scholarly journals On Reliability Estimation of Lomax Distribution under Adaptive Type-I Progressive Hybrid Censoring Scheme

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2903
Author(s):  
Hassan Okasha ◽  
Yuhlong Lio ◽  
Mohammed Albassam
Keyword(s):  
Prior Distribution ◽  
Shape Parameter ◽  
Real Data ◽  
Reliability Estimation ◽  
Loss Functions ◽  
Type I ◽  
Empirical Bayesian ◽  
Asymmetric Loss ◽  
Lomax Distribution ◽  
Bayesian Estimates

Bayesian estimates involve the selection of hyper-parameters in the prior distribution. To deal with this issue, the empirical Bayesian and E-Bayesian estimates may be used to overcome this problem. The first one uses the maximum likelihood estimate (MLE) procedure to decide the hyper-parameters; while the second one uses the expectation of the Bayesian estimate taken over the joint prior distribution of the hyper-parameters. This study focuses on establishing the E-Bayesian estimates for the Lomax distribution shape parameter functions by utilizing the Gamma prior of the unknown shape parameter along with three distinctive joint priors of Gamma hyper-parameters based on the square error as well as two asymmetric loss functions. These two asymmetric loss functions include a general entropy and LINEX loss functions. To investigate the effect of the hyper-parameters’ selections, mathematical propositions have been derived for the E-Bayesian estimates of the three shape functions that comprise the identity, reliability and hazard rate functions. Monte Carlo simulation has been performed to compare nine E-Bayesian, three empirical Bayesian and Bayesian estimates and MLEs for any aforementioned functions. Additionally, one simulated and two real data sets from industry life test and medical study are applied for the illustrative purpose. Concluding notes are provided at the end.

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 Estimating the Entropy for Lomax Distribution Based on Generalized Progressively Hybrid Censoring

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1219 ◽  
Author(s):  
Shuhan Liu ◽  
Wenhao Gui
Keyword(s):  
Incomplete Data ◽  
Real Data ◽  
Unknown Parameters ◽  
Data Set ◽  
Hybrid Censoring ◽  
Lomax Distribution ◽  
Bayesian Estimates ◽  
Squared Error ◽  
General Entropy Loss Function ◽  
Two Parameter

As it is often unavoidable to obtain incomplete data in life testing and survival analysis, research on censoring data is becoming increasingly popular. In this paper, the problem of estimating the entropy of a two-parameter Lomax distribution based on generalized progressively hybrid censoring is considered. The maximum likelihood estimators of the unknown parameters are derived to estimate the entropy. Further, Bayesian estimates are computed under symmetric and asymmetric loss functions, including squared error, linex, and general entropy loss function. As we cannot obtain analytical Bayesian estimates directly, the Lindley method and the Tierney and Kadane method are applied. A simulation study is conducted and a real data set is analyzed for illustrative purposes.

scholarly journals E-Bayesian and Bayesian Estimation for the Lomax Distribution under Weighted Composite LINEX Loss Function

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Afrah Al-Bossly
Keyword(s):  
Bayesian Estimation ◽  
Loss Function ◽  
Shape Parameter ◽  
Likelihood Estimation ◽  
Loss Functions ◽  
Bayesian Estimator ◽  
Linex Loss Function ◽  
Lomax Distribution ◽  
Linex Loss ◽  
Asymmetric Loss Function

The main contribution of this work is the development of a compound LINEX loss function (CLLF) to estimate the shape parameter of the Lomax distribution (LD). The weights are merged into the CLLF to generate a new loss function called the weighted compound LINEX loss function (WCLLF). Then, the WCLLF is used to estimate the LD shape parameter through Bayesian and expected Bayesian (E-Bayesian) estimation. Subsequently, we discuss six different types of loss functions, including square error loss function (SELF), LINEX loss function (LLF), asymmetric loss function (ASLF), entropy loss function (ENLF), CLLF, and WCLLF. In addition, in order to check the performance of the proposed loss function, the Bayesian estimator of WCLLF and the E-Bayesian estimator of WCLLF are used, by performing Monte Carlo simulations. The Bayesian and expected Bayesian by using the proposed loss function is compared with other methods, including maximum likelihood estimation (MLE) and Bayesian and E-Bayesian estimators under different loss functions. The simulation results show that the Bayes estimator according to WCLLF and the E-Bayesian estimator according to WCLLF proposed in this work have the best performance in estimating the shape parameters based on the least mean averaged squared error.

scholarly journals Bayesian Modeling of 3-Component Mixture of Exponentiated Inverted Weibull Distribution under Noninformative Prior

2020 ◽  
Vol 2020 ◽  
pp. 1-11 ◽  
Author(s):  
Ammara Nawaz Cheema ◽  
Muhammad Aslam ◽  
Ibrahim M. Almanjahie ◽  
Ishfaq Ahmad
Keyword(s):  
Weibull Distribution ◽  
Research Work ◽  
Real Data ◽  
Loss Functions ◽  
Type I ◽  
Bayes Estimators ◽  
Component Mixture ◽  
Type I Censoring ◽  
Censoring Technique ◽  
The Impact

