The empirical Bayes estimators of the rate parameter of the inverse gamma distribution with a conjugate inverse gamma prior under Stein's loss function

Sun, Jiankun; Zhang, Ying-Ying; Sun, Yajuan

doi:10.1080/00949655.2020.1858299

Cited by 5 publications

(4 citation statements)

References 33 publications

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“…Since the estimators

and

and the PESLs

and

depend on

and

, where

12

and

0

, we can plot the surfaces of the estimators and the PESLs on the domain

via the R function persp3d() in the R package rgl (see ( Adler and Murdoch, 2017 ; Zhang et al, 2017 ; Zhang et al, 2019 ; Sun et al, 2021 )). We remark that the R function persp() in the R package graphics can not add another surface to the existing surface, but persp3d() can.…”

Section: Numerical Simulationsmentioning

confidence: 99%

“…In these cases, we should not choose the squared error loss function, but choose a loss function which penalizes gross overestimation and gross underestimation equally, that is, an action a will incur an infinite loss when it tends to 0 or ∞ . Stein’s loss function has this property, and thus it is recommended to use for the positive restricted parameter space

by many authors (see for example ( James and Stein, 1961 ; Petropoulos and Kourouklis, 2005 ; Oono and Shinozaki, 2006 ; Bobotas and Kourouklis, 2010 ; Zhang, 2017 ; Xie et al, 2018 ; Zhang et al, 2019 ; Sun et al, 2021 )). In the normal model with a known mean μ , our parameters of interest are θ = σ 2 (a variance parameter) and θ = σ (a scale parameter).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Zhang¹,

Rong²,

Li³

2022

Front. Big Data

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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.

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“…Since the estimators

and

and the PESLs

and

depend on

and

, where

12

and

0

, we can plot the surfaces of the estimators and the PESLs on the domain

Section: Numerical Simulationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Zhang¹,

Rong²,

Li³

2022

Front. Big Data

Self Cite

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show abstract

“…According to the hierarchical Bayesian model, 60 the posterior distribution of influence line identification can be constructed as follows:

p ((),,|, c, σ^{2}, μ^{2}, {boldR}_{boldm}) \propto p (()|,, {boldR}_{boldm}, c, σ^{2}) p ((), σ^{2}) p (()|, c, μ^{2}) p ((), μ^{2})

p (()|,, {boldR}_{boldm}, c, σ^{2})

and

p (()|, c, μ^{2})

are determined by Equations (7) and (8). For the convenience of calculation, the prior probability density functions of parameters

σ^{2}

and

μ^{2}

are determined by the principle of conjugate priori and are expressed by the inverse gamma distribution 61 :

p ((), σ^{2}) \propto \frac{{β_{0}}^{α_{0}}}{normalΓ ((), α_{0})} σ^{- 2 ((), α_{0} goodbreak+ 1)} e^{- β_{0} σ^{- 2}}

p ((), μ^{2}) \propto \frac{{β_{1}}^{α_{1}}}{normalΓ ((), α_{1})} μ^{- 2 ((), α_{1} + 1)} e^{- β_{1} μ^{- 2}}

where

((),, α_{0}, β_{0})

and

((),, α_{1}, β_{1})

are the hyperparameters of the inverse gamma distributions.…”

Section: Bil Identification Based On Bayesian Regularizationmentioning

confidence: 99%

“…Þare determined by Equations ( 7) and (8). For the convenience of calculation, the prior probability density functions of parameters σ 2 and μ 2 are determined by the principle of conjugate priori and are expressed by the inverse gamma distribution 61 :…”

Section: Posterior Probability Density Functionmentioning

confidence: 99%

A statistical influence line identification method using Bayesian regularization and a polynomial interpolating function

Chen

Zhao

Yan

et al. 2022

Structural Contr & Hlth

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As inherent characteristics of bridge structures, influence lines have been successfully applied in the fields of model updating, damage detection, and condition evaluation. The fast and accurate identification of a bridge influence line (BIL) is the premise and foundation of the above applications. BIL identification can be regarded as a typically ill-posed problem for which it is usually necessary to establish a regularization model to identify the model parameters and reconstruct the BIL. In this study, a BIL identification method that can automatically determine the regularization coefficient and quantify the uncertainties of BIL identification results is proposed. To accommodate the uncertainties involved in the measurements as well as the modeling error, an interpolation function-aided influence line model is embedded into the Bayesian framework with Gaussian prior distribution. The most probable values (MPVs) and variance of the interpolation function coefficients are derived analytically and then further used to infer the posterior probability density function of the influence line. Numerical example of a concrete continuous beam and field test for a box girder bridge show the accuracy, efficiency and qualitative evaluation of the proposed method. The results indicate that Bayesian regularization can be used to select the optimal regularization coefficient more accurately and effectively than traditional methods. More importantly, the uncertainty quantification for the influence line can qualitatively reflect the accuracy of the results as well as the effects of the parameters of the BIL identification model.

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The empirical Bayes estimators of the parameter of the uniform distribution with an inverse gamma prior under Stein’s loss function

Sun

Zhang

Sun

2022

Communications in Statistics - Simulation and Computation

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The empirical Bayes estimators of the rate parameter of the inverse gamma distribution with a conjugate inverse gamma prior under Stein's loss function

Cited by 5 publications

References 33 publications

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

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

A statistical influence line identification method using Bayesian regularization and a polynomial interpolating function

The empirical Bayes estimators of the parameter of the uniform distribution with an inverse gamma prior under Stein’s loss function

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