Search

The record values data have received the attention of researchers in statistics for over seven decades. Through these decades the records have played a significant and widely utilized role for statistical inference in parameter estimation, predicting future values, hypothesis tests, as well as stressstrength tests, and characterizing distributions. In this paper, the types of record values, some distributional properties, and statistical inferences of record values and their applications are reviewed. The purpose of this paper is to shed light on the role of record values in statistical inference. Therefore, we will examine this issue from two perspectives, the first perspective being estimation and the second perspective being prediction. These are through some of the most important lifetime distributions are Exponential, Weibull, Gumbel, Geometric, Pareto, Generalized exponential, Rayleigh, Lomax, and Nadarajah-Haghighi distributions. I hope that the findings of this paper will be useful for researchers in various fields and lead to further enhancement of research in record values theory and its applications. [ABSTRACT FROM AUTHOR]

In this paper, we focus on Bayesian estimation of the parameters in the geometric process (GP) in which the first occurrence time of an event is assumed to have Weibull distribution. The Bayesian estimators are derived based on both symmetric (Squared Error) and asymmetric (General Entropy, LINEX) loss functions. Since the Bayesian estimators of unknown parameters cannot be obtained analytically, Lindley's approximation and the Markov Chain Monte Carlo (MCMC) methods are applied to compute the Bayesian estimates. Furthermore, by using the MCMC methods, credible intervals of the parameters are constructed. Maximum likelihood (ML) estimators are also derived for unknown parameters. The confidence intervals of the parameters are obtained based on an asymptotic distribution of ML estimators. Moreover, the performances of the proposed Bayesian estimators are compared with the corresponding ML, modified moment and modified maximum likelihood estimators through an extensive simulation study. Finally, analyses of two different real data sets are presented for illustrative purposes. [ABSTRACT FROM AUTHOR]

The competing risks model plays an important role in the statistical analysis of engineering, econometric, biological and other fields. A tested product may fail from different causes. Therefore, these failure causes as if competing with each other to bring about the failure of tested products (an experimental unit). Hence, in the statistical literature, this is known as the competing risk problem and it has been studied quite extensively by several researchers when the lifetime of the product is the latent failure time of the first failure cause among all the possible failure causes. In this paper, a competing risks model based on a generalized progressive hybrid type-I (GPH type-I) censoring is considered when the latent lifetime distributions of failure causes are inverse Weibull (IW) distributed and partially observed. We established various estimation methods including Maximum likelihood estimates (MLEs) and Bayes estimates (BEs). MLEs with the corresponding asymptotic confidence intervals are obtained. Bayes estimates of the parameters are obtained based on squared error loss (SEL) function under the assumption of independent gamma priors. Furthermore, we applied Markov Chain Monte Carlo (MCMC) techniques to compute BEs and to calculate the credible intervals. Finally, simulation studies and real data set are used for illustrative purpose. [ABSTRACT FROM AUTHOR]

In this paper, a general exponential form of the underlying distribution and a general conjugate prior are used to discuss the maximum likelihood and Bayesian estimation based on an adaptive progressive censored sample. A general procedure for deriving the point and interval Bayesian prediction of the future progressive censored from the same sample as well as that from an unobserved future sample is also developed. The Weibull, Pareto, and Burr Type-XII distributions are then used as illustrative examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here. [ABSTRACT FROM AUTHOR]

There are many situations in reliability or medical studies, where items may fail due to one of several causes. Further, censoring is unavoidable in any life testing. In this paper attempts to estimate problem for the competing risks model where the data are adaptive type-II progressive hybrid censored scheme and follow the inverse Weibull distribution. In this regards, we obtain the maximum likelihood estimation of the parameters and the asymptotic confidence intervals for the unknown parameters. Further, Bayes estimates of the parameters which obtained based on squared error and LINEX loss functions under the assumptions of independent gamma priors. For Bayesian estimation, we take advantage of Markov Chain Monte Carlo techniques by applying the Metropolis-Hasting algorithm under the square error and LINEX loss function to derive Bayesian estimators. Finally, two data sets with a Monte Carlo simulation study and a real data set of the electronic applications are analyzed for illustrative purposes. [ABSTRACT FROM AUTHOR]

