LINEX Loss Function Research Articles

Bayesian and non‐Bayesian inference of the process capacity index Spk′$S^{\prime }_{pk}$ under progressive type‐II right censored scheme

AbstractProcess capability indices (PCIs), are widely used to gauge how well a process performs within its requirements. While greater PCIs are indicative of improved process “quality,” they do not necessarily translate into lower rejection rates. Therefore, using a loss‐based PCI to gauge process capability makes more sense. In this paper, the PCI has been considered for normal random variable. The article attempts to study the classical and the Bayesian estimation of for type‐II progressive right censored sample under symmetric as well as asymmetric loss functions, namely squared error loss function, and LINEX loss function, respectively. The Markov chain Monte Carlo simulation technique has been efficiently used here to have the approximate solution for . Through an extensive Monte Carlo simulation study along with two real life examples related to electronic industry, we compare the performances of the classical and the Bayes estimates based on symmetric and asymmetric loss functions and compared among the asymptotic confidence interval and highest posterior density credible intervals, in terms of average width and corresponding coverage probabilities of , respectively.

Statistical inference of entropy functions of generalized inverse exponential model under progressive type-II censoring test.

This article explores the estimation of Shannon entropy and Rényi entropy based on the generalized inverse exponential distribution under the condition of stepwise Type II truncated samples. Firstly, we analyze the maximum likelihood estimation and interval estimation of Shannon entropy and Rényi entropy for the generalized inverse exponential distribution. In this process, we use the bootstrap method to construct confidence intervals for Shannon entropy and Rényi entropy. Next, we select the gamma distribution as the prior distribution and apply the Lindley approximation algorithm to calculate `estimates of Shannon entropy and Rényi entropy under different loss functions including Linex loss function, entropy loss function, and DeGroot loss function respectively. Afterwards, simulation is used to calculate estimates and corresponding mean square errors of Shannon entropy and Rényi entropy in GIED model. The research results show that under DeGroot loss function, estimation accuracy of Shannon entropy and Rényi entropy for generalized inverse exponential distribution is relatively high, overall Bayesian estimation performs better than maximum likelihood estimation. Finally, we demonstrate effectiveness of our estimation method in practical applications using a set of real data.

Optimal variable acceptance sampling plan for exponential distribution using Bayesian estimate under Type-I hybrid censoring

Acceptance sampling plans with censoring schemes are crucial for improving quality control by efficiently managing incomplete information. This approach improves cost and time effectiveness compared to traditional methods, providing a more accurate assessment of product quality. In this study, a variable acceptance sampling plan under Type-I hybrid censoring is designed for a lot of independent and identical units with exponential lifetimes using Bayesian estimation of the mean life. This novel approach diverges from conventional methods in acceptance sampling plans, which rely on maximum likelihood estimation and the minimization of Bayes risk. Bayesian estimation is obtained using both squared error loss and Linex loss functions. Under each method, a nonlinear optimization problem is solved to minimize the testing cost, and the optimal values of the plan parameters are determined. The proposed plans are illustrated using various numerical examples, with each plan presented in tables. The acceptance sampling plan using the squared error loss function proves to be more cost-effective than the plan using the Linex loss function. A comparative analysis of the proposed plans with existing work in the literature demonstrates that our cost is much lower than the cost of existing plans using maximum likelihood estimation. Additionally, a real-life case study is conducted to validate the approach.

On bayesian bonus-malus premium under linex loss function with applications

The majority of bonus-malus systems (BMS) determine the premium for each policyholder solely based on their claim frequency. Adopting this method would be unjust. A driver who files a claim for small damage cannot be subjected to the same penalties as a driver responsible for a claim involving significant damage. To achieve a balance in the portfolio between good and bad drivers, a new portion of the premium is computed as a compromise. To do this, the claim severity is created and assessed to provide a fair premium. The consideration was determined by taking into account both the quantity and magnitude of the claims. This work assumes that the frequency of claims follows a Poisson-Akash distribution, but the severity of claims follows a newly proposed distribution known as the Inverse-Gamma Lindley distribution. The bonus-malus premiums were computed using the Bayesian approach for both the frequency and severity of claims. The premium is computed using the asymmetric Linex loss function for both the frequency and severity of claims in the bonus-malus system. Illustrative instances utilizing an actual dataset for the application of BMS are presented, showcasing the use of claim frequency alone and the combination of claim frequency and claim severity. The R software is employed for these demonstrations. We concluded this paper with a discussion comparing the premiums obtained in each case.

