Birnbaum-Saunders Distribution Research Articles

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.

Probability of maximum values of stress concentration factors in tubular DKT-joints reinforced with FRP under axial loads

In the paper, the most suitable probability distributions for SCFmax in DKT-Joints stiffened with FRP under axial loading conditions are studied. To this end, 1296 models were generated and then analyzed in the ANSYS to provide five reliable samples. Created FE models were validated using previous laboratory and FE data. Afterward, to predict the appropriate probability distribution models, 12 different kinds of distributions were utilized to check the fit distribution curve to the samples. Next, the accuracy and reliability of the distribution curves were assessed with the help of goodness-of-fit tests, such as the Kolmogorov-Smirnov (KS) and the Chi-squared (CS). The results of the analysis reveal that the Birnbaum-Saunders distribution is the best option. Furthermore, based on the probability graphs, it was concluded that the developed distribution is suitable enough for the given data sets.

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.

Birnbaum-Saunders parameters estimation using simple random sampling and ranked set sampling

The two-parameter Birnbaum-Saunders distribution is widely used in reliability analysis. In this paper, several Birnbaum-Saunders parameters estimation using simple random sampling (SRS), ranked set sampling (RSS) and a RSS version based on the order statistic that maximizes the Fisher information for a fixed set size (RSSF) are respectively considered. Theoretical properties of the suggested estimators are compared with its counterpart estimators using SRS by numerical simulation and a real data application. The numerical results and real data application show that the suggested estimators using RSS and RSSF can be real competitors for those using SRS.

Characterizations of continuous log-symmetric distributions based on properties of order statistics

The class of log-symmetric distributions is a generalization of log-normal distribution and includes some well-known distributions such as log-normal, log-logistic, log-Laplace, log-Cauchy, log-power-exponential, log-student-t, log-slash, and Birnbaum-Saunders distributions. In this paper, several characterization results are obtained for log-symmetric distributions based on moments of some functions of the parent distribution and also on the basis of some properties of order statistics. Specifically, when X is identical in distribution with a decreasing continuous function h ( X ) , then a relationship is established between upper and lower order statistics which is then utilized to construct characterization results for log-symmetric distributions in terms of functions of order statistics. The established results can be used for constructing a goodness-of-fit test for log-symmetric distributions.

Modeling Metallic Fatigue Data Using the Birnbaum–Saunders Distribution

This work employs the Birnbaum–Saunders distribution to model the fatigue-life of metallic materials under cyclic loading and compares it with the normal distribution. Fatigue-limit models are fitted to three datasets of unnotched specimens of 75S-T6 aluminum alloys and carbon laminate with different loading types. A new equivalent stress definition that accounts for the effect of the experiment type is proposed. The results show that the Birnbaum–Saunders distribution consistently outperforms the normal distribution in fitting the fatigue data and provides more accurate predictions of fatigue-life and survival probability.

Birnbaum Saunders distribution for imprecise data: statistical properties, estimation methods, and real life applications

A neutrosophic statistic is a random variable and it has a neutrosophic probability distribution. So, in this paper, we introduce the new neutrosophic Birnbaum–Saunders distribution. Some statistical properties are derived, using Mathematica 13.1.1 and R-Studio Software. Two different estimation methods for parameters estimation are introduced for new distribution: maximum likelihood estimation method and Bayesian estimation method. A Monte-Carlo simulation study is used to investigate the behavior of parameters estimates of new distribution, compare the performance of different estimates, and compare between our distribution and the classical version of Birnbaum–Saunders. Finally, study the validity of our new distribution in real life.

Estimation of the percentile of Birnbaum-Saunders distribution and its application to PM2.5 in Northern Thailand.

The Birnbaum-Saunders distribution plays a crucial role in statistical analysis, serving as a model for failure time distribution in engineering and the distribution of particulate matter 2.5 (PM2.5) in environmental sciences. When assessing the health risks linked to PM2.5, it is crucial to give significant weight to percentile values, particularly focusing on lower percentiles, as they offer a more precise depiction of exposure levels and potential health hazards for the population. Mean and variance metrics may not fully encapsulate the comprehensive spectrum of risks connected to PM2.5 exposure. Various approaches, including the generalized confidence interval (GCI) approach, the bootstrap approach, the Bayesian approach, and the highest posterior density (HPD) approach, were employed to establish confidence intervals for the percentile of the Birnbaum-Saunders distribution. To assess the performance of these intervals, Monte Carlo simulations were conducted, evaluating them based on coverage probability and average length. The results demonstrate that the GCI approach is a favorable choice for estimating percentile confidence intervals. In conclusion, this article presents the results of the simulation study and showcases the practical application of these findings in the field of environmental sciences.

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.

