Exponentiated Rayleigh Distribution Research Articles

Likelihood and pivotal-based reliability inference for inverse exponentiated Rayleigh distribution under block progressive censoring

ABSTRACT In this paper, focus is on the estimation of survival characteristics, specifically the reliability function, hazard rate function, median time to failure, and differences in different test facilities, using block progressive censored data. Estimations are carried out through both maximum likelihood and pivotal methods, assuming the lifetime distribution of test units follows an inverse exponentiated Rayleigh distribution. The paper derives maximum likelihood estimators for unknown parameters, exploring their existence and uniqueness properties. Approximate confidence intervals for survival characteristics are constructed using the delta method and likelihood theory. Moreover, point and generalized confidence interval estimators are developed through a pivotal quantity-based method. A simulation study is conducted to compare the performance of the proposed approaches, revealing that the pivotal quantity-based approach yields superior estimation results. Finally, the proposed estimates are applied to analyze two real datasets, illustrating their practical applicability.

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.

Reliability inference based on inverted exponentiated Rayleigh lifetime distribution under unified hybrid censored scheme

In this study, we investigated the Inverted Exponentiated Rayleigh Distribution (IERD), a significant and efficient continuous lifetime distribution commonly applied in lifespan research. Our focus was on estimating unknown parameters for a two-parameter inverted exponentiated Rayleigh distribution using unified hybrid censored data. We considered both maximum likelihood and Bayesian estimation approaches. Specifically, we employed the Gibbs within Metropolis–Hastings samplers method to develop approximate Bayes estimators utilizing informative and non-informative priors, along with symmetric and asymmetric loss functions. In addition, we utilized Markov chain Monte Carlo (MCMC) samples to derive maximum posterior density credible intervals. Simulation experiments were conducted to assess the efficacy of the proposed methodologies, and actual data analysis was performed to validate the proposed estimators.

Utilizing Empirical Bayes Estimation to Assess Reliability in Inverted Exponentiated Rayleigh Distribution with Progressive Hybrid Censored Medical Data

This study addresses the issue of estimating the shape parameter of the inverted exponentiated Rayleigh distribution, along with the assessment of reliability and failure rate, by utilizing Type-I progressive hybrid censored data. The study explores the estimators based on maximum likelihood, Bayes, and empirical Bayes methodologies. Additionally, the study focuses on the development of Bayes and empirical Bayes estimators with balanced loss functions. A concrete example based on actual data from the field of medicine is used to illustrate the theoretical insights provided in this study. Monte Carlo simulations are employed to conduct numerical comparisons and evaluate the performance and accuracy of the estimation methods.

Inference and optimal censoring scheme for a competing-risks model with type-II progressive censoring

ABSTRACT For statistical inference of competing risks under type-II progressive censoring, lifetimes are modeled by an inverted exponential Rayleigh distribution, which allows the use of a non-monotonic hazard function. Maximum-likelihood estimators for all parameters exist and are unique. The Newton-Raphson algorithm and maximum stochastic expectation each provide estimates. Confidence intervals result from the Fisher matrix and the asymptotic normality of maximum-likelihood estimators. For small samples, the Bootstrap estimators of the parameters do not need to be asymptotically normal. In addition, the Monte Carlo method with the Metropolis-Hastings algorithm and importance sampling allow for Bayesian estimation, with the associated highest posterior-density intervals. The Bayesian method takes account of prior information, contrary to the frequentist method. The Bootstrap method improves the precision of the estimation, especially in the case of small sample sizes. The estimated range obtained by Bootstrap is between 20% and 60% smaller than that obtained by maximum likelihood. Frequentist and Bayesian estimations using the inverted exponentiated Rayleigh distribution under type-II progressive censoring allow for fitting empirical mouse mortality data and obtaining parameter estimates of this distribution. A quantile-dependent criterion and a quantile-independent criterion are used to determine the optimal censoring and to design the experiment.

