Squared Error Loss Function Research Articles

Estimation and Optimal Censoring Plan for a New Unit Log-Log Model via Improved Adaptive Progressively Censored Data

To gather enough data from studies that are ongoing for an extended duration, a newly improved adaptive Type-II progressive censoring technique has been offered to get around this difficulty and extend several well-known multi-stage censoring plans. This work, which takes this scheme into account, focuses on some conventional and Bayesian estimation missions for parameter and reliability indicators, where the unit log-log model acts as the base distribution. The point and interval estimations of the various parameters are looked at from a classical standpoint. In addition to the conventional approach, the Bayesian methodology is examined to derive credible intervals beside the Bayesian point by leveraging the squared error loss function and the Markov chain Monte Carlo technique. Under varied settings, a simulation study is carried out to distinguish between the standard and Bayesian estimates. To implement the proposed procedures, two actual data sets are analyzed. Finally, multiple precision standards are considered to pick the optimal progressive censoring scheme.

Statistical inference of inverted Nadarajah–Haghighi distribution under type-II generalized hybrid censoring competing risks data

Tahir et al. (J Stat Comput Simul 88(14):2775–2798, 2018) introduced the inverse Nadarajah–Haghighi distribution (INHD) and demonstrated its ability to model positive real data sets with decreasing and upside-down bathtub hazard rate shapes. This article focuses on the inference of unknown parameters using a generalized Type-II hybrid censoring scheme (GT-II HCS) for the INHD in the presence of competing risks. The maximum likelihood (ML) and Bayes approaches are used to estimate the model parameters. Based on the squared error loss function, we compute Bayes estimates using Markov Chain Monte Carlo (MCMC) by applying Metropolis-Hasting (M-H) algorithm. Furthermore, the asymptotic confidence intervals, bootstrap confidence intervals (BCIs) and the highest posterior density (HPD) credible intervals are constructed. Using real data sets and simulation studies, we examined the introduced methods of inference with different sample sizes.

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.

Bayesian Approximation Methods of the Estimation for Generalized Exponential Distribution and Neutrosophic Approximation Methods

In this article, we used approximate methods to obtain Bayes method estimations for the shape and scale parameters of the generalized exponential distribution, as three approximation methods were employed: Lindley approximation and neutrosophic Lindley approximation, Gibbs sampling and neutrosophic Gibbs sampling, the most important samples based on the gamma informative prior under the squared error loss function. Through different simulation experiments a comparison was made between those estimators of these three approximate methods, from the simulation results we found a relative preference for the important sampling method over the other two methods. The results of simulation experiments were also confirmed by applying these approximate methods to real data representing the operating times of one of the machines of the publishing, printing, and translation house in Baghdad. On the other hand, we apply the same method to the neutrosophic exponential distribution, and the results will be compared to the classical case.

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>

Bayesian estimation of 3-component mixture geometric distribution under different loss functions

This article focuses on discrete survival data analysis, employing a doubly Type-I censoring scheme (DT1SC) under a Bayesian framework for parameters estimation in a 3-component mixture geometric (3-CMG) distribution. The non-informative (uniform) prior is utilized under squared error, DeGroot, and precautionary loss functions. The time is considered as discrete in this paper, presenting a departure from continuous approaches in survival analysis. The methodology is tailored to address challenges that traditional survival analysis encounters when faced with limited or missing information. The proposed method effectively handles the complexities arising from incomplete or missing data, providing a robust mechanism for accurate parameter estimation. Through extensive simulations and application to real-world datasets, our approach demonstrates its effectiveness in managing uncertainties, particularly in scenarios involving sparse or missing data. This work represents a noteworthy contribution to methodological advancements in survival analysis, offering a valuable tool for researchers and practitioners navigating intricate dynamics within the 3-CMG under DT1CS with time being considered as discrete.

Inference of stress-strength reliability based on adaptive progressive type-Ⅱ censing from Chen distribution with application to carbon fiber data

