Estimation of stress-strength reliability R=P(X>Y) based on Weibull record data in the presence of inter-record times
TL;DR: In this article, a considerable body of literature has been recently devoted to the inference problem of the reliability parameter R = P ( X > Y ) based on record data, and the records as well as the corresponding inter-record times to develop inference procedures for R assuming X and Y come from Weibull distribution.
Abstract: A considerable body of literature has been recently devoted to the inference problem of the reliability parameter R = P ( X > Y ) based on record data. In this article, we consider the records as well as the corresponding inter-record times to develop inference procedures for R assuming X and Y come from Weibull distribution. The maximum likelihood estimator of R and its corresponding confidence interval are determined. Bayesian analyses involving Tierney and Kadane’s approximation and Metropolis-Hasting samplers are used to estimate R based on LINEX and square error loss functions. In addition, the estimation problem of R is discussed in the models with known shape parameters. To compare all methods developed here, numerical simulation is carried out. Finally, different real data sets are analyzed.
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TL;DR: In this paper , the authors proposed the Bayesian estimators of Kumaraswamy distribution parameters through the use of type-II censoring data, along with several loss functions, namely, linex loss function (LLF), weighted linex (WLLF) and composite linex function (CLLF).
Abstract: The Kumaraswamy distribution (KD) is widely applied for modeling data in practical domains, such as medicine, engineering, economics, and physics. The present work proposes the Bayesian estimators of KD parameters through the use of type-II censoring data. Both E-Bayesian and Bayesian estimation approaches are briefly described, along with several loss functions, namely, linex loss function (LLF), weighted linex loss function (WLLF), and composite linex loss function (CLLF). In the Bayesian framework, gamma distribution has been utilized as a conjugate distribution in view of finding theoretical results. The E-Bayesian estimators for the hyperparameter using different distributions are developed. Moreover, a novel loss function referred to as the weighted composite loss function (WCLLF) in the estimation perspective is established. Finally, the Monte Carlo simulation approach is carried out to reveal that the new suggested loss function outperforms several counterparts in determining the shape parameter of the KD.
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TL;DR: In this article , a linear, exponential loss function (LXLF) was proposed to estimate the dependability value and variables of the Inverse Weibull Distribution (IWD) according to Lowest Record Values.
Abstract: The main objective of this article is to develop a linear, exponential loss function (LXLF) to the dependability value and variables of the Inverse Weibull Distribution (IWD) according to Lowest Record Values. By weighting the LXLF to construct current function loss, the modified linear, exponential loss function (WLXLF) was named. Then use WLXLF to estimate the parameters and reliability functions of the (IWD). After that, we evaluated how well the prediction made in this article performed against the Bayesian estimator using the symmetric and asymmetric loss functions and maximum likelihood estimation (MLE). The credibility range for the values of the parameters and durability function has been created using the parametric bootstrap approach, and the outcomes have been contrasted via a Monte Carlo simulation. An actual data set from a realistic situation was utilized to explain the concept. The computer simulation results demonstrated that the suggested method, performed as expected, was the best for estimating the scale parameter and reliability function. The proposed technique performed well in estimating the shape parameter based on real data and had an acceptable performance for estimating scale parameters and reliability functions.
TL;DR: In this article , a generalized inferential approach is proposed for estimating stress strength reliability (SSR), where two types of pivotal quantities are constructed respectively, and then the generalized point and interval estimates for SSR are proposed in consequence, where the associated Monte Carlo sampling approach is provided for computation.
Abstract: This paper explores estimation of stress-strength reliability based on upper record values. When the strength and stress variables follow unit-Burr Ⅲ distributions, a generalized inferential approach is proposed for estimating stress-strength reliability (SSR). Under the common strength and stress parameter case, two types of pivotal quantities are constructed respectively, and then the generalized point and interval estimates for SSR are proposed in consequence, where the associated Monte-Carlo sampling approach is provided for computation. In addition, when strength and stress variables feature unequal model parameters, different generalized point and confidence interval estimates are also established in this regard. Extensive simulation studies are conducted to examine the behavior of proposed methods. Finally, a real-life data example is presented for illustration.
TL;DR: In this paper , the reliability of a multicomponent stress-strength (MSS) model with inverse Topp-Leone distributions was investigated. And the maximum likelihood and uniformly minimum variance unbiased estimates for the reliability were obtained explicitly.