Inference on exponentiated Rayleigh distribution with constant stress partially accelerated life tests under progressive type-II censoring

This study aims to explore the issues of evaluating the parameters and the accelerating factor based on constant stress for partially accelerating life tests when the potential failure times have an exponentiated Rayleigh distribution. Within the framework of progressive Type-II censoring schemes, we employ the Newton-Raphson algorithm as an iterative methodology to gain the maximum likelihood estimates, accompanied by proof of the existence of these point estimators. We also construct asymptotic confidence intervals for interested parameters and acceleration factors by utilizing the asymptotical characteristics of the maximum likelihood estimators. The Bayesian estimations of unknown parameters are derived by using the independent gamma priors and dependent Gamma-Dirichlet prior on the basis of square error and relatively smooth LINEX loss functions, respectively. Furthermore, we adopt the importance sampling method to compute Bayesian point estimates and the credible intervals with the highest posterior density. To validate the effectiveness of the suggested approaches, a series of simulated experiments are carried out. Lastly, we conduct analyzes on two actual datasets to show the applicability of the suggested techniques.

A Bayesian wavelet shrinkage rule under LINEX loss function

A Bayesian wavelet shrinkage rule under LINEX loss function

Bayesian estimation and prediction under progressive-stress accelerated life test for a log-logistic model

Accelerated life tests play a very critical role in reliability analysis because highly reliable products are being produced with recent advanced technologies to sustain the market demand and competition. A progressive-stress accelerated life test is one of the kinds of accelerated life tests that allows applied stress to change continuously. Considering the importance of progressive-stress accelerated life tests, this paper deals with progressive-stress accelerated life tests when the lifetimes of units follow the log-logistic distribution and are progressively type-II censored where the associated scale parameter conforms to the inverse power law. Different estimators of model parameters are derived using maximum likelihood and Bayesian methods. Interval estimation is also considered. In the sequel, approximate confidence, bootstrap, and Bayes credible intervals are constructed. Bayes estimators are obtained under squared error, LINEX, and entropy loss functions using proper and improper prior distributions. A simulation study is conducted based on various censoring schemes. The coverage percentages and interval widths are computed via Monte Carlo simulations. Bayes predictive estimates and intervals are also obtained. Finally, two different accelerated life test data are analyzed for illustration purposes.

Estimation of Marshall–Olkin Extended Generalized Extreme Value Distribution Parameters under Progressive Type-II Censoring by Using a Genetic Algorithm

In this article, we consider the statistical analysis of the parameter estimation of the Marshall–Olkin extended generalized extreme value under liner normalization distribution (MO-GEVL) within the context of progressively type-II censored data. The progressively type-II censored data are considered for three specific distribution patterns: fixed, discrete uniform, and binomial random removal. The challenge lies in the computation of maximum likelihood estimations (MLEs), as there is no straightforward analytical solution. The classical numerical methods are considered inadequate for solving the complex MLE equation system, leading to the necessity of employing artificial intelligence algorithms. This article utilizes the genetic algorithm (GA) to overcome this difficulty. This article considers parameter estimation through both maximum likelihood and Bayesian methods. For the MLE, the confidence intervals of the parameters are calculated using the Fisher information matrix. In the Bayesian estimation, the Lindley approximation is applied, considering LINEX loss functions and square error loss, suitable for both non-informative and informative contexts. The effectiveness and applicability of these proposed methods are demonstrated through numerical simulations and practical real-data examples.