This paper deals with the classical and Bayesian estimations of step-stress partially accelerated life test model under type-I censoring for the inverse Weibull lifetime distribution. In classical estimation, the maximum likelihood estimates of the distribution parameters and the acceleration factor were obtained. In addition, approximate confidence intervals of the parameters were constructed based on the asymptotic distribution of the maximum likelihood estimators. Under Bayesian inference, besides the Lindley and Tierney–Kadane approximation posterior expectation methods, which yielded point estimates of the distribution parameters and the acceleration factors under square error loss function, we also applied the Gibbs sampling method, in order to construct credible intervals of these parameters together with their point estimates. Finally, Monte Carlo simulations were conducted to compare the performances of the above estimation methods. [ABSTRACT FROM AUTHOR]

In this paper, based on progressively Type-II censored samples, the problem of estimation of multicomponent stress–strength reliability under generalized half-normal (GHN) distribution is considered. The reliability of a k-component stress-strength system is estimated when both stress and strength variates are assumed to have a GHN distribution with various cases of same and different shape and scale parameters. Different methods such as the maximum likelihood estimates (MLEs) and Bayes estimation are discussed. The expectation maximization algorithm and approximate maximum likelihood methods are proposed to compute the MLE of reliability. The Lindley's approximation method, as well as Metropolis–Hastings algorithm, are applied to compute Bayes estimates. The performance of the proposed procedures is also demonstrated via a Monte Carlo simulation study and an illustrative example. [ABSTRACT FROM AUTHOR]

In this paper, we consider the estimation of the multicomponent reliability by assuming the inverted exponentiated Rayleigh distribution. Both stress and strength are assumed to have an inverted exponentiated Rayleigh distribution with common scale parameter. The random variable representing the stress experienced by the system and representing the strength of system available to overcome the stress. The system works flawlessly only if at least out of strength variables exceed the random stress. The multicomponent reliability of the system is given by. We estimate by using frequentist and Bayesian approaches. Bayes estimates of have been obtained by using Markov Chain Monte Carlo methods since joint posteriors of the parameters does not have the explicit forms. We also construct asymptotic and highest probability density credible intervals for. The behavior of the proposed estimators is studied on the basis of estimated risks through Monte Carlo simulations. Finally, a data set is analyzed for illustrative purposes. Abbreviations: PDF: Probability Density Function; CDF: Cumulative Density Function; IER: Inverted Exponentiated Rayleigh; MLE: Maximum likelihood estimators; HPD: Highest Posterior Density; UBT: Upside down Bathtub; HNC: Head and Neck Cancer data; MCMC: Monte-Carlo Markov Chains; CP: Coverage Probability; KS: Kolmogorov - Smirnov. [ABSTRACT FROM AUTHOR]

In this paper, a study of the stress-strength reliability model is introduced subject to the exponentiated power generalized Weibull distribution. The maximum likelihood estimator for the stress-strength reliability function is deduced. The asymptotic confidence interval for the stressstrength reliability function is derived. The fuzzy stressstrength reliability function is discussed using the triangular membership function. A Bayesian estimator for the stressstrength reliability function is deduced. A real data application is introduced to show the results for the stress-strength model based on real data and compare the use of exponentiated power generalized Weibull distribution with existing distributions. [ABSTRACT FROM AUTHOR]

In this paper, a competing risks model is considered under adaptive type-I progressive hybrid censoring scheme (AT-I PHCS). The lifetimes of the latent failure times have Weibull distributions with the same shape parameter. We investigate the maximum likelihood estimation of the parameters. Bayes estimates of the parameters are obtained based on squared error and LINEX loss functions under the assumption of independent gamma priors. We propose to apply Markov Chain Monte Carlo (MCMC) techniques to carry out a Bayesian estimation procedure and in turn calculate the credible intervals. To evaluate the performance of the estimators, a simulation study is carried out. [ABSTRACT FROM AUTHOR]

Products come from different lines with the same facility are tested under comparative life tests which known with the jointly censoring scheme. In this paper, two sets of products under the same facility have Weibull lifetime distributions are selected to test under Type- I generalized hybrid censoring scheme (GHCS). The observed censoring data are used to build the maximum likelihood (ML) estimators as well as approximate confidence intervals for the model parameters. Also, Bayes estimators with the help of MCMC methods are discussed. The analysis of simulated data set with Monte Carlo simulation study is used to illustrate and compare the theoretical results. Finally, a brief comment is summarized in concluding section. [ABSTRACT FROM AUTHOR]

