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 under Different Loss Functions Using Gamma Prior for the Case of Exponential Distribution

2017 ◽  
Vol 9 (1) ◽  
pp. 67-78
Author(s):  
M. R. Hasan ◽  
A. R. Baizid
Keyword(s):  
Bayesian Estimation ◽  
Exponential Distribution ◽  
Loss Function ◽  
Mean Squared Error ◽  
Simulated Data ◽  
Life Time ◽  
Loss Functions ◽  
Quadratic Loss ◽  
Squared Error Loss Function ◽  
Squared Error

The Bayesian estimation approach is a non-classical estimation technique in statistical inference and is very useful in real world situation. The aim of this paper is to study the Bayes estimators of the parameter of exponential distribution under different loss functions and compared among them as well as with the classical estimator named maximum likelihood estimator (MLE). Since exponential distribution is the life time distribution, we have studied exponential distribution using gamma prior. Here the gamma prior is used as the prior distribution of exponential distribution for finding the Bayes estimator. In our study we also 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. We have used simulated data using R-coding to find out the mean squared error (MSE) of different loss functions and hence found that non-classical estimator is better than classical estimator. Finally, mean square error (MSE) of the estimators of different loss functions are presented graphically.

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 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 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 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 Bayesian Analysis of the Weibull Paired Comparison Model Using Numerical Approximation

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Khalil Ullah ◽  
Muhammad Aslam
Keyword(s):  
Loss Function ◽  
Goodness Of Fit ◽  
Numerical Approximation ◽  
Paired Comparison ◽  
Estimation Procedure ◽  
Loss Functions ◽  
Quadratic Loss ◽  
Chi Square ◽  
Squared Error Loss Function ◽  
Comparison Model

The method of paired comparisons (PC) is widely used to rank items using sensory evaluations. The PC models are developed to provide basis for such comparisons. In this study, the Weibull PC model is analyzed under the Bayesian paradigm using noninformative priors and different loss functions, namely, Squared Error Loss Function (SELF), Quadratic Loss Function (QLF), DeGroot Loss Function (DLF), and Precautionary Loss Function (PLF). Numerical approximation is used to illustrate the entire estimation procedure. A real dataset showing usage preferences for different cellphone brands, Huawei (HW), Samsung (SS), Oppo (OP), QMobile (QM), and Nokia (NK), is used. Quadrature method is used to evaluate the Bayes estimates, their posterior risks, preference probabilities, predictive probabilities, and posterior probabilities to establish and verify ranking order of the competing cellphone brands under study. The results show that the paired comparison model under the study using Bayesian approach involving various loss functions can offer mathematical approach to evaluate cellphone brand preferences. The ranking provided by the model is justifiable according to the usage preference for these cellphone brands. The ranking given by the model indicates that cellphone brand Samsung is preferred the most and QMobile is the least preferred. The plausibility of the model is also assessed using the Chi square test of goodness of fit.

scholarly journals A NOTE ON PROPERTIES OF EXPONENTIAL POWER DISTRIBUTION

2020 ◽  
pp. 145-150
Author(s):  
Akinlolu Olosunde
Keyword(s):  
Exponential Distribution ◽  
Power Distribution ◽  
Real Life ◽  
Distribution Functions ◽  
Exponential Power Distribution ◽  
Double Exponential Distribution ◽  
Life Data ◽  
Real Life Data ◽  
Double Exponential ◽  
Exponential Power

The most common generalization of the normal, Kotz-symmetric and double exponential distribution functions was the exponential power distribution. This distribution had been found useful in modeling real life data as studied in the literature. The present study found it necessary to fill the void in the literature by presenting some properties which characterized exponential power distribution and further made it useful in applications.

scholarly journals On A Shape Parameter of Gompertz Inverse Exponential Distribution Using Classical and Non Classical Methods of Estimation

2020 ◽  
pp. 1-10
Author(s):  
Terna Godfrey Ieren ◽  
Adana’a Felix Chama ◽  
Olateju Alao Bamigbala ◽  
Jerry Joel ◽  
Felix M. Kromtit ◽  
...  
Keyword(s):  
Exponential Distribution ◽  
Loss Function ◽  
Real Life ◽  
Quadratic Loss ◽  
Prior Distributions ◽  
Quadratic Loss Function ◽  
Informative Prior ◽  
Informative Prior Distribution ◽  
Inverse Exponential Distribution ◽  
Bayesian Estimators

The Gompertz inverse exponential distribution is a three-parameter lifetime model with greater flexibility and performance for analyzing real life data. It has one scale parameter and two shape parameters responsible for the flexibility of the distribution. Despite the importance and necessity of parameter estimation in model fitting and application, it has not been established that a particular estimation method is better for any of these three parameters of the Gompertz inverse exponential distribution. This article focuses on the development of Bayesian estimators for a shape of the Gompertz inverse exponential distribution using two non-informative prior distributions (Jeffery and Uniform) and one informative prior distribution (Gamma prior) under Square error loss function (SELF), Quadratic loss function (QLF) and Precautionary loss function (PLF). These results are compared with the maximum likelihood counterpart using Monte Carlo simulations. Our results indicate that Bayesian estimators under Quadratic loss function (QLF) with any of the three prior distributions provide the smallest mean square error for all sample sizes and different values of parameters.

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 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.

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