Birnbaum-Saunders Research Articles

A novel random-effect Birnbaum-Saunders distribution for reliability assessment considering accelerated mechanism equivalence

ABSTRACT Birnbaum-Saunders (B-S) distribution has been widely used in reliability assessment of products. However, the basic B-S distribution fails in characterizing the variation of damage caused by different units. In this paper, a novel random-effect B-S distribution which considers the unit heterogeneity is proposed. First, the novel distribution incorporates the inverse gamma distribution into the shape parameter of B-S distribution, and statistical inferences of the novel distribution are thoroughly investigated. Then, an extension of the proposed B-S distribution is developed to handle accelerated life test (ALT) data. In order to ensure the validity of ALTs, a coefficient of variation hypothesis test method employing the necessary condition for mechanism equivalence of the proposed distribution is executed. To estimate the parameters of the novel B-S distribution with explanatory variables, a two-step algorithm combining analogue least square and expectation maximization (ALS-EM) is proposed based on the data with constant coefficients of variation. A simulation is conducted to verify the effectiveness of the proposed ALS-EM method. Finally, two actual engineering cases are utilized to show the advantages of the random-effect B-S distribution in terms of goodness-of-fit and reliability assessment precision.

Optimal life-testing plans for joint progressively Type-II censored Birnbaum-Saunders distributions with compound optimal criteria

ABSTRACT In this paper, we first derive the maximum likelihood estimates (MLEs) of the parameters of Birnbaum-Saunders (BISA) distributions based on the joint progressive Type-II censored (JPC) samples. We also discuss using the EM algorithm to obtain the MLEs of the model parameters. Then, we determine the single-objective-criterion optimal JPC schemes for BISA distributions based on the cost minimization criterion, A -optimality criterion, and D -optimality criterion by the complete search method, variable neighborhood search (VNS) algorithm and modified VNS (MVNS) algorithm. These algorithms are compared in terms of accuracy and computation time. In addition, we determine the compound-criterion optimal JPC schemes based on different optimal criteria and the reasonably efficient compound optimal JPC schemes for two competing statistical models, such as the inverse Gaussian and BISA models. The advantages of using the compound optimal scheme over the single-objective-criterion optimal scheme are demonstrated through a real-life data set on the micro-indentation test about the hardness of polymeric bone cement.

Pollution concentration monitoring using a new Birnbaum–Saunders control chart

AbstractAir pollution monitoring is an important issue in environmental science. The Birnbaum–Saunders (BS) distribution, originally applied to describe product failure time distribution to fatigue failures and general random wear failures, is also well to describe the pollutant concentration data due to accumulations of various pollutants in the air over time. Sulfur dioxide (SO2) is a critical factor in air pollution. Hence, it is important to monitor its concentration variation for air pollution prevention. Due to the complexity of its distribution form, there is no reliable and easy‐to‐use control chart for monitoring pollutant concentrations based on the BS distribution. We found that the SO2 concentration data follows the BS distribution. In this study, we propose a new median control chart based on the exact sampling distribution of the monitoring statistic to detect shifts in the median of BS distribution. Thus, given the false alarm rate, the control limits for such control charts can be obtained precisely satisfying a preset in‐control average run length using Monte Carlo simulations. The out‐of‐control average run lengths are calculated by simulation to evaluate the detection performance of the proposed chart when the median shifts occur. We further compare the detection performance of the proposed chart and those of the existing control charts based on asymptotic sampling distributions. In order to improve the detection ability of the proposed chart for small median shifts, an exponentially weighted moving average (EWMA) control chart is constructed. The results of numerical analyses demonstrated that the proposed EWMA chart performs much better than all existing control charts for monitoring the median of BS distribution. Finally, the proposed control charts are applied to monitor the median of SO2 concentrations for air pollution control, showing that both charts can effectively detect a shift in the median of SO2 concentrations. The proposed EWMA control chart even detects out a small shift in the median of SO2 concentrations. The results provide a continuous monitoring solution for air pollution prevention.