Generalized Confidence Interval for the Difference Between Percentiles of Birnbaum–Saunders Distributions and Its Application to PM2.5 in Thailand

The Birnbaum–Saunders distribution is of particular interest for statistical inference. This distribution represents the failure time distribution in engineering. In addition, the Birnbaum–Saunders distribution is commonly used in different areas of science and engineering. Percentiles are a frequently employed statistical concept. Percentiles help ascertain the position of an observation concerning the percentage of data points below it. These percentiles serve as indicators of both the central tendency and the dispersion of data. While comparing two data distributions, the mean is typically the most dependable parameter for describing the population. However, in situations where the distribution exhibits significant skewness, percentiles may sometimes offer a more reliable representation. Herein, the confidence intervals for the difference between percentiles of Birnbaum–Saunders distributions were constructed by the generalized confidence interval (GCI) approach, the bootstrap approach, the Bayesian approach, and the highest posterior density (HPD) approach. A Monte Carlo simulation was conducted to evaluate the performance of the confidence intervals. The performance was considered via coverage probability and average width. The findings suggest that utilizing the GCI approach is advisable for estimating confidence intervals for the disparity between two percentiles. Ultimately, the outcomes of the simulation investigation, coupled with an application in the field of environmental sciences, were outlined.

Statistical inference of the Birnbaum-Saunders model using adaptive progressively hybrid censored data and its applications

<abstract><p>Lately, the Birnbaum-Saunders distribution has gained a lot of attention, mainly due to its different density shapes and the non-monotonicity property of its failure rates. This work considered some estimation issues for the Birnbaum-Saunders distribution using adaptive progressive Type-Ⅱ hybrid censoring. Point and interval estimations were performed employing both conventional and Bayesian methodologies. In addition to estimating the model parameters, we obtained point and interval estimates for the reliability and hazard rate functions. We looked at the method of maximum likelihood as a classical approach, and its asymptotic traits were employed to obtain approximate confidence ranges. From a Bayesian point of perspective, we considered the squared error loss function to obtain the point estimates of the various parameters. The Bayes and highest posterior density credible intervals were additionally determined. For the complex form of the posterior distribution, Bayes estimates and credible intervals were computed by sampling from the posterior distribution through the Markov chain Monte Carlo procedure. For assessing the performance of all of these estimators, a Monte Carlo simulation was employed. Several statistical standards were applied to check the effectiveness of various estimates for multiple levels of censoring with small, moderate, and large sample sizes. Finally, two scenarios for applications were given in order to highlight the usefulness of the supplied approaches.</p></abstract>

Regression Models According to Birnbaum-Saunders Distribution

The Birnbaum–Saunders Distribution (BSD) is a positive Chile distribution that is often used to analyze life span data. Today, Birnbaum Sanders Distribution (BSD) has gained increasing popularity in various fields such as air pollution, business, earth sciences, industry and medicine. BSD is a special model due to its specificity. We have presented some results of reconstructed version of the BS distribution and a method of generating random numbers from this distribution. This article examines three regression models according on Birnbaum-Sanders distribution. The first model is derived from the distribution. The second model is derived from parameter re-averaging. The third model is derived through the logarithmic transformation of the response variable. The main purpose of this article is to compare the performance of three Birnbaum-Sanders regression models. Finally, an example of genuine data is presented to compare the regression models.

A flexible partially linear regression with random effects for bimodal data with application in postharvest

In the postharvest area of agricultural products, it is common to have data sets in which the dependent variable in general may not be symmetrical and unimodal, therefore, the use of usual regression models is not always appropriate. In addition, the data can also have repeated measurements over time. With this in mind, the aim of this article is to propose a partially linear regression model with random effects for postharvest bimodal data, this extended regression is based on the Birnbaum–Saunders distribution. Additionally we present different mathematical properties of this new model. To verify the impartiality and accuracy of the estimators, a simulation study was carried out. The parameters of partially linear regression with random effects are estimated using the penalized maximum likelihood. The results of the developed model are presented empirically and indicate that the storage temperature can cause a significant change in the lychee’s respiratory metabolism. Thus, according to the model selection and residue analysis measures, it can be concluded that the proposed model is a statistical tool for postharvest data, providing an adequate analysis.

The Geometric Generalized Birnbaum–Saunders model with long-Term Survivors

Introduction: A cure rate survival model was developed based on the assumption that the number of competing reasons for the event of interest has the Geometric distribution and the time allocated to the event of interest follows the Generalized Birnbaum-Saunders distribution.
 Methods: The Geometric Generalized Birnbaum–Saunders distribution was defined and two useful representations were represented for its density function which contributes to the creation of some mathematical properties. Furthermore, the parameters of the model with cure rate were estimated by using the maximum likelihood method.
 Results: Several simulations were performed and a real data set was analyzed from the medical area for different sample sizes and censoring percentages.In the melanoma data set and regarding the AIC and SBC selection criteria, the Geometric Generalized Birnbaum–Saunders distribution model was preferred and was selected as the appropriate model in the present study.
 Conclusion: Geometric Generalized Birnbaum–Saunders distribution is a highly flexible lifetime model which allows for different degrees of Kurtosis and asymmetry.by considering the advantages of the Geometric Generalized Birnbaum–Saunders distribution model, the model can be implemented as an appropriate alternative to explain or predict the survival time for long-term individuals.

