically from the empirical density of small sample estimates calculated over a whole scene. We show that this method is more robust than procedures where the estimate is calculated in a manually selected region of interest.     

Likelihood ratios for sequential hypothesis testing on Markov sequences

A variety of likelihood ratios are derived for detecting Gauss-Markov and finite-state Markov sequences in additive Gaussian noise. The Bayesian recursions appropriate to related filtering problems are exploited, together with "known-form" likelihood ratios, to obtain the desired results. In the derivation of a discrete-time Gauss-Markov likelihood ratio, a "pure" causal estimator-correlator structure is sought and a "locally stable" state estimator is encountered that is of some interest in its own right. The likelihood ratio is "pure" in the sense that the locally stable estimator is used in precisely the same manner as the stored replica is used in known-form signal detection problems to form the likelihood ratio. Consequently, the likelihood ratio is devoid of the extra data-dependent term that arises whenever one uses least squares state estimators to form the likelihood ratio statistic. The locally stable estimator equalizes, within a constant related to the {em a priori} and {em a posteriori} filtering error covariances, the {em a priori} and {em a posteriori} filtering densities. Heuristically, the estimator is a compromise between the one-step predictor and the filtered estimator of a discrete-time Kalman filter. When the observation noise covariance is unknown, a generalization of the so-called unknown level problem, then a Wishart prior is assigned to the innovations covariance and an integral representation is obtained for the desired likelihood ratio. The representation suggests a parallel structure for approximating the likelihood ratio when the observation noise covariance is unknown. Finally, the likelihood ratio for detecting finite-state Markov sequences is derived to illustrate that in general no "pure" estimator-correlator structure can exist when the state-space is finite.     

Small-Sample Estimation for the Lognormal Distribution With Unit Shape Parameter

Let X be a random variable such that In [(X - ?)/?] has a s-normal distribution with mean zero and variance one. Then X has a 3-parameter lognormal distribution with the third parameter, the shape parameter, fixed at unity. This paper presents the coefficients required to construct the best linear unbiased estimators (BLUEs) of ? and ? for samples of size fifteen and less. The variances and covariances of these estimators are provided. These estimators yield the BLUEs of the mean, standard deviation, and percentiles of X since these quantities are linear functions of ? and ?. Blom's estimators and maximum likelihood estimators compare favorably with the BLUEs.     

The logarithmic series distribution as a failure model from the Bayesian point of view

In this presentation the logorithmic series is studied as a failure model from the Bayesian point of view. It is assumed that the location parameter behaves as a random variable with beta as its prior distribution. Based on this assumption Bayes estimators for the location parameter and reliability function are derived. By using computer simulation we compare the Bayes estimator for the parameter with the corresponding minimum variance unbiased estimator (MVUE) and the Bayes estimator for the reliability with a corresponding unbiased estimator derived from the MVUE of the probability function.     

The Burr type XII distribution as a failure model under various loss functions

Bayesian estimates of the parameter p and the reliability function for the two-parameter Burr type XII failure model under three different loss functions, absolute difference, squared error and logarithmic are derived. It is assumed that the parameter p behaves as a random variable having (i) a gamma prior and (ii) a vague prior. Monte Carlo simulations are presented to compare the Bayesian estimators and the maximum likelihood estimators of the parameter p and the reliability function. The results show that the “popular” squared error loss function is not always the best, and that the other loss functions give comparable results.