scholarly journals Objective bayesian analysis for multiple repairable systems

PLoS ONE ◽  
2021 ◽  
Vol 16 (11) ◽  
pp. e0258581
Author(s):  
Amanda M. E. D’Andrea ◽  
Vera L. D. Tomazella ◽  
Hassan M. Aljohani ◽  
Pedro L. Ramos ◽  
Marco P. Almeida ◽  
...  
Keyword(s):  
Repairable Systems ◽  
Minimal Repair ◽  
Jeffreys Prior ◽  
Unknown Parameters ◽  
Data Set ◽  
Reference Priors ◽  
Unbiased Estimates ◽  
Frequentist Coverage ◽  
The Bayesian Approach ◽  
Bayesian Estimators

This article focus on the analysis of the reliability of multiple identical systems that can have multiple failures over time. A repairable system is defined as a system that can be restored to operating state in the event of a failure. This work under minimal repair, it is assumed that the failure has a power law intensity and the Bayesian approach is used to estimate the unknown parameters. The Bayesian estimators are obtained using two objective priors know as Jeffreys and reference priors. We proved that obtained reference prior is also a matching prior for both parameters, i.e., the credibility intervals have accurate frequentist coverage, while the Jeffreys prior returns unbiased estimates for the parameters. To illustrate the applicability of our Bayesian estimators, a new data set related to the failures of Brazilian sugar cane harvesters is considered.

Reference priors for Bayesian fisheries models

2002 ◽  
Vol 59 (9) ◽  
pp. 1492-1502 ◽  
Author(s):  
Russell B Millar
Keyword(s):  
Prior Knowledge ◽  
Population Analysis ◽  
Production Model ◽  
Model Parameters ◽  
Jeffreys Prior ◽  
Unknown Parameters ◽  
Prior Distributions ◽  
Reference Priors ◽  
Surplus Production ◽  
Surplus Production Model

Bayesian models require the specification of prior distributions for all unknown parameters, and this formal utilization of prior knowledge (if any) can be used to great advantage in some fisheries. However, regardless of whether prior knowledge about model parameters is available, specification of prior distributions is seldom unequivocal. This work addresses the problem of specifying default priors for several common fisheries models. To maintain consistency of terminology with the statistical literature, such priors are herein called reference priors to recognize that they can be interpreted as providing a sensible reference point against which the implications of alternative priors can be compared. Here, the Jeffreys' prior is demonstrated for the Ricker and Beverton–Holt stock–recruit curves, von Bertalanffy growth curve, Schaefer surplus production model, and sequential population analysis. The Jeffreys' priors for relevant derived parameters are demonstrated, including the steepness parameter of the Beverton–Holt stock–recruit curve. The sequential population analysis example is used to show that the Jeffreys' prior should not be automatically accepted as a reference prior in all models—this needs to be decided on a case-by-case basis.

scholarly journals Bayesian Estimation and Prediction of Three-Parameter Complementary Exponential Power Distribution using MCMC Technique

2020 ◽  
Vol 10 (2) ◽  
pp. 164-174
Keyword(s):  
Power Distribution ◽  
Real Data ◽  
Credible Interval ◽  
Simulation Method ◽  
Complete Sample ◽  
Unknown Parameters ◽  
Data Set ◽  
Exponential Power Distribution ◽  
Exponential Power ◽  
Bayesian Estimators

The Markov chain Monte Carlo (MCMC) technique is applied for estimating the Complementary Exponential Power (CEP) distribution's parameters through the analysis of complete sample in this article. With the help of the Bayesian and the Maximum Likelihood techniques, the unknown parameters of the model are estimated. To find Complementary Exponential Power distribution's parameters' Bayesian estimates, a new methodology is developed, via simulation method of MCMC through the application of OpenBUGS platform. To demonstrate under the gamma and uniform sets of priors, a real data set is taken. The generations of posterior MCMC samples is conducted with OpenBUGS software. For analyzing the output of so generated MCMC samples, and studying the statistical properties, distribution's comparison tools and model validation the functions of R have been used. The credible interval and predicted of the reliability, hazard and modal parameters' values are also estimated. We have shown that Bayesian estimators are more efficient than classical estimators for any real data set.