Abstract: In this article, the reliability inference for a multicomponent stress-strength (MSS) model, when both stress and strength random variables follow inverse Topp-Leone distributions, was studied. The maximum likelihood and uniformly minimum variance unbiased estimates for the reliability of MSS model were obtained explicitly. The exact Bayes estimate of MSS reliability was derived the under squared error loss function. Also, the Bayes estimate was obtained using the Monte Carlo Markov Chain method for comparison with the aforementioned exact estimate. The asymptotic confidence interval was determined under the expected Fisher information matrix. Furthermore, the highest probability density credible interval was established through using Gibbs sampling method. Monte Carlo simulations were implemented to compare the different proposed methods. Finally, a real life example was presented in support of the suggested procedures.
02 Jan 2023
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01 Jan 1943TL;DR: Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integral Integral Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequality 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform
Abstract: 0 Introduction 1 Elementary Functions 2 Indefinite Integrals of Elementary Functions 3 Definite Integrals of Elementary Functions 4.Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integrals of Special Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequalities 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform
27,354 citations
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01 Jan 1995TL;DR: Detailed notes on Bayesian Computation Basics of Markov Chain Simulation, Regression Models, and Asymptotic Theorems are provided.
Abstract: FUNDAMENTALS OF BAYESIAN INFERENCE Probability and Inference Single-Parameter Models Introduction to Multiparameter Models Asymptotics and Connections to Non-Bayesian Approaches Hierarchical Models FUNDAMENTALS OF BAYESIAN DATA ANALYSIS Model Checking Evaluating, Comparing, and Expanding Models Modeling Accounting for Data Collection Decision Analysis ADVANCED COMPUTATION Introduction to Bayesian Computation Basics of Markov Chain Simulation Computationally Efficient Markov Chain Simulation Modal and Distributional Approximations REGRESSION MODELS Introduction to Regression Models Hierarchical Linear Models Generalized Linear Models Models for Robust Inference Models for Missing Data NONLINEAR AND NONPARAMETRIC MODELS Parametric Nonlinear Models Basic Function Models Gaussian Process Models Finite Mixture Models Dirichlet Process Models APPENDICES A: Standard Probability Distributions B: Outline of Proofs of Asymptotic Theorems C: Computation in R and Stan Bibliographic Notes and Exercises appear at the end of each chapter.
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Abstract: Algebra of Vectors and Matrices. Probability Theory, Tools and Techniques. Continuous Probability Models. The Theory of Least Squares and Analysis of Variance. Criteria and Methods of Estimation. Large Sample Theory and Methods. Theory of Statistical Inference. Multivariate Analysis. Publications of the Author. Author Index. Subject Index.
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TL;DR: The Markov Chain Monte Carlo Implementation Results Summary and Discussion MEDICAL MONITORING Introduction Modelling Medical Monitoring Computing Posterior Distributions Forecasting Model Criticism Illustrative Application Discussion MCMC for NONLINEAR HIERARCHICAL MODELS.
Abstract: INTRODUCING MARKOV CHAIN MONTE CARLO Introduction The Problem Markov Chain Monte Carlo Implementation Discussion HEPATITIS B: A CASE STUDY IN MCMC METHODS Introduction Hepatitis B Immunization Modelling Fitting a Model Using Gibbs Sampling Model Elaboration Conclusion MARKOV CHAIN CONCEPTS RELATED TO SAMPLING ALGORITHMS Markov Chains Rates of Convergence Estimation The Gibbs Sampler and Metropolis-Hastings Algorithm INTRODUCTION TO GENERAL STATE-SPACE MARKOV CHAIN THEORY Introduction Notation and Definitions Irreducibility, Recurrence, and Convergence Harris Recurrence Mixing Rates and Central Limit Theorems Regeneration Discussion FULL CONDITIONAL DISTRIBUTIONS Introduction Deriving Full Conditional Distributions Sampling from Full Conditional Distributions Discussion STRATEGIES FOR IMPROVING MCMC