In this paper, maximum likelihood and Bayesian methods of estimation are used to estimate the unknown parameters of two Weibull populations with the same shape parameter under joint progressive Type-I (JPT-I) censoring scheme. Bayes estimates of the parameters are obtained based on squared error and LINEX loss functions under the assumption of independent gamma priors. We propose to apply Markov Chain Monte Carlo (MCMC) technique to carry out a Bayesian estimation procedure. The approximate confidence intervals and the credible intervals for the unknown parameters are also obtained. Finally, we analyze a one real data set for illustration purpose. [ABSTRACT FROM AUTHOR]

Joint Type-I progressive hybrid censoring scheme has been proposed to terminate the life-test experiment at maximum time that the experimenter can afford to continue. This article deals with the problem of estimating the two Weibull population parameters with the same shape parameter under joint Type-I progressively hybrid censoring scheme on the two samples using maximum likelihood and Bayesian inferential approaches. Using Fisher information matrix, the two-sided approximate confidence intervals of the unknown quantities are constructed. Under the assumption of independent gamma priors, the Bayes estimators are developed using squared-error loss function. Since the Bayes estimators cannot be expressed in closed forms, hence, Gibbs within Metropolis-Hastings algorithm is proposed to carry out the Bayes estimates and also to construct the corresponding credible intervals. Moreover, some popular joint censoring plans are generalized and can be obtained as a special cases from our results. Monte Carlo simulations are performed to assess the performance of the proposed estimators. To determine the optimal progressive censoring plan, two different optimality criteria are considered. Finally, to show the applicability of the proposed methods in real phenomenon, a real-life data set is analyzed. [ABSTRACT FROM AUTHOR]

Based on record statistics from three-parameter modified Weibull distribution, we consider the problem of estimating the unknown parameters using Bayesian and non-Bayesian approaches. Under a continuous-discrete joint prior distribution, Bayesian estimators and confidence intervals for the shape and scale parameters involved in the underlying model are obtained. In addition, maximum likelihood prediction and Bayesian prediction (either point or interval) of future record statistics based on an informative set of records are developed. Data analyses involving records extracted from a machine used to measure burr and times to breakdown of an insulating fluid between electrodes have been performed. Finally, Monte Carlo simulations are performed to compare the methods developed here. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to suggest and study a new exponentiated general class (EGC) of distributions. Maximum likelihood, Bayesian and empirical Bayesian estimators of the parameter of the EGC of distributions based on lower record values are obtained. Furthermore, Bayesian prediction of future records is considered. Based on lower record values, the exponentiated Weibull distribution, its special cases of distributions and exponentiated Gompertz distribution are applied to the EGC of distributions. [ABSTRACT FROM AUTHOR]

This paper describes the frequentist and Bayesian estimation for the scale parameter λ and shape parameter β of the inverse Weibull (IW) distribution based on adaptive type-II progressive hybrid censoring scheme (AT-II PHCS). We discuss the maximum likelihood estimators (MLEs) and the approximate MLEs, where the MLEs cannot be obtained in closed forms. The Bayes estimates for the IW parameters are obtained based on squared error (SE) loss function by using the approximation form of Lindley (1980). The optimal censoring scheme has been suggested using two different optimality criteria. A real life data set is used for illustration purpose. Finally, the different proposed estimators have been compared through an extensive simulation studies. [ABSTRACT FROM AUTHOR]

In this article, a new modification of the Weibull model with three parameters, the new exponential Weibull distribution (E-WD), is defined. The new model has many statistical advantages, the heavy-tailed behavior and the regular variation property were offered. Many of the important statistical functions of the modified model are presented in closed forms. The flexibility of E-WD has been improved. The proposed model can be used to fit data with different shapes, it can be right-skewed, left-skewed, decreasing, curved and symmetric. Some distribution properties of the proposed model, including moment generating function, characteristic function, moment, quantile and identifiability property, have been derived. In addition to the information generating function, the Shannon entropy and information energy are also discussed. The maximum likelihood approach and Bayesian estimation are used to estimate the distribution parameters. In the Bayesian method, three different loss functions are used. The calculations show the biases and estimated risks to obtain the best estimator. The bootstrap confidence intervals, the asymptotic confidence intervals and the observed variance-covariance matrix are obtained. Metropolis Hastings' MCMC procedure is used for the calculations. We apply the composite distribution to stock data for four variables. The goodness-of-fit results show that the model performs well compared to its competitors. The proposed model can be used for forecasting and decision making. [ABSTRACT FROM AUTHOR]

