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Abstract
The loss function plays an important role in Bayesian analysis and decision theory. In this paper, a new Bayesian approach is introduced for parameter estimation under the asymmetric linear-exponential (LINEX) loss function. In order to provide a robust estimation and avoid making subjective choices, the proposed method assumes that the parameter of the LINEX loss function has a probability distribution. The Bayesian estimator is then obtained by taking the expectation of the common LINEX-based Bayesian estimator over the probability distribution. This alternative proposed method is applied to estimate the exponential parameter by considering three different distributions of the LINEX parameter, and the associated Bayes risks are also obtained in consequence. Extensive simulation studies are conducted in order to compare the performance of the proposed new estimators. In addition, three real data sets are analyzed to investigate the applicability of the proposed results. The results of the simulation and real data analysis show that the proposed estimation works satisfactorily and performs better than the conventional standard Bayesian approach in terms of minimum mean square error and Bayes risk.
Highlights
In Bayesian analysis and decision theory, the loss function plays an important role as it can be used to describe the overestimation and underestimation in analysis
In terms of minimum mean square error (MSE), we can conclude that the Bayesian estimates using the expected LINEX loss function perform better than all other estimates for π3(φ) with d = 5 when β = 0.5;
For avoiding the subjective choice of the LINEX parameter and giving a relatively objective balance between overestimation and underestimation, the proposed alternative method assumes that the parameter φ of the LINEX loss function has a probability distribution in the range of (−∞, ∞)
Summary
In Bayesian analysis and decision theory, the loss function plays an important role as it can be used to describe the overestimation and underestimation in analysis. Compared with the symmetric loss, the asymmetric loss is more realistic and useful in practical applications. Many other authors, including Zellner [1], Chang and Huang [2] and Khatun and Matin [3], pointed out that commonly used symmetric loss functions such as squared error (SE) loss may be inappropriate in practical applications. Asymmetric loss functions are widely applied in statistical inference; see, for example, Ali [4,5]. One of the most useful asymmetric loss functions is called LINEX loss, first introduced by Klebanov [6] and used by Varian [7] in his study of real estate assessment. Let θ be the unknown parameter to be estimated and θbe an estimate of θ; in this case, the LINEX loss function can be written as
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