<abstract><p>In this paper, we used the maximum likelihood estimation (MLE) and the Bayes methods to perform estimation procedures for the reliability of stress-strength $ R = P(Y < X) $ based on independent adaptive progressive censored samples that were taken from the Chen distribution. An approximate confidence interval of $ R $ was constructed using a variety of classical techniques, such as the normal approximation of the MLE, the normal approximation of the log-transformed MLE, and the percentile bootstrap (Boot-p) procedure. Additionally, the asymptotic distribution theory and delta approach were used to generate the approximate confidence interval. Further, the Bayesian estimation of $ R $ was obtained based on the balanced loss function, which came in two versions here, the symmetric balanced squared error (BSE) loss function and the asymmetric balanced linear exponential (BLINEX) loss function. When estimating $ R $ using the Bayesian approach, all the unknown parameters of the Chen distribution were assumed to be independently distributed and to have informative gamma priors. Additionally, a mixture of Gibbs sampling algorithm and Metropolis-Hastings algorithm was used to compute the Bayes estimate of $ R $ and the associated highest posterior density credible interval. In the end, simulation research was used to assess the general overall performance of the proposed estimators and a real dataset was provided to exemplify the theoretical results.</p></abstract>

Bayesian and non-Bayesian two-sample prediction for the Fréchet distribution under progressive type II censoring

In many domains, including medical and technical sciences, prediction based on censored data is a very significant issue. On the other hand, the Fréchet distribution is a continuous model, which has wide applicability in extreme value theory, based on a progressively type-II right censoring scheme. Based on two-sample prediction, Bayesian and non-Bayesian predictions are discussed in this study for the Fréchet distribution. For obtaining prediction bounds as well as Bayes predictive estimations under a squared error loss function for the sth-order statistic in a future random sample drawn independently of the parent population, with an arbitrary progressive censoring scheme, a general form for the lifetime model as the Fréchet model is taken into consideration. As an example, a simulation study was done as examples to demonstrate how the operations are carried out. Moreover, two sets of real data are applied.

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>

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.

Estimation of a New Stress Strength Index for One Parameter Exponential Family

Stress-strength reliability is widely used in manufacturing industry for producing good quality equipment. A new stress-strength index has been introduced when strengths of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> independent components follow exponential distributions with different scale parameters. We obtain the maximum likelihood estimator, a uniformly minimum variance unbiased estimator, a Bayes estimator, and an analogue of the best scale equivariant estimator for the new index. It is shown that the Bayes estimators' limit is a generalized Bayes estimator under the squared error loss function. The asymptotic distribution of the MLE is derived using the multivariate delta method. A detailed comparison of the risk performance of all these estimators is done numerically. The results are illustrated by two real data analyses.

On estimation of P(Y < X) for inverse Pareto distribution based on progressively first failure censored data.

The stress-strength reliability (SSR) model ϕ = P(Y < X) is used in numerous disciplines like reliability engineering, quality control, medical studies, and many more to assess the strength and stresses of the systems. Here, we assume X and Y both are independent random variables of progressively first failure censored (PFFC) data following inverse Pareto distribution (IPD) as stress and strength, respectively. This article deals with the estimation of SSR from both classical and Bayesian paradigms. In the case of a classical point of view, the SSR is computed using two estimation methods: maximum product spacing (MPS) and maximum likelihood (ML) estimators. Also, derived interval estimates of SSR based on ML estimate. The Bayes estimate of SSR is computed using the Markov chain Monte Carlo (MCMC) approximation procedure with a squared error loss function (SELF) based on gamma informative priors for the Bayesian paradigm. To demonstrate the relevance of the different estimates and the censoring schemes, an extensive simulation study and two pairs of real-data applications are discussed.

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.

Stress–Strength Reliability Analysis for Different Distributions Using Progressive Type-II Censoring with Binomial Removal

In the statistical literature, one of the most important subjects that is commonly used is stress–strength reliability, which is defined as δ=PW&lt;V, where V and W are the strength and stress random variables, respectively, and δ is reliability parameter. Type-II progressive censoring with binomial removal is used in this study to examine the inference of δ=PW&lt;V for a component with strength V and being subjected to stress W. We suppose that V and W are independent random variables taken from the Burr XII distribution and the Burr III distribution, respectively, with a common shape parameter. The maximum likelihood estimator of δ is derived. The Bayes estimator of δ under the assumption of independent gamma priors is derived. To determine the Bayes estimates for squared error and linear exponential loss functions in the lack of explicit forms, the Metropolis–Hastings method was provided. Utilizing comprehensive simulations and two metrics (average of estimates and root mean squared errors), we compare these estimators. Further, an analysis is performed on two actual data sets based on breakdown times for insulating fluid between electrodes recorded under varying voltages.