Improved Bayesian Inferences for Right-Censored Birnbaum–Saunders Data

This work focuses on making Bayesian inferences for the two-parameter Birnbaum–Saunders (BS) distribution in the presence of right-censored data. A flexible Gibbs sampler is employed to handle the censored BS data in this Bayesian work that relies on Jeffrey’s and Achcar’s reference priors. A comprehensive simulation study is conducted to compare estimates under various parameter settings, sample sizes, and levels of censoring. Further comparisons are drawn with real-world examples involving Type-II, progressively Type-II, and randomly right-censored data. The study concludes that the suggested Gibbs sampler enhances the accuracy of Bayesian inferences, and both the amount of censoring and the sample size are identified as influential factors in such analyses.

Stress–Strength Reliability of the Type P(X < Y) for Birnbaum–Saunders Components: A General Result, Simulations and Real Data Set Applications

An exact expression for R=P(X<Y) has been obtained when X and Y are independent and follow Birnbaum–Saunders (BS) distributions. Using some special functions, it was possible to express R analytically with minimal parameter restrictions. Monte Carlo simulations and two applications considering real datasets were carried out to show the performance of the BS models in reliability scenarios. The new expressions are accurate and easy to use, making the results of interest to practitioners using BS models.

Comparison performance of the Bayesian Approach with the Weibull and Birnbaum-Saunders distributions in imputation of time-to-event censors.

Almost all survival data is censored, and censor imputation is necessary. This study aimed to investigate the performance of the Bayesian Approach (BA) in the imputation of censored records in simulated and Breast Cancer (BC) data. Due to the difference in the distribution of time to event in survival analysis, two well-known the Weibull and Birnbaum-Saunders (BS) distributions have been used to test the performance of the BA. For each of the censored, 10,000 times were simulated using the BA in R and BUGS software, and their median or mean was imputed instead of each censor. The eligibility of both imputation methods was investigated using different curves, different censoring percentages, and sample sizes, as well as the Deviance Information Criteria (DIC), Effective Sample Size, and the Geweke diagnostic in simulated and especially real BC data. The BC data, which contains 220 patients who were identified and followed up between 2015 and 2023, was made accessible on February 1, 2023. The Kaplan-Meier, the BA, and other survival curves were drawn for the observed times. Findings indicated that the performance of the BA under the Weibull and BS distributions in simulated data is similar. The DIC index in the BC data under the BS distribution (1510) is less than the Weibull distribution (1698). Therefore, the BS distribution is preferred over the Weibull for imputation of censoring times in real BC data.

Multivariate Birnbaum–Saunders distribution based on a skewed distribution and associated EM-estimation

We develop here a multivariate generalization of Birnbaum–Saunders (BS) distribution based on the multivariate skew-normal distribution. Some distributional characteristics and properties are presented, as well as a simple and efficient EM algorithm for the iterative computation of the maximum likelihood (ML) estimates of model parameters, through the hierarchical representation of the proposed model. The standard errors of the maximum likelihood estimates are calculated from the observed Fisher information matrix. Moreover, by using the tools, we present a log-linear regression model, where the the ML estimates are once again obtained using an EM algorithm. Finally, simulation studies and two applications to real data sets are presented for illustrating the model and the inferential results developed here.

Simultaneous Confidence Intervals for All Pairwise Differences between the Coefficients of Variation of Multiple Birnbaum–Saunders Distributions

In situations where several positive random variables cannot be described using symmetrical distributions, a positively asymmetric distribution which has garnered much attention for studying them is the Birnbaum-Saunders (BS) distribution. This distribution was originally proposed to study fatigue over time in materials and has become widely employed for reliability and fatigue studies. In statistics, the coefficient of variation (CV) is employed to measure relative variation. Furthermore, comparing the CVs of several samples from BS distributions is an important approach to assess the variation among them. Herein, we propose estimation methods for the simultaneous confidence intervals (SCIs) for all pairwise differences between the CVs of multiple BS distributions based on the percentile bootstrap, the generalized confidence interval (GCI), the method of variance estimates recovery (MOVER) based on the asymptotic confidence interval (ACI) and GCI, Bayesian credible interval, and the highest posterior density (HPD) interval. The coverage probabilities and average lengths of the proposed methods were examined via a simulation study to determine their performance. The results demonstrate that GCI and the MOVER based on the GCI method provided satisfactory performances in almost every case studied. Particulate matter ≤ 2.5 μm (PM2.5) concentration datasets from three areas in northern Thailand were used to illustrate the effectiveness of the proposed methods.

