Bayesian Analysis of Dynamic Cumulative Residual Entropy for Lindley Distribution.

Some new results on the cumulative residual entropy

To analyze the performance of various estimators, this paper compares Bayesian and classical estimate approaches across a variety of loss functions. The work focuses on informative versus non-informative priors in Bayesian estimation. The calculations are based on a number of loss functions, including the Square Error Loss Function (SELF), Quadratic Loss Function (QLF), Precautionary Loss Function (PLF), and Entropy Loss Function (ELF). The methodology uses simulation techniques as well as real-world datasets to test the performance and robustness of the proposed estimators. The Bayesian estimators, classical estimators, and related Bayes hazards using empirical analysis and extensive simulations to computed. The findings show that the Minimum Mean Square Error estimator (MiniMSE) is more accurate and reliable than other estimators in a variety of settings. Furthermore, for both types of priors, it is demonstrated that the Bayes risk under the Quadratic Loss Function (QLF) is the lowest among all loss functions considered, including SELF, PLF, and ELF. This suggests that, when compared to other loss functions, QLF is a more effective and balanced criterion for making estimation decisions. The study’s findings provide vital new insights into how to choose the optimum estimate techniques and loss functions for statistical decision theory and practical data processing.

Recently, a new class of uncertainty measure, namely the dynamic cumulative residual Renyi's entropy, has been introduced and studied in the literature. In parallel to that here, we have defined cumulative residual Renyi's entropy for a doubly truncated random variable called interval cumulative residual Renyi's entropy (ICRRE). Several properties, including the effect of monotone transformation, and bounds of ICRRE are discussed. Furthermore, it is shown that the proposed measure uniquely determines the distribution function and obtained characterizations of some well-known life distributions. Finally, we have proposed four nonparametric estimators of ICRRE and compared their performance by using the Monte-Carlo simulation study. A real-life data set is also illustrated based on the best-proposed estimator.

In this paper, we developed linear exponential (LINEX) loss function by emerging weights to produce weighted linear exponential (WLINEX) loss function. Then we utilized WLINEX to derive scale parameter and reliability function of the Weibull distribution based on record values when the shape parameter is known. After, we estimated scale parameter and reliability function of Weibull distribution by using maximum likelihood (ML) estimation and by several Bayes estimations. The Bayes estimates were obtained with respect to symmetric loss function (squared error loss (SEL)), asymmetric loss function (LINEX) and asymmetric loss function (WLINEX). The ML and the different Bayes estimates are compared via a Monte Carlo simulation study. The result of simulation mentioned that the proposed WLINEX loss function is promising and can be used in real environment especially at the case of underestimate where it revealed better performance than LINEX loss function for estimating scale parameter.

Bayesian estimators of Gini index and a Poverty measure are obtained in case of Pareto distribution under censored and complete setup. The said estimators are obtained using two noninformative priors, namely, uniform prior and Jeffreys’ prior, and one conjugate prior under the assumption of Linear Exponential (LINEX) loss function. Using simulation techniques, the relative efficiency of proposed estimators using different priors and loss functions is obtained. The performances of the proposed estimators have been compared on the basis of their simulated risks obtained under LINEX loss function.

This study introduces a robust Bayesian framework for simple-step stress accelerated life testing. For modeling lifetime characteristics, this study adopts the Xgamma distribution as the principal life model. Bayesian parameter estimation is conducted using the Markov Chain Monte Carlo method under three distinct loss functions: the squared error loss function, generalized entropy loss function, and linear exponential loss function. The resulting Bayesian estimators are analyzed and compared with those obtained through the classical maximum likelihood estimation method. Finally, the proposed methods are illustrated with the analysis of two real data sets.

This study examines the estimation of the scale parameter of the Weibull-Power Function Distribution (WPFD) using both maximum likelihood and Bayesian approaches. Bayesian estimation is conducted under one informative Gamma prior with hyperparameters , as well as two non-informative priors: the uniform prior and Jeffreys’ prior. For each prior specification, Bayes estimators are derived under squared error, quadratic, and precautionary loss functions. Closed-form expressions for the posterior distributions and corresponding Bayes estimators are obtained. The finite-sample performance of the competing estimators is evaluated through a Monte Carlo simulation study based on 1000 replications. Estimator performance is assessed using mean squared error (MSE), bias, and coverage probability. The results indicate that all estimators are consistent, with bias and MSE decreasing as sample size increases. Across different prior specifications and parameter settings, the Bayesian estimator under the quadratic loss function consistently attains the lowest MSE, yielding reductions of approximately 10-20% relative to the maximum likelihood estimator in small and moderate samples. These findings suggest that Bayesian estimation under quadratic loss provides improved finite-sample efficiency for estimating the WPFD scale parameter, while maintaining asymptotic comparability with the maximum likelihood approach.

