scholarly journals The Bayes Estimators of the Variance and Scale Parameters of the Normal Model With a Known Mean for the Conjugate and Noninformative Priors Under Stein’s Loss

2022 ◽  
Vol 4 ◽  
Author(s):  
Ying-Ying Zhang ◽  
Teng-Zhong Rong ◽  
Man-Man Li
Keyword(s):  
Loss Function ◽  
Bayes Estimator ◽  
Bayes Estimation ◽  
Normal Model ◽  
Bayes Estimators ◽  
Conjugate Prior ◽  
Squared Error Loss Function ◽  
Noninformative Priors ◽  
Scale Parameters ◽  
Variance Parameter

For the normal model with a known mean, the Bayes estimation of the variance parameter under the conjugate prior is studied in Lehmann and Casella (1998) and Mao and Tang (2012). However, they only calculate the Bayes estimator with respect to a conjugate prior under the squared error loss function. Zhang (2017) calculates the Bayes estimator of the variance parameter of the normal model with a known mean with respect to the conjugate prior under Stein’s loss function which penalizes gross overestimation and gross underestimation equally, and the corresponding Posterior Expected Stein’s Loss (PESL). Motivated by their works, we have calculated the Bayes estimators of the variance parameter with respect to the noninformative (Jeffreys’s, reference, and matching) priors under Stein’s loss function, and the corresponding PESLs. Moreover, we have calculated the Bayes estimators of the scale parameter with respect to the conjugate and noninformative priors under Stein’s loss function, and the corresponding PESLs. The quantities (prior, posterior, three posterior expectations, two Bayes estimators, and two PESLs) and expressions of the variance and scale parameters of the model for the conjugate and noninformative priors are summarized in two tables. After that, the numerical simulations are carried out to exemplify the theoretical findings. Finally, we calculate the Bayes estimators and the PESLs of the variance and scale parameters of the S&P 500 monthly simple returns for the conjugate and noninformative priors.

scholarly journals Bayes Estimators of Exponentiated Inverse Rayleigh Distribution using Lindleys Approximation

2021 ◽  
pp. 60-71
Author(s):  
Bashiru Omeiza Sule ◽  
Taiwo Mobolaji Adegoke ◽  
Kafayat Tolani Uthman
Keyword(s):  
Loss Function ◽  
Posterior Distribution ◽  
Real Life ◽  
Rayleigh Distribution ◽  
Loss Functions ◽  
Bayes Estimators ◽  
Shape Parameters ◽  
True Parameter ◽  
Squared Error Loss Function ◽  
Scale Parameters

In this paper, Bayes estimators of the unknown shape and scale parameters of the Exponentiated Inverse Rayleigh Distribution (EIRD) have been derived using both the frequentist and bayesian methods. The Bayes theorem was adopted to obtain the posterior distribution of the shape and scale parameters of an Exponentiated Inverse Rayleigh Distribution (EIRD) using both conjugate and non-conjugate prior distribution under different loss functions (such as Entropy Loss Function, Linex Loss Function and Scale Invariant Squared Error Loss Function). The posterior distribution derived for both shape and scale parameters are intractable and a Lindley approximation was adopted to obtain the parameters of interest. The loss function were employed to obtain the estimates for both scale and shape parameters with an assumption that the both scale and shape parameters are unknown and independent. Also the Bayes estimate for the simulated datasets and real life datasets were obtained. The Bayes estimates obtained under dierent loss functions are close to the true parameter value of the shape and scale parameters. The estimators are then compared in terms of their Mean Square Error (MSE) using R programming language. We deduce that the MSE reduces as the sample size (n) increases.

scholarly journals Bayesian Inference of Burr Type VIII Distribution Based on Censored Samples

2014 ◽  
Vol 2014 ◽  
pp. 1-21
Author(s):  
Navid Feroz
Keyword(s):  
Bayesian Inference ◽  
Loss Function ◽  
Simulation Study ◽  
Bayes Estimation ◽  
Loss Functions ◽  
Quadratic Loss ◽  
Bayes Estimators ◽  
Type Viii ◽  
Censored Samples ◽  
Made In

This paper is concerned with estimation of the parameter of Burr type VIII distribution under a Bayesian framework using censored samples. The Bayes estimators and associated risks have been derived under the assumption of five priors and three loss functions. The comparison among the performance of different estimators has been made in terms of posterior risks. A simulation study has been conducted in order to assess and compare the performance of different estimators. The study proposes the use of inverse Levy prior based on quadratic loss function for Bayes estimation of the said parameter.