scholarly journals Bayesian and E-Bayesian Estimations of Bathtub-Shaped Distribution under Generalized Type-I Hybrid Censoring

Entropy ◽  
2021 ◽  
Vol 23 (8) ◽  
pp. 934
Author(s):  
Yuxuan Zhang ◽  
Kaiwei Liu ◽  
Wenhao Gui
Keyword(s):  
Bayesian Method ◽  
Real Data ◽  
Loss Functions ◽  
Type I ◽  
Statistical Efficiency ◽  
Unknown Parameters ◽  
Data Set ◽  
Hybrid Censoring ◽  
Bayesian Estimations ◽  
Generalized Type

For the purpose of improving the statistical efficiency of estimators in life-testing experiments, generalized Type-I hybrid censoring has lately been implemented by guaranteeing that experiments only terminate after a certain number of failures appear. With the wide applications of bathtub-shaped distribution in engineering areas and the recently introduced generalized Type-I hybrid censoring scheme, considering that there is no work coalescing this certain type of censoring model with a bathtub-shaped distribution, we consider the parameter inference under generalized Type-I hybrid censoring. First, estimations of the unknown scale parameter and the reliability function are obtained under the Bayesian method based on LINEX and squared error loss functions with a conjugate gamma prior. The comparison of estimations under the E-Bayesian method for different prior distributions and loss functions is analyzed. Additionally, Bayesian and E-Bayesian estimations with two unknown parameters are introduced. Furthermore, to verify the robustness of the estimations above, the Monte Carlo method is introduced for the simulation study. Finally, the application of the discussed inference in practice is illustrated by analyzing a real data set.

scholarly journals On the Estimation of Reliability of Weighted Weibull Distribution: A Comparative Study

2016 ◽  
Vol 5 (4) ◽  
pp. 1
Author(s):  
Bander Al-Zahrani
Keyword(s):  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Real Data ◽  
Reliability Function ◽  
Nonparametric Methods ◽  
Empirical Method ◽  
Density Estimator ◽  
Bayes Estimators ◽  
Unknown Parameters ◽  
Data Set

The paper gives a description of estimation for the reliability function of weighted Weibull distribution. The maximum likelihood estimators for the unknown parameters are obtained. Nonparametric methods such as empirical method, kernel density estimator and a modified shrinkage estimator are provided. The Markov chain Monte Carlo method is used to compute the Bayes estimators assuming gamma and Jeffrey priors. The performance of the maximum likelihood, nonparametric methods and Bayesian estimators is assessed through a real data set.

scholarly journals A Bayesian model for binary Markov chains

2004 ◽  
Vol 2004 (8) ◽  
pp. 421-429 ◽  
Author(s):  
Souad Assoudou ◽  
Belkheir Essebbar
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Bayesian Estimation ◽  
Bayesian Model ◽  
Transition Probabilities ◽  
Simulated Data ◽  
Bayesian Estimator ◽  
Jeffreys Prior ◽  
Data Set

This note is concerned with Bayesian estimation of the transition probabilities of a binary Markov chain observed from heterogeneous individuals. The model is founded on the Jeffreys' prior which allows for transition probabilities to be correlated. The Bayesian estimator is approximated by means of Monte Carlo Markov chain (MCMC) techniques. The performance of the Bayesian estimates is illustrated by analyzing a small simulated data set.