Model misspecification of Log-Normal and Birnbaum-Saunders distributions

Model misspecification can be a serious issue in any lifetime data analysis. Assumption of the correct model is very important particularly in prediction for future observations and for estimating the tail probabilities of any lifetime distribution. In this paper we have considered the model misspecification of the log-normal and Birnbaum-Saunders distributions. These two distributions have striking similarities both in terms of the probability density functions and hazard functions, in certain range of parameter values, which makes it extremely difficult to detect the correct model. Hence, to identify the correct model, first the conventional method of ratio of maximized likelihood approach has been tried. The necessary theoretical results have been derived. Based on extensive simulations it has been observed that for certain range of parameter values the model misspecification can be quite high even for very large sample sizes. Some of the other methods like the ratio of maximized product of spacings and minimized Kolmogorov-Smirnov distance, have also been explored. But none of the method performs uniformly better than the other over the entire range of the parameter space. Some counter intuitive results have also been obtained. Hence, we finally conclude that it is extremely difficult to choose the correct model in case of log-normal and Birnbaum-Saunders distribution for certain range of parameter values. Finally, we have suggested a modified ratio of maximized likelihood approach, and the performances are quite satisfactory.

A New Family of Modified Slash Distributions with Applications

This article presents a new family of symmetric heavy-tailed distributions. This model is based on the ratio of two independent random variables; one with a normal distribution in the numerator and another with a Birnbaum–Saunders distribution in the denominator. The result is a new slash-like distribution capable of modeling high levels of kurtosis, so it can be considered as a viable alternative to other heavy-tailed distributions in the literature. Fundamental properties such as density and raw moments are derived. Parameter estimation is performed using the moment and maximum likelihood methods. A simulation study to evaluate the behavior of the estimators is carried out. Finally, the utility of the new distribution is illustrated by fitting two real datasets.

Robust explicit estimators using the power-weighted repeated medians

This paper consists of two parts. The first part of the paper is to propose an explicit robust estimation method for the regression coefficients in simple linear regression based on the power-weighted repeated medians technique that has a tuning constant for dealing with the trade-offs between efficiency and robustness. We then investigate the lower and upper bounds of the finite-sample breakdown point of the proposed method. The second part of the paper is to show that based on the linearization of the cumulative distribution function, the proposed method can be applied to obtain robust parameter estimators for the Weibull and Birnbaum-Saunders distributions that are commonly used in both reliability and survival analysis. Numerical studies demonstrate that the proposed method performs well in a manner that is approximately comparable with the ordinary least squares method, whereas it is far superior in the presence of data contamination that occurs frequently in practice.

Degradation index construction and learning-based prognostics for stochastically deteriorating feedback control systems

Degradation-based prognostics is crucial for the health management of technological systems. In this work, we are interested in the degradation index construction and remaining useful life prognostics for stochastically deteriorating feedback control systems. The main challenges reside in the lack of knowledge about the system structure and the latent internal damage, as well as in the fault tolerance nature of feedback control systems. Our solution is to consider the whole system as a black-box, and use its easy-to-observe reference input/time response output to estimate the system transfer function. The associated H∞ norm, also called maximum energy gain, is defined as a system degradation index. Since the system fault tolerance does not allow to efficiently model the index evolution by common stochastic processes, traditional prognostics based on degradation processes are no longer applicable. To remedy, we propose to fit the system remaining useful life population to the versatile Birnbaum–Saunders distribution, and adopt a segmenting piecewise polynomials algorithm to learn the mapping between the distribution parameters and degradation index from degradation and failure data of similar systems. By this way, the remaining useful life distribution of deteriorating feedback control systems can be predicted in real-time given the system input/output at an inspection time. We numerically experiment our method on a stabilization loop control device driven by proportional–integral–differential controller in an inertial platform. Numerous sensitivity results under various configurations of system characteristics and training data corroborate the outperformance of proposed degradation index and the learning-based prognostics method.

Using Different Loss Function to Estimate the Parameters of Birnbaum-Saunders Distribution by Bayesian Method with Application

The Birnbaum-Saunders distribution is one of the most important distributions that explain fatigue time in general, as it has many engineering and industrial applications. In our Paper we introduce three estimation methods to estimate the parameters of Birnbaum-Saunders parameters, first is the Maximum Likelihood Estimator MLE and second, is Bayes estimation with quadratic loss function BQ and last is the Bayes estimator with weighted loss function BW. simulated data were used as well as real data used which represented by the fatigue time of a concrete block under pressure before its final collapse. It was concluded that the Bayes estimator with squared loss function BQ is the best.

Evaluation of Nakagami and Birnbaum-Saunders probability distribution for wind speed and power estimation

This study compares three probability density functions (PDFs) for understanding and estimating wind power based on wind patterns: the commonly-used Weibull distribution, the relatively-new Birnbaum-Saunders distribution, and the Nakagami distribution. The wind profile of Jumla, Nepal was analyzed using data from 2004 to 2014. The Nakagami distribution performed similarly to the Weibull distribution in terms of understanding wind patterns. However, for estimating wind power, the Nakagami distribution was found to be more effective than the Weibull distribution in most cases. The Birnbaum-Saunders distribution was found to be the least effective of the three PDFs compared.