Introduction Reparameterization Random and Adaptive Direction Sampling Modifying the Stationary Distribution Methods Based on Continuous-Time Processes Discussion IMPLEMENTING MCMC Introduction Determining the Number of Iterations Software and Implementation Output Analysis Generic Metropolis Algorithms Discussion INFERENCE AND MONITORING CONVERGENCE Difficulties in Inference from Markov Chain Simulation The Risk of Undiagnosed Slow Convergence Multiple Sequences and Overdispersed Starting Points Monitoring Convergence Using Simulation Output Output Analysis for Inference Output Analysis for Improving Efficiency MODEL DETERMINATION USING SAMPLING-BASED METHODS Introduction Classical Approaches The Bayesian Perspective and the Bayes Factor Alternative Predictive Distributions How to Use Predictive Distributions Computational Issues An Example Discussion HYPOTHESIS TESTING AND MODEL SELECTION Introduction Uses of Bayes Factors Marginal Likelihood Estimation by Importance Sampling Marginal Likelihood Estimation Using Maximum Likelihood Application: How Many Components in a Mixture? Discussion Appendix: S-PLUS Code for the Laplace-Metropolis Estimator MODEL CHECKING AND MODEL IMPROVEMENT Introduction Model Checking Using Posterior Predictive Simulation Model Improvement via Expansion Example: Hierarchical Mixture Modelling of Reaction Times STOCHASTIC SEARCH VARIABLE SELECTION Introduction A Hierarchical Bayesian Model for Variable Selection Searching the Posterior by Gibbs Sampling Extensions Constructing Stock Portfolios With SSVS Discussion BAYESIAN MODEL COMPARISON VIA JUMP DIFFUSIONS Introduction Model Choice Jump-Diffusion Sampling Mixture Deconvolution Object Recognition Variable Selection Change-Point Identification Conclusions ESTIMATION AND OPTIMIZATION OF FUNCTIONS Non-Bayesian Applications of MCMC Monte Carlo Optimization Monte Carlo Likelihood Analysis Normalizing-Constant Families Missing Data Decision Theory Which Sampling Distribution? Importance Sampling Discussion STOCHASTIC EM: METHOD AND APPLICATION Introduction The EM Algorithm The Stochastic EM Algorithm Examples GENERALIZED LINEAR MIXED MODELS Introduction Generalized Linear Models (GLMs) Bayesian Estimation of GLMs Gibbs Sampling for GLMs Generalized Linear Mixed Models (GLMMs) Specification of Random-Effect Distributions Hyperpriors and the Estimation of Hyperparameters Some Examples Discussion HIERARCHICAL LONGITUDINAL MODELLING Introduction Clinical Background Model Detail and MCMC Implementation Results Summary and Discussion MEDICAL MONITORING Introduction Modelling Medical Monitoring Computing Posterior Distributions Forecasting Model Criticism Illustrative Application Discussion MCMC FOR NONLINEAR HIERARCHICAL MODELS Introduction Implementing MCMC Comparison of Strategies A Case Study from Pharmacokinetics-Pharmacodynamics Extensions and Discussion BAYESIAN MAPPING OF DISEASE Introduction Hypotheses and Notation Maximum Likelihood Estimation of Relative Risks Hierarchical Bayesian Model of Relative Risks Empirical Bayes Estimation of Relative Risks Fully Bayesian Estimation of Relative Risks Discussion MCMC IN IMAGE ANALYSIS Introduction The Relevance of MCMC to Image Analysis Image Models at Different Levels Methodological Innovations in MCMC Stimulated by Imaging Discussion MEASUREMENT ERROR Introduction Conditional-Independence Modelling Illustrative examples Discussion GIBBS SAMPLING METHODS IN GENETICS Introduction Standard Methods in Genetics Gibbs Sampling Approaches MCMC Maximum Likelihood Application to a Family Study of Breast Cancer Conclusions MIXTURES OF DISTRIBUTIONS: INFERENCE AND ESTIMATION Introduction The Missing Data Structure Gibbs Sampling Implementation Convergence of the Algorithm Testing for Mixtures Infinite Mixtures and Other Extensions AN ARCHAEOLOGICAL EXAMPLE: RADIOCARBON DATING Introduction Background to Radiocarbon Dating Archaeological Problems and Questions Illustrative Examples Discussion Index
7,399 citations
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01 Jan 1999TL;DR: This new edition contains five completely new chapters covering new developments and has sold 4300 copies worldwide of the first edition (1999).
Abstract: We have sold 4300 copies worldwide of the first edition (1999). This new edition contains five completely new chapters covering new developments.
6,884 citations