BAYES ESTIMATION OF EXPONENTIALLY DISTRIBUTED SURVIVAL DATA UNDER SYMMETRIC AND ASYMMETRIC LOSS FUNCTIONS

In a research study, population data are often not available, so the population parameter is unknown. Meanwhile, knowledge about the population parameter is needed to know the characteristics of the studied population. Therefore, it is needed to estimate the parameter of the population which can be estimated by sample data. There are several methods of parameter estimation which are generally classified into classical and Bayesian method. This research studied the Bayesian parameter estimation method to determine the parameters of the exponentially distributed survival data associated with the reliability measure of the estimates under symmetric and asymmetric loss functions for complete sample data in a closed form. The symmetric loss functions used in this research are Squared Error Loss Function (SELF) and Minimum Expected Loss Function (MELF). The asymmetric loss functions used are the General Entropy Loss Function (GELF) and Linex Loss Function (LLF). Performance of some loss functions used in this research are then compared through numerical simulation to select the best loss function in determining the parameter estimation of the exponentially distributed survival data. We also studied which loss function is best for underestimation and overestimation modeling. Based on simulation results, the Bayes estimates using MELF is the best method to estimate population parameters of the exponentially distributed survival data for the overestimation modeling, while LLF is the best for the underestimation modeling. We provided direct application in a case study of fluorescence lamp survival data. The results show that the best method to estimate the parameter of the standard fluorescence life data is using LLF for underestimation with and MELF for overestimation with .

The Efficiency of Hazard Rate Preservation Method for Generating Discrete Rayleigh–Lindley Distribution

In this study, we introduce two novel discrete counterparts for the Rayleigh–Lindley mixture, constructed through the application of survival and hazard rate preservation techniques. These two-parameter discrete models demonstrate exceptional adaptability across various data types, including skewed, symmetric, and monotonic datasets. Statistical analyses were conducted using maximum likelihood estimation and Bayesian approaches to assess these models. The Bayesian analysis, in particular, was implemented with the squared error and LINEX loss functions, incorporating a modified Lwin Prior distribution for parameter estimation. Through simulation studies and numerical methods, we evaluated the estimators’ performance and compared the effectiveness of the two discrete adaptations of the Rayleigh–Lindley distribution. The simulations reveal that Bayesian methods are especially effective in this setting due to their flexibility and adaptability. They provide more precise and dependable estimates for the discrete Rayleigh–Lindley model, especially when using the hazard rate preservation method. This method is a compelling alternative to the traditional survival discretization approach, showcasing its significant potential in enhancing model accuracy and applicability. Furthermore, two real data sets are analyzed to assess the performance of each analog.

Exponentiated gamma constant-stress partially accelerated life tests with unified hybrid censored data: Statistical inferences

This study considers one of the partially accelerated life testing models, mainly a constant-stress approach with unified hybrid censored data. The data represents the lifetime of units following the exponentiated gamma distribution (EGD). The estimation techniques for the parameter of EGD and the acceleration factor are discussed, along with the maximum likelihood estimation technique, the Bayesian and E-Bayesian estimation. Utilizing the squared error loss (SEL) and LINEX loss functions, the E-Bayesian and Bayesian estimates are generated. For deriving the estimates of Bayesian and E-Bayesian approaches, we use the MCMC method. Real-world data is also provided for illustration. Furthermore, unified hybrid censoring schemes (HCSs) are examined to determine their effectiveness in the light of the results. In the end, we compare among the proposed methods during their findings. The study's findings demonstrate how well the suggested techniques work when utilizing unified HCS. The study also makes clear how important it is to adopt the CSPALT model, as it yields more failed things in a shorter amount of time than items with typical use situations. In conclusion, it is suggested that this research be used to tasks like industrial product reliability testing.

Investigation the generalized extreme value under liner distribution parameters for progressive type-Ⅱ censoring by using optimization algorithms

<abstract><p>Several random phenomena have been modeled by using extreme value distributions. Based on progressive type-Ⅱ censored data with three different distributions (i.e., fixed, discrete uniform, and binomial random removal), the statistical inference of the generalized extreme value distribution under liner normalization (GEVL distribution) parameters is investigated in this study. Since there is no analytical solution, determining the maximum likelihood parameters for the GEVL distribution is considered to be a problem. Standard numerical methods are frequently insufficient for this dilemma, requiring the use of artificial intelligence algorithms to address this difficulty. Here, nonlinear minimization and a genetic algorithm have been used to tackle that problem. In addition, Lindley approximation and Monte Carlo estimation were implemented via Metropolis-Hastings algorithms to carry out the Bayesian point estimation based on both the squared error loss function and LINEX loss functions. Moreover, the highest posterior density intervals were applied. The proposed theoretical inference techniques have been applied in a numerical simulation and a real-life example.</p></abstract>