Estimation of a Shape Parameter of a Gompertz-lindley Distribution Using Bayesian and Maximum Likelihood Methods

The Gompertz-Lindley distribution is an extension of the Lindley distribution with three parameters. It was found to be more flexible for modeling real life events. The distribution contains two shape parameters and a scale parameter. Despite the necessity of parameter estimation theory in modeling, it has not been shown that a method of estimation method is better for any of these three parameters of the Gompertz-Lindley distribution. This paper identifies the best estimation method for the shape parameter of the Gompertz-Lindley distribution by deriving Bayesian estimators for the shape parameter of the distribution using two non-informative prior distributions (Uniform and Jeffery) and an informative prior (gamma) under squared error loss function (SELF), quadratic loss function (QLF) and precautionary loss function (PLF). These estimators were evaluated and the results compared with the maximum likelihood estimation method using Monte Carlo simulations with the mean square error (MSE) as a criterion for choosing the best estimator.

Bayesian and E-Bayesian estimation based on constant-stress partially accelerated life testing for inverted Topp–Leone distribution

Abstract Accelerated or partially accelerated life tests are particularly significant in life testing experiments since they save time and cost. Partially accelerated life tests are carried out when the data from accelerated life testing cannot be extrapolated to usual conditions. The constant-stress partially accelerated life test is proposed in this study based on a Type-II censoring scheme and supposing that the lifetimes of units at usual conditions follow the inverted Topp–Leone distribution. The Bayes and E-Bayes estimators of the distribution parameter and the acceleration factor are derived. The balanced squared error loss function, which is a symmetric loss function, and the balanced linear exponential loss function, which is an asymmetric loss function, are considered for obtaining the Bayes and E-Bayes estimators. Based on informative gamma priors and uniform hyper-prior distributions, the estimators are obtained. Finally, the performance of the proposed Bayes and E-Bayes estimates is evaluated through a simulation study and an application using real datasets.

Estimation of stress-strength reliability for generalized Gompertz distribution under progressive type-II censoring

In this study, the stress-strength reliability, $R=P(Y

Statistical inference in Burr type XII lifetime model based on progressive randomly censored data

Censoring commonly occurs in real-world scenarios, either intentionally or unintentionally. Unintentional (or accidental) censoring usually happens randomly, i.e., it is out of the experimenters’ control, such as broken equipment, lack of follow-up, etc. Experimenters typically use intentional censoring to save experimental time and cost. In this article, we develop frequentist and Bayesian statistical inferential procedures for the parameters and reliability characteristics of the Burr type XII lifetime model under the Koziol-Green model based on progressive randomly censored data. For the frequentist approach, maximum likelihood methods for point and interval estimation are developed. For the Bayesian approach, the Bayes estimates under the squared error loss function (SELF) are evaluated using the Markov Chain Monte Carlo (MCMC) and Tierney-Kadane (T-K) approximation techniques. A Monte Carlo simulation study is used to assess the performance of the proposed estimation procedures. A real data analysis is performed to illustrate the proposed methods. Moreover, obtaining progressive censoring schemes for experimental planning purposes is also discussed.

A mathematical model predicting the abundance and spatio-temporal dispersal of adfluvial salmonid (Salvelinus malma) fry in a stream

Models describing the spatial and temporal distributions of fishes are necessary for conservation and management of populations and their habitats. The early life stages of fishes are a particularly important period that can influence population dynamics and fisheries productivity. The fry of riverine migratory fish species are particularly influenced by downstream transport although relatively few models have been used to predict spatio-temporal abundance and dispersion of fry in combination with life history parameters in unperturbed habitats. We present a novel model predicting the downstream dispersal of age-0 adfluvial Dolly Varden in a spawning stream draining into Kronotskoye Lake basin, Kamchatka Peninsula, Russia. The model accounted for the number of and distance between known spawning grounds, diel activity, daily fish growth, and intra-specific diversity (i.e., morphotype) to describe how fry abundance changed at a specific stream location over the course of spring and summer. Data were collected using a trapnet monitored periodically during spring and summer to record changes in abundance, size (growth), and life history composition (‘nosed’ or ‘white’ morphotype) of fish between age-0 and -2. The experimental data were fitted to the model using a gradient descent optimization algorithm and a mean squared error loss function was used to verify how well experimental data fit to the theoretical model. The predicted abundance of age-0 fish was 18,831 with a mean dispersion rate of ∼36 m/day and a transit time between spawning area and stream mouth ranging 32–76 days. The catch was mainly comprised (81%) of the slower growing ‘nosed’ morphotype and exhibited variation in abundance over time and decreased considerably by the end of summer. Our comprehensive model will be useful to simulate the downstream dispersal of salmonid fishes whose early life stages inhabit riverine systems.

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.