Confidence Intervals for Common Coefficient of Variation of Several Birnbaum–Saunders Distributions

The Birnbaum–Saunders (BS) distribution, also known as the fatigue life distribution, is right-skewed and used to model the failure times of industrial components. It has received much attention due to its attractive properties and its relationship to the normal distribution (which is symmetric). Furthermore, the coefficient of variation (CV) is commonly used to analyze variation within a dataset. In some situations, the independent samples are collected from different instruments or laboratories. Consequently, it is of importance to make inference for the common CV. To this end, confidence intervals based on the generalized confidence interval (GCI), method of variance estimates recovery (MOVER), large-sample (LS), Bayesian credible interval (BayCrI), and highest posterior density interval (HPDI) methods are proposed herein to estimate the common CV of several BS distributions. Their performances in terms of their coverage probabilities and average lengths were investigated by using Monte Carlo simulation. The simulation results indicate that the HPDI-based confidence interval outperformed the others in all of the investigated scenarios. Finally, the efficacies of the proposed confidence intervals are illustrated by applying them to real datasets of PM10 (particulate matter ≤ 10 μm) concentrations from three pollution monitoring stations in Chiang Mai, Thailand.

Statistical analysis and estimation of the cumulative distribution function of COVID-19 cure duration in Iraq

COVID-19 disease has aggressively affected all aspects of human life since late 2019. Hospital staff have been under unprecedented pressure from a large number of patients, and in some countries, the lack of space for patients at the height of the epidemic has reached a point where hospitals do not have the capacity to accept new patients. Therefore, studying the duration of treatment of COVID-19 patients is very important in managing the ability of treatment staff and hospital facilities. In this paper, the length of hospitalization of all COVID-19 patients in Al-Sadr General Hospital in Al-Amarah, Iraq, is statistically studied from March 2020 to April 2021. The cumulative distribution function (cdf) of the patients’ treatment duration is estimated using Birnbaum-Saunders (B-S) kernel estimator. This estimate allows us to estimate the probability of a patient’s stay in the hospital for a specified period of time. In this paper, we obtain an asymptotic confidence interval for the B-S kernel estimator. However, due to the dependence of the obtained confidence interval on the unknown cdf and its derivatives, we propose a bootstrap algorithm to calculate the confidence interval and use it for the length of hospital stay of COVID-19 patients.

Generalized interval estimation of process capability indices for the Birnbaum–Saunders distribution

AbstractSince sampling variation would lead to the inaccurate assessment of process capability indices (PCIs), the interval estimation of PCIs has received considerable attention recently. The coverage probabilities (CPs) of the widely used bootstrap confidence intervals (BCIs) of PCIs are not sufficiently close to their nominal confidence level. Moreover, the bootstrap method is time‐consuming. This paper develops a procedure for constructing generalized confidence intervals (GCIs) of two widely used percentile‐based PCIs for the Birnbaum–Saunders (BS) distribution. A simulation study is conducted and the results indicate that the proposed GCI outperforms its bootstrap counterparts in terms of the CPs, the average widths (AWs) of the confidence intervals, and the variability of the interval widths. Finally, two real examples are used to illustrate the implementation of the proposed procedure.

Comparison Analysis on the Coefficients of Variation of Two Independent Birnbaum-Saunders Distributions by Constructing Confidence Intervals for the Ratio of Coefficients of Variation

The fatigue failure of materials can be investigated by applying the Birnbaum-Saunders (BS) distribution to fatigue failure datasets. The coefficient of variation (CV) is an important descriptive statistic that is widely used to measure the dispersion of data. In addition, for two independent datasets following BS distributions, the ratio of their CVs can be used to compare their CVs, especially when the difference is small, and constructing confidence intervals for this scenario is of interest in this study. Hence, we propose new confidence intervals for the ratio of the CVs from two BS distributions by using the bootstrap confidence interval (BCI), the fiducial generalized confidence interval (FGCI), a Bayesian credible interval (BayCI), and the highest posterior density (HPD) interval approaches. The performances of the proposed confidence intervals were compared with the generalized confidence interval (GCI) in terms of their coverage probabilities and average lengths via Monte Carlo simulations. The results indicate that the HPD interval outperformed the others when the coverage probabilities and the average lengths were both considered together. The efficacies of the proposed methods and GCI are illustrated using real datasets of the fatigue life of 6061-T6 aluminum coupons.