Estimation of multicomponent stress–strength reliability for exponentiated Gumbel distribution

In this paper, the stress–strength reliability R s , k of a multicomponent s-out-of-k system for exponentiated Gumbel distribution is considered. An s-out-of-k system means a system with total k components and the system can survive only when atleast s of the total components function properly. The ability of the system to overcome the experiencing stress with its strength is termed as its stress–strengh reliability. The maximum likelihood estimator and Bayes estimator for R s , k are obtained. The Bayes estimators are obtained using Markov chain Monte Carlo(MCMC) method under both symmetric and asymmetric loss functions. The loss functions we considered are squared error loss function, LINEX loss function and entropy loss function. The asymptotic, bootstrap and highest posterior density(HPD) confidence intervals for R s , k are also obtained. A simulation study is conducted for evaluating the efficiency of the estimators derived in this paper. Real data sets are also considered for illustration.

Monitoring of semiconductor manufacturing process on Bayesian AEWMA control chart under paired ranked set sampling schemes

Quality control often employs memory-type control charts, including the exponentially weighted moving average (EWMA) and Shewhart control charts, to identify shifts in the location parameter of a process. This article pioneers a new Bayesian Adaptive EWMA (AEWMA) control chart, built on diverse loss functions (LFs) such as the square error loss function (SELF) and the Linex loss function (LLF). The proposed chart aims to enhance the process of identifying small to moderate as well as significant shifts in the mean, signifying a notable advancement in the field of quality control. These are implemented utilizing an informative prior for both posterior and posterior predictive distributions, employing various paired ranked set sampling (PRSS) schemes. The effectiveness of the suggested chart is appraised using average run length (ARL) and the standard deviation of run length (SDRL). Monte Carlo simulations are employed to contrast the recommended approach against other control charts. The outcomes demonstrate the dignitary performance of the recommended chart in identifying out-of-control signals, especially applying PRSS designs, in comparison to simple random sampling (SRS). Finally, a practical application was conducted in the semiconductor manufacturing context to appraise the efficacy of the offered chart using various paired ranked set sampling strategies. The results reveal that the suggested control chart performed well in capturing the out-of-control signals far better than the already in use control charts. Overall, this study interposes a new technique with diverse LFs and PRSS designs, improving the precision and effectiveness in detecting process mean shifts, thereby contributing to advancements in quality control and process monitoring.

Estimating the scale parameters of several exponential distributions under order restriction

. In the present work, we have investigated the problem of estimating the parameters of several exponential distributions with ordered scale parameters under linex loss function. The study includes the cases of known as well as unknown location parameters. For both cases, we have considered class of equivariant estimators. Then, sufficient conditions have been obtained under which the equivariant class of estimators improves upon the usual estimators. Using the established results, it has been shown that the restricted maximum likelihood estimator is inadmissible. Finally, for both cases, simulation study has been conducted to compare the risk performance of the proposed estimators.

Monitoring the process mean under the Bayesian approach with application to hard bake process

This study introduces the Bayesian adaptive exponentially weighted moving average (AEWMA) control chart within the framework of measurement error, examining two separate loss functions: the squared error loss function and the linex loss function. We conduct an analysis of the posterior and posterior predictive distributions utilizing a conjugate prior. In the presence of measurement error (ME), we employ a linear covariate model to assess the control chart's effectiveness. Additionally, we explore the impacts of measurement error by investigating multiple measurements and a method involving linearly increasing variance. We conduct a Monte Carlo simulation study to assess the control chart's performance under ME, examining its run length profile. Subsequently, we offer a specific numerical instance related to the hard-bake process in semiconductor manufacturing, serving to verify the functionality and practical application of the suggested Bayesian AEWMA control chart when confronted with ME.