Univariate and bivariate extensions of the generalized exponential distributions

2021 ◽  
Vol 71 (6) ◽  
pp. 1581-1598
Author(s):  
Vahid Nekoukhou ◽  
Ashkan Khalifeh ◽  
Hamid Bidram
Keyword(s):  
Em Algorithm ◽  
Exponential Distribution ◽  
Bivariate Distribution ◽  
Generalized Exponential Distribution ◽  
Exponential Distributions ◽  
Unknown Parameters ◽  
Data Set ◽  
New Class ◽  
Special Cases ◽  
Univariate Distribution

Abstract The main aim of this paper is to introduce a new class of continuous generalized exponential distributions, both for the univariate and bivariate cases. This new class of distributions contains some newly developed distributions as special cases, such as the univariate and also bivariate geometric generalized exponential distribution and the exponential-discrete generalized exponential distribution. Several properties of the proposed univariate and bivariate distributions, and their physical interpretations, are investigated. The univariate distribution has four parameters, whereas the bivariate distribution has five parameters. We propose to use an EM algorithm to estimate the unknown parameters. According to extensive simulation studies, we see that the effectiveness of the proposed algorithm, and the performance is quite satisfactory. A bivariate data set is analyzed and it is observed that the proposed models and the EM algorithm work quite well in practice.

scholarly journals THE EFFECTS OF SUCCESSIVE THINNINGS ON THE HYPSOMETRIC FUNCTION FOR Pinus oocarpa STANDS

FLORESTA ◽  
2011 ◽  
Vol 41 (2) ◽  
Author(s):  
Sebastião Do Amaral Machado ◽  
Dalmo Arantes de Barros ◽  
José Roberto Soares Scolforo ◽  
Fausto Weimar Acerbi Junior
Keyword(s):  
Standard Error ◽  
Unbiased Estimate ◽  
Sao Paulo ◽  
Average Height ◽  
São Paulo ◽  
Data Set ◽  
Management Regimes ◽  
Pinus Oocarpa ◽  
Unbiased Estimates ◽  
Multiple Determination

This research aimed to test the hypothesis that successive thinnings mischaracterize the hypsometric function, and that after some thinnings the average height of the plots or of the stand () is an unbiased estimate of the remaining tree heights (hi), that is, ĥi =  . The data were obtained from Duraflora S. A. in Agudos, State of São Paulo. This data set came from the measurement of diameters and respective heights of 1,100 trees equitably distributed in 11 different treatments with ages ranging from 5 to 25 years, and number of thinnings varying between 0 (zero) and 6. The Stoffels model, previously fitted to all treatments, was used to observe the behavior of the following factors: coefficient of multiple determination (R²), standard error of estimate in percent (Syx %), distribution of residuals, and the significance of the b0 and b1 coefficients in each one of the equations. After tests it was confirmed the hypothesis that average height is a precise estimate of hi for the oldest treatments with more than 4 thinnings. This means low Syx % and unbiased estimates, that is, good distribution of residuals.Keywords: Hypsometric curves; management regimes; plantations; Pinus oocarpa.ResumoEfeitos de desbastes sucessivos sobre a função hipsométrica em povoamentos de Pinus oocarpa Shiede. Esta pesquisa teve como objetivo testar a hipótese que desbastes sucessivos descaracterizam a função hipsométrica e que após alguns desbastes a altura média da parcela ou do povoamento () é uma boa estimativa das demais alturas (ĥi), isto é, ĥi = . A base de dados utilizada para testar a hipótese formulada proveio de plantios de Pinus oocarpa, pertencentes à empresa Duraflora S.A., situada no município de Agudos, sudoeste do Estado de São Paulo. Esta base consistiu da altura e diâmetro de 1100 árvores distribuídas igualmente em 11 tratamentos, com variação de idades e número de desbastes. Visando atingir aos objetivos, foi primeiramente ajustado o modelo de Stoffels, separadamente para cada um dos 11 tratamentos, com o fim de observar a tendência dos coeficientes de determinação múltiplos (R2), do erro padrão da estimativa em porcentagem (Syx %), a distribuição dos resíduos e a significância dos coeficientes de interseção (b0) e de inclinação (b1) em cada uma das equações. Após proceder as análises conclui-se que a medida que se aumenta o número de desbastes os R² vão se tornando cada vez mais baixos , porém os Syx % foram sempre inferiores a 10%. Palavras-chave: Curva hipsométrica; regimes de manejo; plantações; Pinus oocarpa. 