Bayesian study of 3-component mixture modeling of exponentiated inverted Weibull distribution under right type I censoring technique is conducted in this research work. The posterior distribution of the parameters is obtained assuming the noninformative (Jeffreys and uniform) priors. The different loss functions (squared error, quadratic, precautionary, and DeGroot loss function) are used to obtain the Bayes estimators and posterior risks. The performance of the Bayes estimators through posterior risks under the said loss functions is investigated through simulation process. Real data analysis of tensile strength of carbon fiber is also applied for 3 components to conclude the presentation of Bayes estimators. The limiting expressions are also elaborated for Bayes estimators and posterior risks in this study. The impact of some test termination times and sample sizes is reported on Bayes estimators.

scholarly journals Bayesian Approach in Estimation of Shape Parameter of an Exponential Inverse Exponential Distribution

2020 ◽  
pp. 13-27
Author(s):  
Bashiru Omeiza Sule ◽  
Taiwo Mobolaji Adegoke
Keyword(s):  
Exponential Distribution ◽  
Gamma Distribution ◽  
Loss Function ◽  
Posterior Distribution ◽  
Prior Distribution ◽  
Shape Parameter ◽  
Real Life ◽  
Loss Functions ◽  
Jeffreys Prior ◽  
True Parameter

Aims: This study aimed to obtain the shape parameter of an Exponential Inverted Exponential distribution using different prior distributions under different loss functions. Methodology: The Bayes’ theorem was adopted to obtain the posterior distribution of the shape parameter of an Exponential inverted Exponential distribution for both non-information prior (such as Jeffreys prior, Hartigen prior and Uniform prior) and an informative prior (such as Gamma distribution and chi-square distribution). Different loss functions (such as Entropy loss function, Square error loss function, Al-Bayyati’s loss function and Precautionary loss function) were employed to obtain the estimate parameter of the shape parameter with an assumption that the scale parameter is known. Results: The posterior distribution of the shape parameter of an Exponential Inverted Exponential distribution follows a Gamma distribution for all the prior distribution in the study. Also the Bayes estimate for the simulated datasets and real life dataset were obtained. Conclusion: The Bayes’ estimates for different prior distribution under different loss functions are close to the true parameter value of the shape parameter. The estimators are then compared in terms of their Mean Square Error (MSE) which is computed using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

scholarly journals Estimation of Information Measures for Power-Function Distribution in Presence of Outliers and Their Applications

2021 ◽  
Vol 21 (No.1) ◽  
pp. 1-25
Author(s):  
Amal Soliman Hassan ◽  
Elsayed Ahmed Elsherpieny ◽  
Rokaya Elmorsy Mohamed
Keyword(s):  
Maximum Likelihood ◽  
Power Function ◽  
Real Data ◽  
The Other ◽  
Loss Functions ◽  
Squared Error Loss Function ◽  
Bayesian Estimates ◽  
Function Distribution ◽  
Power Function Distribution ◽  
Bayesian Estimators

The measure of entropy has an undeniable pivotal role in the field of information theory. This article estimates the Rényi and q-entropies of the power function distribution in the presence of s outliers. The maximum likelihood estimators as well as the Bayesian estimators under uniform and gamma priors are derived. The proposed Bayesian estimators of entropies under symmetric and asymmetric loss functions are obtained. These estimators are computed empirically using Monte Carlo simulation based on Gibbs sampling. Outcomes of the study showed that the precision of the maximum likelihood and Bayesian estimates of both entropies measures improves with sample sizes. The behavior of both entropies estimates increase with number of outliers. Further, Bayesian estimates of the Rényi and q-entropies under squared error loss function are preferable than the other Bayesian estimates under the other loss functions in most of cases. Eventually, real data examples are analyzed to illustrate the theoretical results.

scholarly journals Bayesian Estimation of the Parameters of Discrete Weibull Type (I) Distribution

2020 ◽  
Vol 18 (2) ◽  
pp. 2-13
Author(s):  
Samir Kamel Ashour ◽  
Mohamed Salem Abdelwahab Muiftah
Keyword(s):  
Weibull Distribution ◽  
Bayesian Estimation ◽  
Shape Parameter ◽  
Random Variables ◽  
Independent Random Variables ◽  
Parameter Estimates ◽  
Type I ◽  
Sample Sizes ◽  
Bayesian Estimates ◽  
Distribution Parameters