In this paper, we make a statistical inference for the Gompertz distribution based on multiply Type-I censoreddata, where we determine a censoring time point for each unit in the life time test. Estimation of the parameters is obtainedusing maximum likelihood method and Bayesian method, also the asymptotic con dence intervals for the parameters areconstructed based on the approximate of the Fisher information matrix. The necessary condition for existence and uniquenessof the maximum likelihood estimators is discussed. The Bayesian estimates are obtained depending on the squared error lossfunction, linear exponential loss function and the generalized entropy loss function. The One-sample Bayesian predictionintervals are constructed for the unobserved lifetimes in the same sample. A real data example is presented to illustrate themethods of inference developed here. Finally, the simulation study is executed to compare the performance of the proposedmethods.

Statistical inference based on lower record values for the inverse Weibull distribution under the modified loss function

In this research, Log logistic distribution under Bayesian paradigm is studied. Posterior distribution has been derived using Uniform and Jeffrey’s’ priors. The three loss functions taken up are squared error loss function (SELF), weighted balanced loss function (WBLF) and precautionary loss function (PLF). The performance of an estimator is assessed on the basis of its relative posterior risk. An extensive simulation was carried out to illustrate the applicability of this research and to compare the performance of different estimators. The study indicated that for estimation of the shape parameter using Bayesian estimation technique, the precautionary loss function can effectively be employed. Keywords : Bayesian estimation, Log logistic distribution, Uniform prior, Jeffrey’s prior, loss functions.

Most of the data arising in reliability can be modelled by Pareto distribution. In the present paper, we proposed the Bayes estimation of dynamic cumulative residual entropy for the classical Pareto distribution under upper record values. This measure performs important roles in reliability and survival analysis to model and analyse the data. A class of informative and non-informative priors has been assumed to derive the corresponding posterior distributions. The Bayes estimators (BEs) and the associated posterior risks have been calculated under different symmetric and asymmetric loss functions. We demonstrate the use of the proposed Bayesian estimation procedure with the average July temperatures data of Neuenburg, Switzerland, during the period 1864–1993. The performance of the BEs has been evaluated and compared under a comprehensive simulation study. The purpose is to find out the combination of a loss function and a prior having the minimum Bayes risk and hence producing the best results.

In the classical Taguchi quality model, the symmetric quadratic loss function has been used to measure the loss of quality. However, there are a number of situations in which the symmetric quadratic loss may be inappropriate. In this paper, we proposed an asymmetric loss function, called linear exponential (LINEX) loss function, to determine optimum process parameters for the product quality. When the coefficient of LINEX loss function is small, it will be close to the quadratic loss. Moreover, the trade-off problem between quality and cost will be discussed.

This paper investigates the problem of entropy estimation for the generalized Rayleigh distribution under progressively type-II censored samples. Based on progressively type-II censored samples, we first discuss the maximum likelihood estimation and interval estimation of Shannon entropy for the generalized Rayleigh distribution. Then, we explore the Bayesian estimation problem of entropy under three types of loss functions: K-loss function, weighted squared error loss function, and precautionary loss function. Due to the complexity of Bayesian estimation computation, we use the Lindley approximation and MCMC method for calculating Bayesian estimates. Finally, using a Monte Carlo statistical simulation, we compare the mean square errors to examine the superiority of maximum likelihood estimation and Bayesian estimation under different loss functions. An actual example is provided to verify the feasibility and practicality of various estimations.

The effect of different loss functions on Bayesian estimation and its posterior risk is studied in this paper. The definition of quasi-modified squared error loss function is proposed based on the modified squared error loss function, and the formulas of Bayesian estimation and its posterior risk under the quasi-modified squared error loss function are developed, respectively. Moreover, the Bayesian estimations and its posterior risks of binomial distribution parameter under different loss functions are introduced. Monte Carlo simulation and application examples are provided for illustrative purpose, and the results are compared on the basis of posterior risk and mean square error (MSE). Finally, the Bayesian estimations and the Bayesian credible intervals are obtained, respectively, by MCMC method (also using OpenBUGS).

ABSTRACTBayesian analysis of 3-component mixture model, of exponentiated Weibull distribution under type-I right censoring scheme, is explored in this study. With the help of non-informative (uniform and Jeffreys) priors and loss functions (e.g. squared error loss function, quadratic loss function, precautionary loss function and DeGroot loss function), Bayes estimators and posterior risks are derived. The Bayes estimators and posterior risks are observed as a function of the test termination time. Simulation study as well as a practical example is given in this study.