Confidence Intervals for Comparing the Variances of Two Independent Birnbaum–Saunders Distributions

Fatigue in a material occurs when it is subjected to fluctuating stress and strain, which usually results in failure due to the accumulated damage. In statistics, asymmetric distribution, which is commonly used for describing the fatigue life of materials, is the Birnbaum–Saunders (BS) distribution. This distribution can be transform to the normal distribution, which is symmetrical. Furthermore, variance is used to examine the dispersion of the fatigue life data. However, comparing the variances of two independent samples that follow BS distributions has not previously been reported. To accomplish this, we propose methods for providing the confidence interval for the ratio of variances of two independent BS distributions based on the generalized fiducial confidence interval (GFCI), a Bayesian credible interval (BCI), and the highest posterior density (HPD) intervals based on a prior distribution with partial information (HPD-PI) and a proper prior with known hyperparameters (HPD-KH). A Monte Carlo simulation study was carried out to examine the efficacies of the methods in terms of their coverage probabilities and average lengths. The simulation results indicate that the HPD-PI performed satisfactorily for all sample sizes investigated. To illustrate the efficacies of the proposed methods with real data, they were also applied to study the confidence interval for the ratio of the variances of two 6061-T6 aluminum coupon fatigue-life datasets.

Analysis of medians under two-way model with and without interaction for Birnbaum–Saunders distributed response

The Birnbaum–Saunders (BS) distribution, well-known as the fatigue-life distribution, has been used in numerous disciplines ranging from engineering to medical sciences. In this article, we develop a test for analysis of medians for BS distributed response to assess the impact of two interacting factors on the median, where no test is presently available. The proposed integrated likelihood ratio test (ILRT) eliminates the nuisance shape parameters by integrating them out. The second-order accurate asymptotic chi-square distribution of ILRT is derived. An in-depth simulation study strongly supports its excellent performance even under small group sizes. Furthermore, ILRT developed under the one-way model is found uniformly superior over its peers, is straightway extendable under general multiway setup, and has potential to be extended to other non-normal response variables. Its genuine need in industry, where non-normal responses are commonly encountered, is highlighted through analysis of three real data sets: ILRT strongly picked out the deposition time as influential factor in epitaxial layer experiment, revealed significant impact of spools on fiber life for the failure times of Kevlar 49 fiber data, and gave more accurate parameter estimates in delivery time data experiment, as assessed by various model adequacy tools, where its competitors failed to deliver desired results.

Multi-damage accelerated life test based on the Birnbaum-Saunders reliability evaluation model

ABSTRACT The present study aims to reveal the deterioration law and service life of concrete durability in the saline soil area of western China. For this purpose, the indoor accelerated tests with multiple damage factors including compound salt erosion, dry-wet cycle, freeze-thaw damage, and solar radiation were designed. The deterioration mechanisms of concrete durability were analysed from both macroscopic and microscopic aspects. The Birnbaum-Saunders (B-S) model was selected for modeling, and parameters were estimated by various methods such as moment estimation method (MEM), inverse estimation method (IMEM), quantile estimation Method (QEM), regression estimation method (REM), and maximum likelihood method (MIM). The results showed that under the action of multiple damage factors, the surface of the concrete gradually becomes rough and pitted from the initial smooth and flat surface. As the concrete’s interior gel is not dense, rod-shaped, and needle-shaped corrosion products were formed. For the two-parameter model, the reliability curve, density curve, and failure rate curve obtained by the QEM differ the most from the other 5 types of parameter estimation methods. When the MEM is used for parameter estimation, the difference between the two-parameter model and the three-parameter model under the same reliability level is relatively small.

Optimal designs of constant‐stress accelerated life‐tests for one‐shot devices with model misspecification analysis

AbstractThe design of constant‐stress accelerated life‐test (CSALT) is important in reliability estimation. In reliability studies, practitioners usually rely on underlying distribution to design CSALTs. However, model misspecification analysis of optimal designs has not been examined extensively. This paper considers one‐shot device testing data by assuming gamma, Weibull, lognormal and Birnbaum–Saunders (BS) lifetime distributions, which are popular lifetime distributions in reliability studies. We then investigate the effect of model misspecification between these lifetime distributions in the design of optimal CSALTs, in which the asymptotic variance of the estimate of reliability of the device at a specific mission time is minimized subject to a prefixed budget and a termination time of the life‐test. The inspection frequency, number of inspections at each stress level, and allocation of the test devices are determined in optimal design for one‐shot device testing. Finally, a numerical example involving a grease‐based magnetorheological fluids (G‐MRF) data set is used to illustrate the developed methods. Results suggest the assumption of lifetime distribution as Weibull or lognormal to be more robust to model misspecification, while the assumption of gamma lifetime distribution seems to be the most non‐robust (or most sensitive) one.