Bayesian estimation of the continuous Weibull distribution parameters was studied by Ahmad and Ahmad (2013) under the assumption of knowing the shape parameter. Bayesian estimates are considered here of the parameters of the discrete Weibull Type I [DW(I)] distribution and are obtained under two different assumptions: when the shape parameter is known, and when both parameters are independent random variables. A Mathcad program is performed to simulate data from the DW(I) distribution considering different values of the parameters and different sample sizes, and to obtain Bayesian parameter estimates. The resulted estimates are compared to the ML and proportion estimates obtained by Khan et al. (1989).

scholarly journals Optimal Sample Size for the Birnbaum–Saunders Distribution under Decision Theory with Symmetric and Asymmetric Loss Functions

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 926
Author(s):  
Eliardo Costa ◽  
Manoel Santos-Neto ◽  
Víctor Leiva
Keyword(s):  
Sample Size ◽  
Real Data ◽  
Loss Functions ◽  
Theoretic Approach ◽  
Optimal Sample Size ◽  
Asymmetric Loss ◽  
Optimal Sample ◽  
Decision Theoretic Approach ◽  
Potential Applications ◽  
Asymmetrical Model

The fatigue-life or Birnbaum–Saunders distribution is an asymmetrical model that has been widely applied in several areas of science and mainly in reliability. Although diverse methodologies related to this distribution have been proposed, the problem of determining the optimal sample size when estimating its mean has not yet been studied. In this paper, we derive a methodology to determine the optimal sample size under a decision-theoretic approach. In this approach, we consider symmetric and asymmetric loss functions for point and interval inference. Computational tools in the R language were implemented to use this methodology in practice. An illustrative example with real data is also provided to show potential applications.

EXPONENTIATED HALF-LOGISTIC LOMAX DISTRIBUTION WITH PROPERTIES AND APPLICATION

2019 ◽  
Vol XVI (2) ◽  
pp. 1-11
Author(s):  
Farrukh Jamal ◽  
Hesham Mohammed Reyad ◽  
Soha Othman Ahmed ◽  
Muhammad Akbar Ali Shah ◽  
Emrah Altun
Keyword(s):  
Real Data ◽  
Continuous Model ◽  
Model Parameters ◽  
Data Set ◽  
Lomax Distribution ◽  
Mathematical Properties ◽  
Proposed Model ◽  
Probability Weighted Moment ◽  
Record Statistics ◽  
Maximum Likelihood Criterion

A new three-parameter continuous model called the exponentiated half-logistic Lomax distribution is introduced in this paper. Basic mathematical properties for the proposed model were investigated which include raw and incomplete moments, skewness, kurtosis, generating functions, Rényi entropy, Lorenz, Bonferroni and Zenga curves, probability weighted moment, stress strength model, order statistics, and record statistics. The model parameters were estimated by using the maximum likelihood criterion and the behaviours of these estimates were examined by conducting a simulation study. The applicability of the new model is illustrated by applying it on a real data set.

scholarly journals A comprehensive evaluation of methods for Mendelian randomization using realistic simulations and an analysis of 38 biomarkers for risk of type 2 diabetes

2021 ◽  
Author(s):  
Guanghao Qi ◽  
Nilanjan Chatterjee
Keyword(s):  
Type 2 Diabetes ◽  
Mendelian Randomization ◽  
Association Studies ◽  
Real Data ◽  
Causal Effects ◽  
Type I ◽  
Genome Wide Association Studies ◽  
Simulation Studies ◽  
Sample Sizes

Abstract Background Previous studies have often evaluated methods for Mendelian randomization (MR) analysis based on simulations that do not adequately reflect the data-generating mechanisms in genome-wide association studies (GWAS) and there are often discrepancies in the performance of MR methods in simulations and real data sets. Methods We use a simulation framework that generates data on full GWAS for two traits under a realistic model for effect-size distribution coherent with the heritability, co-heritability and polygenicity typically observed for complex traits. We further use recent data generated from GWAS of 38 biomarkers in the UK Biobank and performed down sampling to investigate trends in estimates of causal effects of these biomarkers on the risk of type 2 diabetes (T2D). Results Simulation studies show that weighted mode and MRMix are the only two methods that maintain the correct type I error rate in a diverse set of scenarios. Between the two methods, MRMix tends to be more powerful for larger GWAS whereas the opposite is true for smaller sample sizes. Among the other methods, random-effect IVW (inverse-variance weighted method), MR-Robust and MR-RAPS (robust adjust profile score) tend to perform best in maintaining a low mean-squared error when the InSIDE assumption is satisfied, but can produce large bias when InSIDE is violated. In real-data analysis, some biomarkers showed major heterogeneity in estimates of their causal effects on the risk of T2D across the different methods and estimates from many methods trended in one direction with increasing sample size with patterns similar to those observed in simulation studies. Conclusion The relative performance of different MR methods depends heavily on the sample sizes of the underlying GWAS, the proportion of valid instruments and the validity of the InSIDE assumption. Down-sampling analysis can be used in large GWAS for the possible detection of bias in the MR methods.

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