scholarly journals A Comparison between Maximum Likelihood and Bayesian Estimation Methods for a Shape Parameter of the Weibull-Exponential Distribution

2018 ◽  
pp. 1-12
Author(s):  
Terna G. Ieren ◽  
Pelumi E. Oguntunde
Keyword(s):  
Maximum Likelihood ◽  
Exponential Distribution ◽  
Loss Function ◽  
Shape Parameter ◽  
Estimation Methods ◽  
Quadratic Loss ◽  
Posterior Distributions ◽  
Bayes Estimators ◽  
Method Of Maximum Likelihood ◽  
Parameter Values

We considered the Bayesian analysis of a shape parameter of the Weibull-Exponential distribution in this paper. We assumed a class of non-informative priors in deriving the corresponding posterior distributions. In particular, the Bayes estimators and associated risks were calculated under three different loss functions. The performance of the Bayes estimators was evaluated and compared to the method of maximum likelihood under a comprehensive simulation study. It was discovered that for the said parameters to be estimated, the quadratic loss function under both uniform and Jeffrey’s priors should be used for decreasing parameter values while the use of precautionary loss function can be preferred for increasing parameter values irrespective of the variations in sample size.  

scholarly journals The Bayesian Estimation for The Shape Parameter of The Power Function Distribution (PFD-I) to Use Hyper Prior Functions

2021 ◽  
Vol 27 (127) ◽  
pp. 229-252
Author(s):  
Jinan Abbas Naser Al-obedy
Keyword(s):  
Maximum Likelihood ◽  
Exponential Distribution ◽  
Power Function ◽  
Loss Function ◽  
Shape Parameter ◽  
Loss Functions ◽  
Erlang Distribution ◽  
Bayes Estimators ◽  
Function Distribution ◽  
Power Function Distribution

The objective of this study is to examine the properties of Bayes estimators of the shape parameter of the Power Function Distribution (PFD-I), by using two different prior distributions for the parameter θ and different loss functions that were compared with the maximum likelihood estimators. In many practical applications, we may have two different prior information about the prior distribution for the shape parameter of the Power Function Distribution, which influences the parameter estimation. So, we used two different kinds of conjugate priors of shape parameter θ of the Power Function Distribution (PFD-I) to estimate it. The conjugate prior function of the shape parameter θ was considered as a combination of two different prior distributions such as gamma distribution with Erlang distribution and Erlang distribution with exponential distribution and Erlang distribution with non-informative distribution and exponential distribution with the non-informative distribution. We derived Bayes estimators for shape parameter θ of the Power Function Distribution (PFD-I) according to different loss functions such as the squared error loss function (SELF), the weighted error loss function (WSELF) and modified linear exponential (MLINEX) loss function (MLF), with two different double priors. In addition to the classical estimation (maximum likelihood estimation). We used simulation to get the results of this study, for different cases of the shape parameter of the Power Function Distribution used to generate data for different samples sizes.

scholarly journals Bayesian Analysis of Weibull-Lindley Distribution Using Different Loss Functions

2020 ◽  
pp. 28-41
Author(s):  
Innocent Boyle Eraikhuemen ◽  
Olateju Alao Bamigbala ◽  
Umar Alhaji Magaji ◽  
Bassa Shiwaye Yakura ◽  
Kabiru Ahmed Manju
Keyword(s):  
Bayesian Analysis ◽  
Loss Function ◽  
Bayesian Methods ◽  
Shape Parameter ◽  
Loss Functions ◽  
Quadratic Loss ◽  
Function Estimation ◽  
Lindley Distribution ◽  
Method Of Maximum Likelihood ◽  
Parameter Values

In the present paper, a three-parameter Weibull-Lindley distribution is considered for Bayesian analysis. The estimation of a shape parameter of Weibull-Lindley distribution is obtained with the help of both the classical and Bayesian methods. Bayesian estimators are obtained by using Jeffrey’s prior, uniform prior and Gamma prior under square error loss function, quadratic loss function and Precautionary loss function. Estimation by the method of Maximum likelihood is also discussed. These methods are compared by using mean square error through simulation study with varying parameter values and sample sizes.

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 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.

A bivariate Kumaraswamy-exponential distribution with application

2019 ◽  
Vol 69 (5) ◽  
pp. 1185-1212
Author(s):  
Hassan S. Bakouch ◽  
Fernando A. Moala ◽  
Abdus Saboor ◽  
Haniya Samad
Keyword(s):  
Maximum Likelihood ◽  
Exponential Distribution ◽  
Information Matrix ◽  
Bivariate Distribution ◽  
Conditional Density ◽  
Estimation Methods ◽  
Density Functions ◽  
Marginal Density ◽  
Data Set ◽  
Expected Information

Abstract In this paper, we introduce a new bivariate Kumaraswamy exponential distribution, whose marginals are univariate Kumaraswamy exponential. Some probabilistic properties of this bivariate distribution are derived, such as joint density function, marginal density functions, conditional density functions, moments and stress-strength reliability. Also, we provide the expected information matrix with its elements in a closed form. Estimation of the parameters is investigated by the maximum likelihood, Bayesian and least squares estimation methods. A simulation study is carried out to compare the performance of the estimators by estimation methods. Further, one data set have been analyzed to show how the proposed distribution works in practice.

scholarly journals Bayesian Inference on the Shape Parameter and Future Observation of Exponentiated Family of Distributions

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Sanku Dey ◽  
Sudhansu S. Maiti
Keyword(s):  
Loss Function ◽  
Shape Parameter ◽  
Quadratic Loss ◽  
Posterior Density ◽  
Scale Invariant ◽  
Data Set ◽  
Future Observation ◽  
Hpd Intervals ◽  
Highest Posterior Density ◽  
Family Of Distributions

The Bayes estimators of the shape parameter of exponentiated family of distributions have been derived by considering extension of Jeffreys' noninformative as well as conjugate priors under different scale-invariant loss functions, namely, weighted quadratic loss function, squared-log error loss function and general entropy loss function. The risk functions of these estimators have been studied. We have also considered the highest posterior density (HPD) intervals for the parameter and the equal-tail and HPD prediction intervals for future observation. Finally, we analyze one data set for illustration.

scholarly journals Bayesian and Maximum Likelihood Estimation of the Shape Parameter of Exponential Inverse Exponential Distribution: A Comparative Approach

2020 ◽  
pp. 28-43
Author(s):  
Innocent Boyle Eraikhuemen ◽  
Fadimatu Bawuro Mohammed ◽  
Ahmed Askira Sule
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Bayesian Estimation ◽  
Exponential Distribution ◽  
Shape Parameter ◽  
Likelihood Estimation ◽  
Monte Carlo Study ◽  
Comparative Approach ◽  
The Mean ◽  
Inverse Exponential Distribution

This paper aims at making Bayesian analysis on the shape parameter of the exponential inverse exponential distribution using informative and non-informative priors. Bayesian estimation was carried out through a Monte Carlo study under 10,000 replications. To assess the effects of the assumed prior distributions and loss function on the Bayesian estimators, the mean square error has been used as a criterion. Overall, simulation results indicate that Bayesian estimation under QLF outperforms the maximum likelihood estimation and Bayesian estimation under alternative loss functions irrespective of the nature of the prior and the sample size. Also, for large sample sizes, all methods perform equally well.

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.

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