Bayesian Estimation for the Coefficients of Variation of Birnbaum–Saunders Distributions

The Birnbaum–Saunders (BS) distribution, which is asymmetric with non-negative support, can be transformed to a normal distribution, which is symmetric. Therefore, the BS distribution is useful for describing data comprising values greater than zero. The coefficient of variation (CV), which is an important descriptive statistic for explaining variation within a dataset, has not previously been used for statistical inference on a BS distribution. The aim of this study is to present four methods for constructing confidence intervals for the CV, and the difference between the CVs of BS distributions. The proposed methods are based on the generalized confidence interval (GCI), a bootstrapped confidence interval (BCI), a Bayesian credible interval (BayCI), and the highest posterior density (HPD) interval. A Monte Carlo simulation study was conducted to evaluate their performances in terms of coverage probability and average length. The results indicate that the HPD interval was the best-performing method overall. PM 2.5 concentration data for Chiang Mai, Thailand, collected in March and April 2019, were used to illustrate the efficacies of the proposed methods, the results of which were in good agreement with the simulation study findings.

Building flexible regression models: including the Birnbaum-Saunders distribution in the gamlss package

Generalized additive models for location, scale and shape (GAMLSS) are a very flexible statistical modeling framework, being an important generalization of the well-known generalized linear models and generalized additive models. Their main advantage is that any probability distribution (that does not necessarily belong to the exponential family) can be considered to model the response variable and different regression structures can be fitted in each of its parameters. Currently, there are more than 100 distributions that are already implemented in the gamlss package in R software. Nevertheless, researchers can implement different distributions if they are not yet available, e.g., the Birnbaum-Saunders (BS) distribution, which is widely used in fatigue studies. In this paper we make available all codes regarding the inclusion of the BS distribution in the gamlss package, and then present a simple application related to air quality data for illustration purposes

Fiducial Inference on the Right Censored Birnbaum–Saunders Data via Gibbs Sampler

In this article, we implement a flexible Gibbs sampler to make inferences for two-parameter Birnbaum–Saunders (BS) distribution in the presence of right-censored data. The Gibbs sampler is applied on the fiducial distributions of the BS parameters derived using the maximum likelihood, methods of moments, and their bias-reduced estimates. A Monte-Carlo study is conducted to make comparisons between these estimates for Type-II right censoring with various parameter settings, sample sizes, and censoring percentages. It is concluded that the bias-reduced estimates outperform the rest with increasing precision. Higher sample sizes improve the overall accuracy of all the estimates while the amount of censoring shows a negative effect. Further comparisons are made with existing methods using two real-world examples.

Stochastic Study on Discontinuous Slow Crack Growth Kinetics from an Arbitrarily Located Defect of Polyethylene Based on the Crack Layer Theory

The major failure aspect of high-density polyethylene pipes under the service condition is the discontinuous slow crack growth (DSCG) from the initial defects inside the pipe wall. In this study, the DSCG kinetics for an internal eccentric crack is theoretically simulated by developing a crack-layer growth model for an eccentric crack. The present model precisely mimics the experimental DSCG kinetics for the eccentric crack, and it also estimates the final failure time accurately. Furthermore, to investigate the reliability of the DSCG-dominated failure concerning the uncertainties related to the initial crack, a stochastic study on the lifetime distribution due to the probabilistic distribution of the initial crack size is performed. Additionally, lifetime distribution fitting using the Birnbaum–Saunders (B-S) distribution function and the maximum likelihood estimation method is conducted for various initial crack locations, sizes, and applied stresses. The B-S function accurately describes the simulated lifetime distribution, and the equations for estimating the scale and shape parameters of the B-S function with regard to the initial crack distributions at various crack locations and remote stress levels are presented.