Bayesian Concepts for the Estimation of Reliability in the Weibull Life-Testing Model

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

In this paper, we consider the Bayesian inference on the stress-strength parameter \(R = P(Y < X)\), when X and Y follow independent Weibull distributions. We have considered different cases. It is assumed that the random variables X and Y have different scale parameters and (a) a common shape parameter or (b) different shape parameters. Moreover, both stress and strength may depend on some known covariates also. When the two distributions have a common shape parameter, Bayesian inference on R is obtained based on the assumption that the shape parameter has a log-concave prior, and given the shape parameter, the scale parameters have Dirichlet-Gamma prior. The Bayes estimate cannot be obtained in closed form, and we propose to use Gibbs sampling method to compute the Bayes estimate and also to compute the associated highest posterior density (HPD) credible interval. The results have been extended when the covariates are also present. We further consider the case when the two shape parameters are different. Simulation experiments have been performed to see the effectiveness of the proposed methods. One data set has been analyzed for illustrative purposes and finally, we conclude the paper.

The objective of this study is to examine the properties of Bayes estimators of the shape parameter of the Power Function Distribution (PFD-I), by using two different prior distributions for the parameter θ and different loss functions that were compared with the maximum likelihood estimators. In many practical applications, we may have two different prior information about the prior distribution for the shape parameter of the Power Function Distribution, which influences the parameter estimation. So, we used two different kinds of conjugate priors of shape parameter θ of the Power Function Distribution (PFD-I) to estimate it. The conjugate prior function of the shape parameter θ was considered as a combination of two different prior distributions such as gamma distribution with Erlang distribution and Erlang distribution with exponential distribution and Erlang distribution with non-informative distribution and exponential distribution with the non-informative distribution. We derived Bayes estimators for shape parameter θ of the Power Function Distribution (PFD-I) according to different loss functions such as the squared error loss function (SELF), the weighted error loss function (WSELF) and modified linear exponential (MLINEX) loss function (MLF), with two different double priors. In addition to the classical estimation (maximum likelihood estimation). We used simulation to get the results of this study, for different cases of the shape parameter of the Power Function Distribution used to generate data for different samples sizes.

Introduction Assessing objective measures of sleep fragmentation could yield important features reflecting impaired sleep quality in people with insomnia. Survival analysis allows the specific examination of the stability of NREM sleep, REM sleep and wake. The objective of this study was to assess the differences between survival dynamics of NREM sleep, REM sleep and wake between people with insomnia and healthy controls. Methods We analyzed polysomnography recordings from 88 people with insomnia and 92 healthy controls. For each participant, survival dynamics of REM sleep, NREM sleep and wake were represented using Weibull distributions. We used lasso penalized linear regression to analyze the difference between participant groups with respect to the Weibull scale and shape parameters, while correcting for age, sex, total sleep time and relevant interaction effects. Because comparisons were done for scale and shape parameters of REM sleep, NREM sleep and wake, a Bonferroni correction was applied, resulting in an alpha value of 0.05/6 = 0.0083. Results Significant effects of group were found for the NREM scale parameter (unstandardized model coefficient B=-0.79, t=-3.0, p=0.0035), and for the scale and shape parameters of wake (scale parameter B=7.6, t=2.8, p=0.0065; shape parameter B=0.20, t=2.9, p=0.0048). Results indicated that people with insomnia had less stable NREM sleep and more stable wake after sleep onset compared to healthy controls. Additionally, the altered distribution of wake segment lengths indicated an increased difficulty to fall asleep after longer awakenings in the insomnia group. However, these differences were mainly observed in younger participants. Significant effects of group for the survival parameters of REM sleep were not found. Conclusion As illustrated by our results, survival analysis can be very useful for disentangling different types of sleep fragmentation in people with insomnia. For instance, the current findings suggest that people with insomnia have an increased fragmentation of NREM sleep, but not necessarily of REM sleep. Additional research into the underlying mechanisms of NREM sleep fragmentation could possibly lead to a better understanding of impaired sleep quality in people with insomnia, and consequently to improved treatment. Support (if any):

In fatigue testing of metals or composite materials under different levels of stress, the underlying distributions are often assumed to be Weibull with a common shape parameter. The problems of estimation and hypothesis tests of the common shape parameter and scale parameters in the Weibull distributions are considered in this paper. The results are: 1) the maximum likelihood equation for estimating the shape parameter, 2) a s-confidence interval and hypothesis test for the shape parameter, 3) s-bias factors and variances of the pooled estimator of shape parameter, and 4) s-confidence interval and hypothesis test for the scale parameters. Examples are given for composite material applications. Results are based on Monte Carlo simulation methods.

We shall consider the estimation for the parameter and the right tail probability in a general exponential distribution. We also shall consider the estimation of the reliability P(X < Y ) and the skewness trends of the density function of the ratio X=(X+Y) for two independent general exponential variables each having different shape parameters and known scale parameter. We then shall consider the estimation of the failure rate average and the hazard function for a general exponential variable having the density function with the unknown shape and known scale parameters, and for a bivariate density induced by the general exponential density.