Bayesian AEWMA control chart under ranked set sampling with application to reliability engineering

The article introduces a novel Bayesian AEWMA Control Chart that integrates different loss functions (LFs) like the square error loss function and Linex loss function under an informative prior for posterior and posterior predictive distributions, implemented across diverse ranked set sampling (RSS) designs. The main objective is to detect small to moderate shifts in the process mean, with the average run length and standard deviation of run length serving as performance measures. The study employs a hard bake process in semiconductor production to demonstrate the effectiveness of the proposed chart, comparing it with existing control charts through Monte Carlo simulations. The results underscore the superiority of the proposed approach, particularly under RSS designs compared to simple random sampling (SRS), in identifying out-of-control signals. Overall, this study contributes a comprehensive method integrating various LFs and RSS schemes, offering a more precise and efficient approach for detecting shifts in the process mean. Real-world applications highlight the heightened sensitivity of the suggested chart in identifying out-of-control signals compared to existing Bayesian charts using SRS.

Monitoring of manufacturing process using bayesian EWMA control chart under ranked based sampling designs

Control charts, including exponentially moving average (EWMA) , are valuable for efficiently detecting small to moderate shifts. This study introduces a Bayesian EWMA control chart that employs ranked set sampling (RSS) with known prior information and two distinct loss functions (LFs), the Square Error Loss function (SELF) and the Linex Loss function (LLF), for posterior and posterior predictive distributions. The chart's performance is assessed using average run length (ARL) and standard deviation of run length (SDRL) profiles, and it is compared to the Bayesian EWMA control chart based on simple random sampling (SRS). The results indicate that the proposed control chart detects small to moderate shifts more effectively. The application in semiconductor manufacturing provides concrete evidence that the Bayesian EWMA control chart, when implemented with RSS schemes, demonstrates a higher degree of sensitivity in detecting deviations from normal process behavior. Comparison to the Bayesian EWMA control chart using SRS, it exhibits a superior ability to identify and flag instances where the manufacturing process is going out of control. This heightened sensitivity is critical for promptly addressing and rectifying issues, which ultimately contributes to improved quality control in semiconductor production.

Bayesian Estimation and Prediction for Inverse Power Maxwell Distribution with Applications to Tax Revenue and Health Care Data

This article discusses the problem of estimating the unknown model parameters as well as prediction of future observations from inverse power Maxwell distribution. The maximum likelihood method is applied for estimating the model parameters using Newton-Rapson iterative procedures. The existence and uniqueness of maximum likelihood estimates are established using Cauchy-Schwartz inequality. Approximate confidence intervals are constructed using Fisher information matrix. Using independent gamma informative priors, the Bayes estimates of unknown model parameters are obtained under squared error and Linex loss functions. Two approximation techniques namely: Lindley’s approximation and Metropolis-Hastings within Gibbs sampler algorithm have been employed to derive the Bayes estimators and also to construct the associate highest posterior density credible intervals. Based on the informative (observed) sample, Bayesian prediction, predictive density, and predictive intervals are derived for future observation and decision. The performance of proposed methods are evaluated though a Monte Carlo simulation experiment. Two real-life datasets related to tax revenue and heath are incorporated to show the practical utility of proposed methodology in real phenomenon.

Bayesian and non-Bayesian inference for inverse Weibull model based on jointly type-II hybrid censoring samples with modeling to physics data

In recent years, there has been a lot of interest in the research of cooperative censoring schemes. In this work, we compared the relative benefits of two competing length-of-life products using inverse Weibull lifetime products with a joint type-II hybrid censoring scheme (JHC-Type II). We initially examined the maximum likelihood estimators and their confidence intervals (CIs) for the unknown parameters based on JHC-Type II. Then, under the premise of independent gamma priors, we offer Bayes estimates of the parameters using squared error loss and LINEX loss functions. We used the Markov chain Monte Carlo method to create credible intervals and Bayesian estimates. Based on the parametric bootstrapping techniques known as Boot-p and Boot-t, we create two bootstrapping CIs. In addition, we do a Monte Carlo simulation experiment to track how well the aforementioned approaches work and to determine the corresponding confidence and credible intervals. Finally, to show how the approaches covered in this paper might be used, we consider a real physical dataset.