In this article, we present a new statistical modification of the Weibull model for updating the flexibility, called the generalized Weibull-Weibull distribution. The new modification of the Weibull model is defined and studied in detail. Some mathematical and statistical functions are studied, such as the quantile function, moments, the information generating measure, the Shannon entropy and the information energy. The joint distribution functions of the two marginal univariate models via the Copula model are provided. The unknown parameters are estimated using the maximum likelihood method and Bayesian method via Monte Carlo simulations. The Bayesian approach is discussed using three different loss functions: the quadratic error loss function, the LINEX loss function, and the general entropy loss function. We perform some numerical simulations to show how interesting the theoretical results are. Finally, the practical application of the proposed model is illustrated by analyzing two applications in the actuarial and engineering fields using corporate data to show the elasticity and advantage of the proposed generalized Weibull-Weibull model. The practical applications show that proposed model is very suitable for modeling actuarial and technical data sets and other related fields. [ABSTRACT FROM AUTHOR]

A new life performance index is proposed for evaluating the quality of lifetime products. The maximum likelihood estimation method and the Bayesian approaches using informative and non-informative prior distributions are utilized to infer the parameters of the Weibull distribution and the proposed new life performance index under a Type-I hybrid censoring scheme. Monte Carlo simulation results show that two Bayesian approaches outperform the maximum likelihood estimation method in terms of the measures of relative bias, relative mean square error, and coverage probability for the point and confidence interval estimators, respectively. The Bayesian approach using a non-informative prior distribution is recommended if the knowledge of setting up the hyper-parameters in the informative prior distribution is not available. Two real data sets are provided for illustration. [ABSTRACT FROM AUTHOR]

In this paper, the optimum test plan and statistical inference of 3-step stepstress accelerated life tests (SSALT) under progressive Type-I censoring is studied. It is assumed that the lifetime of a test unit follow a Weibull distribution with mean lifetime of a unit is a log-quadratic function of stress level. The maximum likelihood and Bayesian method are used to obtain the point and interval estimates of the model parameters under progressive Type-I censoring. The Bayes estimates are obtained using Markov Chain Monte Carlo (MCMC) simulation based on Gibbs sampling. The optimum 3-step SSALT plan under progressive Type-I censoring is developed by minimizing asymptotic variance of the maximum likelihood estimators of log of mean life at the design stress. Finally, the numerical study is presented to illustrate the proposed study. [ABSTRACT FROM AUTHOR]

In this paper, for Weibull subfamily of Morgenstern family, the joint density of the concomitants of generalized order statistics (GOS 's) is used to obtain the maximum likelihood estimates (MLE) and Bayes estimates for the distribution parameters. Applications of these results for concomitants of order statistics are presented. [ABSTRACT FROM AUTHOR]

In the realm of astronomy, the two-parameter log-normal distribution has ominous implications. In this article, we propose a new version of the two-parameter log-normal distribution with an application to astronomical data. More precisely, a new modulating parameter is added to the two-parameter log-normal distribution through the use of the Topp-Leone generator of distributions. The moments, quantile function, several reliability measures, and other significant aspects of the proposed distribution are investigated. The maximum likelihood approach and a Bayesian technique are both utilized to estimate the unknown parameters. In addition, we present a parametric regression model and a Bayesian regression method. A simulation study is carried out to assess the long-term performance of the estimators of the distribution parameters. Two real datasets are employed to show the applicability of this new distribution. The efficiency of the newly added parameter is tested by utilizing the likelihood ratio test. The parametric bootstrap approach is also utilized to determine the adequacy of the suggested model for the datasets. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the problem of point and interval estimations for the modified Weibull distribution (MWD) using progressively type-II censored sample. The maximum likelihood (ML), Bayes, and parametric bootstrap methods are used for estimating the unknown parameters as well as some lifetime parameters (reliability and hazard functions). Also, we propose to apply Markov chain Monte Carlo (MCMC) technique to carry out a Bayesian estimation procedure. Bayes estimates and the credible intervals are obtained under the assumptions of informative and noninformative priors. The results of Bayes method are obtained under both the balanced squared error loss (bSEL) and balanced linear-exponential (bLINEX) loss. We show that these loss functions are more general, which include the MLE and both symmetric and asymmetric Bayes estimates as special cases. Finally, Two real data sets have been analyzed for illustrative purposes. [Copyright &y& Elsevier]