Bayesian sensitivity analysis in elliptical linear regression models

As an important reliability metric in disc drive industries, the annual failure rate (AFR), estimated from reliability demonstration tests (RDTs), is often used to make business decisions. Due to limited test sample size and short test duration, estimating the AFR from direct test data using Weibull distribution fit may result in a wide confidence interval, raising concerns regarding the uncertainty of the AFR estimate. To improve the confidence of the estimate, this article presents a Bayesian posterior analysis approach to estimate the AFR. Prior distributions of the Weibull distribution's shape and scale parameters are presented based on historic test data. An analysis of real RDT data demonstrates that the Bayesian posterior estimate of the AFR can produce a significantly narrower confidence interval than direct Weibull fitting, promoting higher confidence to accept the AFR estimate. To evaluate the significance of the prior distribution parameters on the AFR estimate, a sensitivity analysis is presented with respect to each individual parameter. The results indicate that the standard deviation value of either β or η has a more significant impact on the AFR estimate than the mean value.

The two‐parameter Weibull distribution (2PWD), similar to an instantaneous unit hydrograph (IUH), is parameterized in terms of Horton catchment ratios on the basis of a geomorphologic model of catchment response. For this the shape and scale parameters of the Weibull distribution are expressed analytically in terms of Horton's catchment ratios. The two parameters of the IUH derived using Nash's model, which is a two‐parameter gamma distribution (2PGD), are also expressed analytically in terms of Horton ratios. The performance of the proposed methods is tested for describing a synthetic unit hydrograph (SUH) under limited data conditions. A comparison is made with the unit hydrographs derived from the event data of two real catchments, and with the existing geomorphological based 2PGD for developing SUH given by Rosso (1984). The sensitivity analysis of the 2PWD to the nondimensional parameter β of the UH (a product of peak discharge and time to peak) shows β to be more sensitive to the shape parameter a than the scale parameter b. Further examination to find any similarity between the behavior of 2PWD and 2PGD showed that a in 2PWD corresponds to the shape parameter n in the 2PGD, and b behaves similar to the scale parameter k in the 2PGD. Finally, practical applicability of the proposed approach to ungauged catchments is tested using field data.

ECONOMIC STATISTICAL DESIGN OF X̄ CONTROL CHARTS FOR SYSTEMS WITH WEIBULL IN-CONTROL TIMES

Article history: Received 1 January 2010 Received in revised form 8 June 2010 Accepted 9 June 2010 Available online 9 June 2010 This paper conducts a simulation study of the effects of violating the ANOVA normality assumption in the presence of Weibull data. Twelve specific Weibull distributions, characterizing the life data of a variety of real-world products and systems, are investigated. Confidence intervals on test significance and power are generated and compared against intervals from normally distributed data. The ANOVA procedure is found to be robust in the majority of cases. Furthermore, a designed experiment is conducted to isolate the effects of the Weibull shape and scale parameters within the preceding study. The shape parameter is found to have a significant effect on significance and power, whereas the scale parameter does not have a significant effect at the target α = 0.05 test significance level. © 2010 Growing Science Ltd. All rights reserved.

Weibull distribution is one of the continous probability density function. This distribution is known as a flexible distribution. One of it’s flexibility is can be transformed into other distributions such as exponential distribution depends on parameter selected. Similar to other distribution, Weibull distribution also characterized by cumulative distribution function, mean, variance and moment generating function. One of the well-known application of this distribution is in the field of quality control utilizing reliability data and the well-known tool in quality control is control chart. Therefore, reliability data is not follow normal distribution then Shewhart control chart cannot be applied. To solve this problem, control chart can be formed with the control limit is obtained by Bayesian method. In applying Bayesian method, prior distribution of shape parameter is assumed to be uniform distribution and variables for reliability is assumed to be invers of Weibull distribution. The prior distribution is combine with likelihood function then posterior distribution is obtained which in then used as the control limits by find it’s mean.

Aim of study: The aim of this work was to model diameter distributions of Quercus suber stands. The ultimate goal was to construct models enabling the development of more affordable forest inventory methods. This is the first study of this type on cork oak forests in the area.Area of study: The area of study is “Los Alcornocales” Natural Park (Cádiz-Málaga, Spain).Material and methods: The diameter distributions of 100 permanent plots were modelled with the two-parameter Weibull function. Distribution parameters were fitted with the non-linear regression, maximum likelihood, moment and percentile-based methods. Goodness of fit with the different methods was compared in terms of number of plots rejected by the Kolmogorov-Smirnov test, bias, mean square error and mean absolute error. The scale and shape parameters in the Weibull function were related to the stand variables by using the parameter prediction model.Main results: The best fitting was obtained with the non-linear regression approach, using as initial values those obtained by maximum likelihood method, the percentage of rejections by the Kolmogorov-Smirnov test was 2% of the total number of cases. The scale parameter (b) was successfully modelled in terms of the quadratic mean diameter under cork (R2 adj = 0.99). The shape parameter (c) was modelled by using maximum diameter, minimum diameter and plot elevation (R2 adj = 0.40).Research highlights: The proposed model diameter distribution can be a highly useful tool for the inventorying and management of cork oak forests.Key words: maximum likelihood method; moment method; non linear regression approach; parameter prediction model; percentile method; scale parameter; shape parameter.

Weibull renewal function has attracted a lot of attention because the Weibull distribution describes in a relatively simple analytical manner a wide range of realistic behavior and its shape and scale parameters can be readily determined with graphical or statistical procedure. On the other hand, there are no closed form analytical solutions for the Weibull renewal function except for the special case of exponential distribution. Bounds, approximations, and tables are usually used. In this article, some approximations based on Normal approximation of Weibull distribution are studied. Such a procedure, which is different from that in the existing literature, is shown to be good for Weibull renewal function when the shape parameter is of moderate or large size. Series truncation expression and approximation bounds can be obtained as well. Numerical examples and comparisons are shown to illustrate the procedure.

Synthesizing dual-track road elevation data