Estimation and prediction of modified progressive hybrid censored data from inverted exponentiated Rayleigh distribution

ABSTRACT In this paper, statistical inference of an inverted exponentiated Rayleigh model is studied when the failure times are obtained under a modified progressive hybrid censoring. The maximum likelihood estimators of the model parameters together with associated existence and uniqueness are established, and approximate confidence intervals are constructed based on asymptotic theory. Alternatively, generalized point and interval estimates for unknown parameters are also constructed based on the proposed pivotal quantities for comparison. In addition, predictive intervals of remaining useful life from the inverted exponentiated Rayleigh distribution are also constructed under classical and generalized inferential approaches, respectively. Finally, extensive simulation studies are carried out to compare the performance of the proposed methods, and two real-life examples are analyzed for illustration. The numerical results indicate that both traditional likelihood and generalized inferential methods work satisfactorily, and that our proposed generalized approach appears much more appealing and are superior to classical results.

Stress-strength reliability estimation for the inverted exponentiated Rayleigh distribution under unified progressive hybrid censoring with application

<abstract> <p>In this paper, we studied the estimation of a stress-strength reliability model ($R = P(X>Y)$) based on inverted exponentiated Rayleigh distribution under the unified progressive hybrid censoring scheme (unified PHCS). The maximum likelihood estimates of the unknown parameters were obtained using the stochastic expectation-maximization algorithm (stochastic EMA). The asymptotic confidence intervals were also created. Under squared error and Linex and generalized entropy loss functions, the Gibbs sampler together with Metropolis-Hastings algorithm was provided to compute the Bayes estimates and the credible intervals. Extensive simulations were performed to see the effectiveness of the proposed estimation methods. Also, parallel to the development of reliability studies, it is necessary to study its application in different sciences such as engineering. Therefore, droplet splashing data under two nozzle pressures were proposed to exemplify the theoretical outcomes.</p> </abstract>

Some Bayes estimators for Exponentiated Rayleigh Distribution under Doubly Type II Censored Data

Bayesian analysis for the shape parameter of Exponentiated Rayleigh distribution or (Burr type x) has been considered in this paper, under two priors distributions [Inverted Levy and Quasi] using doubly type II censored data, with two loss functions [Weighted and Precautionary], we conducted a simulation study that evaluate the effectiveness of four estimators by six presumption experiments for varied values of distribution parameters and three sample sizes using mean squared error [MSE]. In general the best estimator of the shape parameter of the (ERD) using Doubly type II censored data is under Inverted Levy prior under Precautionary loss function in all sample sizes

Reliability analysis for stress–strength model from inverted exponentiated Rayleigh based on the hybrid censored sample

PurposeThe study based on the estimation of the stress–strength reliability parameter plays a vital role in showing system efficiency. In this paper, considering independent strength and stress random variables distributed as inverted exponentiated Rayleigh model, the author have developed estimation procedures for the stress–strength reliability parameter R = P(X>Y) under Type II hybrid censored samples.Design/methodology/approachThe maximum likelihood and Bayesian estimates of R based on Type II hybrid censored samples are evaluated. Because there is no closed form for the Bayes estimate, the author use the Metropolis–Hastings algorithm to obtain approximate Bayes estimate of the reliability parameter. Furthermore, the author construct the asymptotic confidence interval, bootstrap confidence interval and highest posterior density (HPD) credible interval for R. The Monte Carlo simulation study has been conducted to compare the performance of various proposed point and interval estimators. Finally, the validity of the stress–strength reliability model is demonstrated via a practical case.FindingsThe performance of various point and interval estimators is compared via the simulation study. Among all proposed estimators, Bayes estimators using MHG algorithm show minimum MSE for all considered censoring schemes. Furthermore, the real data analysis indicates that the splashing diameter decreases with the increase of MPa under different hybrid censored samples.Originality/valueThe frequentist and Bayesian methods are developed to estimate the associated parameters of the reliability model under the hybrid censored inverted exponentiated Rayleigh distribution. The application of the proposed stress–strength reliability model will help the reliability engineers and also other scientists to estimate the system reliability.