scholarly journals Bayesian Estimation of Inequality and Poverty Indices in Case of Pareto Distribution Using Different Priors under LINEX Loss Function

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Kamaljit Kaur ◽  
Sangeeta Arora ◽  
Kalpana K. Mahajan
Keyword(s):  
Loss Function ◽  
Pareto Distribution ◽  
Loss Functions ◽  
Conjugate Prior ◽  
Jeffreys Prior ◽  
Linex Loss Function ◽  
Poverty Measure ◽  
Noninformative Priors ◽  
Simulation Techniques ◽  
Linex Loss

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.

scholarly journals Bayesian Estimation of the Exponential Parameter under a Multiply Type-II Censoring Scheme

2016 ◽  
Vol 36 (3) ◽  
Author(s):  
Umesh Singh ◽  
Anil Kumar
Keyword(s):  
Monte Carlo ◽  
Loss Function ◽  
Scale Parameter ◽  
Type Ii ◽  
Bayes Estimators ◽  
Conjugate Prior ◽  
Exponential Parameter ◽  
Type Ii Censoring ◽  
Error Loss ◽  
Censoring Scheme

This paper provides the estimation of the scale parameter of the exponential distribution under multiply type-II censoring. Using generalized non-informative prior and natural conjugate prior, Bayes estimator and approximate Bayes estimators of the scale parameter have been obtained under square error loss function. The proposed Bayes estimators and approximate Bayes estimators are compared with the estimators proposed by Singh et al. (2005) and Balasubramanian and Balakrishnan (1992) on the basis of theirsimulated risks under square error loss function of 1000 randomly generated Monte Carlo samples.

Designing optimal estimators for fish stock assessment

1998 ◽  
Vol 55 (2) ◽  
pp. 376-386 ◽  
Author(s):  
Patrick L Cordue
Keyword(s):  
Loss Function ◽  
Stock Assessment ◽  
Estimation Procedure ◽  
Small Sample ◽  
Bayes Estimation ◽  
Fish Stock ◽  
Linear Transformations ◽  
Squared Error Loss Function ◽  
Error Loss ◽  
Fish Stock Assessment

Many estimation procedures are used in the provision of fisheries stock assessment advice. Most procedures use estimators that have optimal large-sample characteristics, but these are often applied to small-sample data sets. In this paper, a minimum integrated average expected loss (MIAEL) estimation procedure is presented. By its design a MIAEL estimator has optimal characteristics for the type of data it is applied to, given that the model assumptions of the particular problem are satisfied. The estimation procedure is developed within a decision-theoretic framework and illustrated with a Bernoulli and a fisheries example. MIAEL estimation is related to optimal Bayes estimation, as both procedures seek an estimator that minimizes an integrated loss function. In most fisheries applications a global MIAEL estimator will be difficult to determine, and a MIAEL estimator will need to be found within a given class of estimators. "Squared f-error," a generalization of the common squared error loss function is defined. It is shown that an estimator can be improved (for a given squared f-error loss function) by using its best linear transformation which is the MIAEL estimator within the class of linear transformations (in f space).

scholarly journals Statistical Inference of the Rayleigh Distribution Based on Progressively Type II Censored Competing Risks Data

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 898 ◽  
Author(s):  
Hongyi Liao ◽  
Wenhao Gui
Keyword(s):  
Competing Risks ◽  
Loss Function ◽  
Information Matrix ◽  
Real Life ◽  
Rayleigh Distribution ◽  
Bayes Estimation ◽  
Type Ii ◽  
Data Set ◽  
Squared Error Loss Function ◽  
Highest Posterior Density

A competing risks model under progressively type II censored data following the Rayleigh distribution is considered. We establish the maximum likelihood estimation for unknown parameters and compute the observed information matrix and the expected Fisher information matrix to construct the asymptotic confidence intervals. Moreover, we obtain the Bayes estimation based on symmetric and non-symmetric loss functions, that is, the squared error loss function and the general entropy loss function, and the highest posterior density intervals are also derived. In addition, a simulation study is presented to assess the performances of different methods discussed in this paper. A real-life data set analysis is provided for illustration purposes.

scholarly journals The Comparison Between the MLE and Standard Bayes Estimators of the Reliability Function of Exponential Distribution