scholarly journals Inference of stress-strength reliability for two-parameter of exponentiated Gumbel distribution based on lower record values

PLoS ONE ◽  
2021 ◽  
Vol 16 (4) ◽  
pp. e0249028
Author(s):  
Ehsan Fayyazishishavan ◽  
Serpil Kılıç Depren
Keyword(s):  
Information Matrix ◽  
Gumbel Distribution ◽  
Sampling Technique ◽  
Real Data ◽  
Record Values ◽  
Unknown Parameters ◽  
Data Set ◽  
Credible Intervals ◽  
Lower Records ◽  
Two Parameter

The two-parameter of exponentiated Gumbel distribution is an important lifetime distribution in survival analysis. This paper investigates the estimation of the parameters of this distribution by using lower records values. The maximum likelihood estimator (MLE) procedure of the parameters is considered, and the Fisher information matrix of the unknown parameters is used to construct asymptotic confidence intervals. Bayes estimator of the parameters and the corresponding credible intervals are obtained by using the Gibbs sampling technique. Two real data set is provided to illustrate the proposed methods.

scholarly journals Computation of posterior distribution in Bayesian analysis – application in an intermittently used reliability system

2002 ◽  
Vol 21 (3) ◽  
pp. 78-82
Author(s):  
V. S.S. Yadavalli ◽  
P. J. Mostert ◽  
A. Bekker ◽  
M. Botha
Keyword(s):  
Posterior Distribution ◽  
Jeffreys Prior ◽  
Posterior Density ◽  
Unknown Parameters ◽  
Stationary Rate ◽  
Analysis Application ◽  
Definition Of ◽  
Hpd Intervals ◽  
Highest Posterior Density ◽  
Bayes Procedure

Bayesian estimation is presented for the stationary rate of disappointments, D∞, for two models (with different specifications) of intermittently used systems. The random variables in the system are considered to be independently exponentially distributed. Jeffreys’ prior is assumed for the unknown parameters in the system. Inference about D∞ is being restrained in both models by the complex and non-linear definition of D∞. Monte Carlo simulation is used to derive the posterior distribution of D∞ and subsequently the highest posterior density (HPD) intervals. A numerical example where Bayes estimates and the HPD intervals are determined illustrates these results. This illustration is extended to determine the frequentistical properties of this Bayes procedure, by calculating covering proportions for each of these HPD intervals, assuming fixed values for the parameters.

scholarly journals Inferences for Weibull parameters under progressively first-failure censored data with binomial random removals

2020 ◽  
Vol 9 (1) ◽  
pp. 47-60
Author(s):  
Samir K. Ashour ◽  
Ahmed A. El-Sheikh ◽  
Ahmed Elshahhat
Keyword(s):  
Monte Carlo ◽  
Censored Data ◽  
Real Data ◽  
Unknown Parameters ◽  
Data Set ◽  
Monte Carlo Simulation Study ◽  
Squared Error Loss Function ◽  
Bayes Estimates ◽  
Monte Carlo Techniques ◽  
Credible Intervals

In this paper, the Bayesian and non-Bayesian estimation of a two-parameter Weibull lifetime model in presence of progressive first-failure censored data with binomial random removals are considered. Based on the s-normal approximation to the asymptotic distribution of maximum likelihood estimators, two-sided approximate confidence intervals for the unknown parameters are constructed. Using gamma conjugate priors, several Bayes estimates and associated credible intervals are obtained relative to the squared error loss function. Proposed estimators cannot be expressed in closed forms and can be evaluated numerically by some suitable iterative procedure. A Bayesian approach is developed using Markov chain Monte Carlo techniques to generate samples from the posterior distributions and in turn computing the Bayes estimates and associated credible intervals. To analyze the performance of the proposed estimators, a Monte Carlo simulation study is conducted. Finally, a real data set is discussed for illustration purposes.

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