In this paper, a competing risks model is studied when the latent failure times follow Weibull distribution. When the failure times are observed under generalized progressive hybrid censoring and the causes of failure are partially observed, the maximum likelihood estimators of the model parameters are established together with associated existence and uniqueness, and the approximate confidence intervals are constructed based on large sample theory. Bayes estimators and associated credible intervals are obtained under fairly general priors. Moreover, classical and Bayesian inferences are also discussed when there is an order restriction on the scale parameters of the Weibull distributions. Finally, a simulation study and a real data example are presented for illustration. • Generalized progressive hybrid censoring is considered for Weibull competing risks model. • Partially causes of failure are observed for censored data. • Order restriction information for scale parameters is considered. • Maximum likelihood estimators are established. • Bayesian estimates are derived via MCMC sampling method. • Data example on electric appliances data is illustrated. [ABSTRACT FROM AUTHOR]

Constant stress partially accelerated life tests are studied according to exponentiated Weibull distribution. Grounded on multiple censoring, the maximum likelihood estimators are determined in connection with unknown distribution parameters and accelerated factor. The confidence intervals of the unknown parameters and acceleration factor are constructed for large sample size. However, it is not possible to obtain the Bayes estimates in plain form, so we apply a Markov chain Monte Carlo method to deal with this issue, which permits us to create a credible interval of the associated parameters. Finally, based on constant stress partially accelerated life tests scheme with exponentiated Weibull distribution under multiple censoring, the illustrative example and the simulation results are used to investigate the maximum likelihood, and Bayesian estimates of the unknown parameters. [ABSTRACT FROM AUTHOR]

The use of finite mixture distributions, to control for unobserved heterogeneity, has become increasingly popular among those estimating dynamic discrete choice models. One of the barriers to using mixture models is that parameters that could previously be estimated in stages must now be estimated jointly: using mixture distributions destroys any additive reparability of the log likelihood function. In this research, Bayesian estimators have been obtained for the parameters of the mixture of exponentiated Weibull distribution when sample is available from censoring scheme. The maximum likelihood estimators of the parameters and the asymptotic variance covariance matrix have been obtained by Elshahat and Mahmoud (2016). Bayes and approximate Bayes (Lindley's form) estimators have been developed under squared error loss function as well as under LINEX loss function using non-informative type of priors for the parameters will be obtained. A numerical illustration for these new results is given. [ABSTRACT FROM AUTHOR]

Many studies have underlined the importance of the log-normal distribution in the modeling of phenomena occurring in biology. With this in mind, in this article we offer a new and motivated transformed version of the log-normal distribution, primarily for use with biological data. The hazard rate function, quantile function, and several other significant aspects of the new distribution are investigated. In particular, we show that the hazard rate function has increasing, decreasing, bathtub, and upside-down bathtub shapes. The maximum likelihood and Bayesian techniques are both used to estimate unknown parameters. Based on the proposed distribution, we also present a parametric regression model and a Bayesian regression approach. As an assessment of the longstanding performance, simulation studies based on maximum likelihood and Bayesian techniques of estimation procedures are also conducted. Two real datasets are used to demonstrate the applicability of the new distribution. The efficiency of the third parameter in the new model is tested by utilizing the likelihood ratio test. Furthermore, the parametric bootstrap approach is used to determine the effectiveness of the suggested model for the datasets. [ABSTRACT FROM AUTHOR]

Using progressive first-failure censored samples, we mainly study the inferences of the unknown parameters and the reliability and failure functions of the Inverted Exponentiated Half-Logistic distribution. The progressive first-failure censoring is an extension and improvement of progressive censoring, which is of great significance in the field of lifetime research. Besides maximum likelihood estimation, we use Bayesian estimation under unbalanced and balanced losses: General Entropy loss function, Squared Error loss function and Linex loss function. Approximate explicit expression of Bayesian estimation is given using Lindley approximation method for point estimation and Metropolis-Hastings method for point and interval estimation. Bayesian credible intervals and asymptotic confidence intervals are derived in the form of average length and coverage probability. To show the research effects, a simulation study and practical data analysis are carried out. Finally, we discuss the optimal censoring mode under four different criteria. [ABSTRACT FROM AUTHOR]