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 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 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 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 Analysis of a Shape Parameter of the Weibull-Frechet Distribution

2018 ◽  
pp. 1-19
Author(s):  
Terna Godfrey Ieren ◽  
Angela Unna Chukwu
Keyword(s):  
Loss Function ◽  
Shape Parameter ◽  
Real Life ◽  
Loss Functions ◽  
Quadratic Loss ◽  
Bayes Estimators ◽  
Informative Priors ◽  
Fréchet Distribution ◽  
Minimum Bayes Risk ◽  
Frechet Distribution

In this paper, we estimate a shape parameter of the Weibull-Frechet distribution by considering the Bayesian approach under two non-informative priors using three different loss functions. We derive the corresponding posterior distributions for the shape parameter of the Weibull-Frechet distribution assuming that the other three parameters are known. The Bayes estimators and associated posterior               risks have also been derived using the three different loss functions. The performance of the Bayes estimators are evaluated and compared using a comprehensive simulation study and a real life application to find out the combination of a loss function and a prior having the minimum Bayes risk and hence producing the best results. In conclusion, this study reveals that in order to estimate the parameter in question, we should use quadratic loss function under either of the two non-informative priors used in this study.  

scholarly journals On estimation of Frechet distribution with known shape using Bayesian analysis under informative priors

2017 ◽  
Vol 5 (2) ◽  
pp. 141
Author(s):  
Wajiha Nasir
Keyword(s):  
Bayesian Analysis ◽  
Loss Function ◽  
Simulation Study ◽  
Posterior Distribution ◽  
Loss Functions ◽  
Quadratic Loss ◽  
Bayes Estimators ◽  
Informative Priors ◽  
Fréchet Distribution ◽  
Frechet Distribution

In this study, Frechet distribution has been studied by using Bayesian analysis. Posterior distribution has been derived by using gamma and exponential. Bayes estimators and their posterior risks has been derived using five different loss functions. Elicitation of hyperparameters has been done by using prior predictive distributions. Simulation study is carried out to study the behavior of posterior distribution. Quasi quadratic loss function and exponential prior are found better among all.

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 A Comparison of Bayes Estimators for the parameter of Rayleigh Distribution with Simulation

2018 ◽  
Vol 24 (106) ◽  
pp. 49
Author(s):  
جنان عباس ناصر
Keyword(s):  
Loss Function ◽  
Prior Distribution ◽  
Mean Squared Error ◽  
Rayleigh Distribution ◽  
Square Root ◽  
Bayes Estimators ◽  
Squared Error Loss Function ◽  
Squared Error Loss ◽  
True Value ◽  
Squared Error

   A comparison of double informative and non- informative priors assumed for the parameter of Rayleigh distribution is considered. Three different sets of double priors are included, for a single unknown parameter of Rayleigh distribution. We have assumed three double priors: the square root inverted gamma (SRIG) - the natural conjugate family of priors distribution, the square root inverted gamma – the non-informative distribution, and the natural conjugate family of priors - the non-informative distribution as double priors .The data is generating form three cases from Rayleigh distribution for different samples sizes (small, medium, and large). And Bayes estimators for the parameter is derived under a squared error loss function and weighted squared error loss function) in the cases of the three different sets of prior distributions .Simulations is employed to obtain results. And determine the best estimator according to the smallest value of mean squared error and weighted mean squared error. We found  that the best estimation for the parameter for all sample sizes (n) , when the double prior distribution for  is SRIG - the natural conjugate family of priors distribution with values (a=5, b=0.5, =8, =0.5) and (a=8, b=1, =5, =1) for the  true value of  respectively .Also ,we obtained the best estimation for  when the double prior distribution for  is the natural conjugate family of priors-non-informative distribution with values(=0.5, =5, c=1) for  the true value of ().  

scholarly journals Bayes Estimation under Conjugate Prior for the Case of Laplace Double Exponential Distribution

2018 ◽  
Vol 40 (1) ◽  
pp. 151-168
Author(s):  
Md Habibur Rahman ◽  
MK Roy
Keyword(s):  
Bayesian Estimation ◽  
Exponential Distribution ◽  
Loss Function ◽  
Real Life ◽  
Bayes Estimation ◽  
Loss Functions ◽  
Quadratic Loss ◽  
Squared Error Loss Function ◽  
Double Exponential Distribution ◽  
Double Exponential

The Bayesian estimation approach is a non-classical device in the estimation part of statistical inference which is very useful in real world situation. The main objective of this paper is to study the Bayes estimators of the parameter of Laplace double exponential distribution. In Bayesian estimation loss function, prior distribution and posterior distribution are the most important ingredients. In real life we try to minimize the loss and want to know some prior information about the problem to solve it accurately. The well known conjugate priors are considered for finding the Bayes estimator. In our study we have used different symmetric and asymmetric loss functions such as squared error loss function, quadratic loss function, modified linear exponential (MLINEX) loss function and non-linear exponential (NLINEX) loss function. The performance of the obtained estimators for different types of loss functions are then compared among themselves as well as with the classical maximum likelihood estimator (MLE). Mean Square Error (MSE) of the estimators are also computed and presented in graphs. The Chittagong Univ. J. Sci. 40 : 151-168, 2018

scholarly journals Bayesian Estimation of the Parameter of the p-Dimensional Size-Biased Rayleigh Distribution

2017 ◽  
Vol 56 (1) ◽  
pp. 88-91
Author(s):  
Arun Kumar Rao ◽  
Himanshu Pandey ◽  
Kusum Lata Singh
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Bayesian Estimation ◽  
Density Function ◽  
Rayleigh Distribution ◽  
Survival Model ◽  
Loss Functions ◽  
Bayes Estimators ◽  
Hazard Functions ◽  
Squared Error

In this paper, we have derived the probability density function of the size-biased p-dimensional Rayleigh distribution and studied its properties. Its suitability as a survival model has been discussed by obtaining its survival and hazard functions. We also discussed Bayesian estimation of the parameter of the size-biased p-dimensional Rayleigh distribution. Bayes estimators have been obtained by taking quasi-prior. The loss functions used are squared error and precautionary.

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