New Mathematical Properties For Rayleigh distribution

Regression analysis is one of the most commonly statistical techniques used for analyzing data in different fields. And used to fit the relation between the dependent variable and the independent variables require strong assumption to be met in the model. Generalized linear models (GLMs) allow the extension of linear modeling ideas to a wider class of response types, such as count data or binary responses. Many statistical methods exist for such data types, but the advantage of the GLM approach is that it unites a seemingly disparate collection of response types under a common modeling methodology. So, the problem of the current research is to try to provide a new mathematical property for Exponentiated Rayleigh distribution, and it was one of the most important properties that was studied is to determine Harmonic Mean, as well as calculating the Quantile function, Moments of Residual life (MRL), Reversed Residual Life, Mean of Residual life. The study also presented the probability density function (pdf) and cumulative distribution function according to linear representations.

Scheming of reliability sampling plans with some lifetime distributions based on percentiles estimation using intervened Poisson characteristics

In industry, the practice of determining the performance of a product by intervening with the normal working conditions of the manufacturing process is an acceptable method. When reliability is a key quality factor, the reliability single sampling plan is generally used to optimise sample size and estimate time characteristics. The application of Intervened Poisson Distribution (IPD) and some of the lifetime distributions are used in this article to implement a reliability sampling plan. The IPD can be used as a model for counting the defective items in an appropriate situation and the defectiveness will be determined with the help of lifetime distributions. The minimum sample size required to accept a lot and the estimation of the true percentile time of the accepted lots according to the magnitude of the intervention parameter can be found using IPD associated with the Rayleigh Distribution (RD), Exponentiated Rayleigh Distribution (ERD) and Inverse Rayleigh Distribution (IRD). The minimum sample size is necessary to ensure the specified life percentile is obtained under a consumer risk. The operating characteristic (OC) values and estimated percentile time of an accepted lot are presented.

A new versatile modification of the Rayleigh distribution for modeling COVID-19 mortality rates

The aim of this paper is to specify a new flexible statistical model to analyze COVID-19 mortality rates in Italy and Canada. A new versatile lifetime distribution with four parameters is proposed by using the exponentiated generalized class of distributions and the gull alpha power Rayleigh distribution to form the exponentiated generalized gull alpha power Rayleigh (EGGAPR) distribution. This new distribution is characterized by a tractable cumulative distribution function. To estimate the unknown parameters of the proposed distribution the maximum likelihood estimation method is used. In evaluating the effectiveness of the MLE method graphical displays of the Monte Carlo simulation are presented. The EGGAPR distribution is compared to its sub-models which include the exponentiated gull alpha Rayleigh distribution, the gull alpha Rayleigh distribution, exponentiated generalized Rayleigh distribution, exponentiated Rayleigh distribution and the Rayleigh distribution. Different measures of goodness-of-fit are used to investigate whether the EGGAPR distribution is more flexible and fit than its sub-models in modeling COVID-19 mortality rates.

Statistical Inference of Inverted Exponentiated Rayleigh Distribution under Joint Progressively Type-II Censoring.

Inverted exponentiated Rayleigh distribution is a widely used and important continuous lifetime distribution, which plays a key role in lifetime research. The joint progressively type-II censoring scheme is an effective method used in the quality evaluation of products from different assembly lines. In this paper, we study the statistical inference of inverted exponentiated Rayleigh distribution based on joint progressively type-II censored data. The likelihood function and maximum likelihood estimates are obtained firstly by adopting Expectation-Maximization algorithm. Then, we calculate the observed information matrix based on the missing value principle. Bootstrap-p and Bootstrap-t methods are applied to get confidence intervals. Bayesian approaches under square loss function and linex loss function are provided respectively to derive the estimates, during which the importance sampling method is introduced. Finally, the Monte Carlo simulation and real data analysis are performed for further study.