2019 ◽  
Vol 32 (1) ◽  
pp. 103
Author(s):  
Mohammed Jamel Ali ◽  
Hazim Mansoor Gorgees
Keyword(s):  
Loss Function ◽  
Reliability Function ◽  
Loss Functions ◽  
Mean Square ◽  
Bayes Estimators ◽  
Squared Error Loss Function ◽  
Monte Carlo Simulation Technique ◽  
Error Loss ◽  
Squared Error ◽  
The One

     In this paper, a Monte Carlo Simulation technique is used to compare the performance of MLE and the standard Bayes estimators of the reliability function of the one parameter exponential distribution.Two types of loss functions are adopted, namely, squared error  loss function (SELF) and modified square error loss function (MSELF) with informative and non- informative prior. The criterion integrated mean square error (IMSE) is employed to assess the performance of such estimators .

scholarly journals Bayesian Estimation of Parameters of Weibull Distribution Using Linex Error Loss Function

2020 ◽  
Vol 9 (2) ◽  
pp. 38
Author(s):  
Josphat. K. Kinyanjui ◽  
Betty. C. Korir
Keyword(s):  
Weibull Distribution ◽  
Loss Function ◽  
Scale Parameter ◽  
Risk Function ◽  
Bayes Estimators ◽  
Squared Error Loss Function ◽  
Squared Error Loss ◽  
Error Loss ◽  
Squared Error ◽  
Linex Loss

This paper develops a Bayesian analysis of the scale parameter in the Weibull distribution with a scale parameter  θ  and shape parameter  β (known). For the prior distribution of the parameter involved, inverted Gamma distribution has been examined. Bayes estimates of the scale parameter, θ  , relative to LINEX loss function are obtained. Comparisons in terms of risk functions of those under LINEX loss and squared error loss functions with their respective alternate estimators, viz: Uniformly Minimum Variance Unbiased Estimator (U.M.V.U.E) and Bayes estimators relative to squared error loss function are made. It is found that Bayes estimators relative to squared error loss function dominate the alternative estimators in terms of risk function.

scholarly journals The Comparison Between Standard Bayes Estimators of the Reliability Function of Exponential Distribution

2018 ◽  
Vol 31 (3) ◽  
pp. 135 ◽  
Author(s):  
Mohammed Jamel Ali ◽  
Hazim Mansoor Gorgees ◽  
Adel Abdul Kadhim Hussein
Keyword(s):  
Exponential Distribution ◽  
Loss Function ◽  
Reliability Function ◽  
Loss Functions ◽  
Bayes Estimators ◽  
Squared Error Loss Function ◽  
Monte Carlo Simulation Technique ◽  
Error Loss ◽  
Squared Error ◽  
The One

   In this paper, a Monte Carlo Simulation technique is used to compare the performance of the standard Bayes estimators of the reliability function of the one parameter exponential distribution .Three types of loss functions are adopted, namely, squared error  loss function (SELF) ,Precautionary error loss function (PELF) andlinear exponential error  loss function(LINEX) with informative and non- informative prior .The criterion integrated mean square error (IMSE) is employed to assess the performance of such estimators

scholarly journals Comparison of Maximum Likelihood and some Bayes Estimators for Maxwell Distribution based on Non-informative Priors

2013 ◽  
Vol 10 (2) ◽  
pp. 480-488 ◽  
Author(s):  
Baghdad Science Journal
Keyword(s):  
Maximum Likelihood ◽  
Loss Function ◽  
Maxwell Distribution ◽  
Bayes Estimators ◽  
Jeffreys Prior ◽  
Likelihood Estimator ◽  
Squared Error Loss Function ◽  
Squared Error Loss ◽  
Error Loss ◽  
Squared Error

In this paper, Bayes estimators of the parameter of Maxwell distribution have been derived along with maximum likelihood estimator. The non-informative priors; Jeffreys and the extension of Jeffreys prior information has been considered under two different loss functions, the squared error loss function and the modified squared error loss function for comparison purpose. A simulation study has been developed in order to gain an insight into the performance on small, moderate and large samples. The performance of these estimators has been explored numerically under different conditions. The efficiency for the estimators was compared according to the mean square error MSE. The results of comparison by MSE show that the efficiency of Bayes estimators of the shape parameter of the Maxwell distribution decreases with the increase of Jeffreys prior constants. The results also show that values of Bayes estimators are almost close to the maximum likelihood estimator when the Jeffreys prior constants are small, yet they are identical in some certain cases. Comparison with respect to loss functions show that Bayes estimators under the modified squared error loss function has greater MSE than the squared error loss function especially with the increase of r.

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