ON THE PROPERTIES AND APPLICATIONS OF A NEW EXTENSION OF EXPONENTIATED RAYLEIGH DISTRIBUTION

Statistical distributions already in existence are not the most appropriate model that adequately describes real-life data such as those obtained from experimental investigations. Therefore, there are needs to come up with their extended forms to give substitutive adaptable models. By adopting the method of Transformed-Transformer family of distributions, an extension of Exponentiated Rayleigh distribution titled Gompertz- Exponentiated Rayleigh (GOM-ER) distribution was proposed and proved to be valid. Some properties of the new distribution including random number generator, quartiles, distribution of smallest and largest order statistics, reliability function, hazard rate function, cumulative or integrated hazard function, odds function, non-central moments, moment generating function, mean, variance and entropy measures were derived. Using the methods of maximum likelihood and maximum product of spacing, the four unknown parameters were estimated. Shapes of the hazard function depicts that GOM-ER is a distribution that is strictly increasing while those of the PDF depicts that GOM-ER can be skewed or symmetrical. Two datasets were fitted to determine the flexibility of GOM-ER. Simulation study evaluates the consistency, accuracy and unbiasedness of the GOM-ER parameter estimates obtained from the two frequentist estimation methods adopted.

Estimation of reliability in a multicomponent stress-strength model for inverted exponentiated Rayleigh distribution under progressive censoring

We consider estimation of the multicomponent stress-strength reliability for inverted exponentiated Rayleigh distributions under progressive Type II censoring. It is assumed that stress and strength variables follow inverted exponentiated Rayleigh distributions with a common scale parameter. Point and interval estimates of the reliability are obtained using maximum likelihood and Bayesian approaches when common parameter is unknown. Bayes estimates are derived using Lindley approximation and Markov chain Monte Carlo methods. The case of known common parameter is also considered. Then uniformly minimum variance unbiased estimator of the reliability is derived. We have also computed the exact Bayes estimates under the squared error loss function. The asymptotic and HPD intervals of the reliability are constructed under this case also. Proposed methods are compared numerically using simulations and comments are obtained. Finally, a real data set is analyzed for illustration purposes.

Pivotal Inference for the Inverted Exponentiated Rayleigh Distribution Based on Progressive Type-II Censored Data

In this article, the pivotal inference is proposed to estimate the two unknown parameters of the inverse exponentiated Rayleigh distribution based on progressive censored data. We derive the point estimator and construct an interval estimator using the pivotal quantity method. To compare the performance of this proposed method and the traditional maximum likelihood estimation method, a simulation study is conducted. The simulation results show that the proposed method performs better in terms of bias and mean squared error. Finally, a real dataset is used to illustrate the proposed approaches.

Skip-lot sampling plan of type SkSP-T for life testes based on percentiles of exponentiated Rayleigh distribution

This paper developed a scheming procedure of skip-lot sampling plan of type SkSP-T with single sampling plan for life tests based on the median percentiles of exponentiated Rayleigh distribution and applied it to determine the lifetime of any product. This procedure finds minimum sample size at required producer and consumer quality levels. In this study, life testing plans based on 50th percentile (median) is measured. Tables are simulated for various parametric values of SkSP-T and exponentiated Rayleigh distribution.

Estimation for a simple step–stress model with type–II hybrid censored data from the exponentiated Rayleigh distribution

Estimation for a simple step–stress model with type–II hybrid censored data from the exponentiated Rayleigh distribution

Relations for Moments of Dual Generalized Order Statistics from Exponentiated Rayleigh Distribution and Associated Inference

In this paper we obtain exact expressions and some recurrence relations satisfied by single and product moments of dual generalized order statistics from exponentiated Rayleigh distribution. These relations are deduced for moments of order statistics and lower record values. Further, conditional expectation, recurrence relations for single moments and truncated moment are used to characterize this distribution.

Evaluating the lifetime performance index with the inverted exponentiated Rayleigh distribution based on censored sample

Lifetime performance index is a powerful and effective way to analyze whether a product achieves the specified standards. In this paper, we investigate the lifetime performance index for the inverted exponential Rayleigh distribution using progressive type II censored sample data. The censored sample is able to greatly save the cost of the experiment and speed up the experiment. We derive the estimation value of lifetime performance index using the maximum likelihood method, and conduct the hypothesis test. Based on extensive numerical simulation, the power function is utilized to assess effectiveness of hypothesis testing. The simulation results show that lifetime performance index is good for determining whether the lifetime of the product reaches the criterion. Finally, a practical dataset is provided to give a demonstration for the procedures of lifetime performance evaluation.