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A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function

A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced... 1IntroductionEstimating the mean parameters is one of the most often encountered difficulties in multivariate statistical analysis. Various studies have dealt with this issue in the context of MVN distribution. When the dimensionality of the parameter space is greater than three, the efficiency of the MLE approach is not fulfilled. There are certain limitations to this approach, which have been shown by Stein [1] and James and Stein [2].A common strategy for enhancing the MLE is the shrinkage estimation approach, which reduces the components of the MLE to zero. The shrinkage estimation approach has been used for enhancing different estimators, such as ordinary least squares estimator [3], and preliminary test and Stein-type shrinkage ridge estimators in robust regression [4]. In the context of enhancing the mean of the MVN distribution, Khursheed [5] studied the domination and admissibility properties of the MLE of a family of shrinkage estimators. Baranchik [6] and Shinozaki [7] also studied the minimaxity of some shrinkage estimators. In addition, several studies have examined the minimaxity and domination properties for various shrinkage estimators under the Bayesian framework, including Efron and Morris [8,9], Berger and Strawderman [10], Benkhaled and Hamdaoui [11], Hamdaoui et al. [12,13], and Zinodiny et al. [14]. Most of these studies have used the quadratic loss function to compute the risk function.This paper introduces a new class of shrinkage estimators that dominate the James-Stein estimator and the MLE. In order to get a competitive estimator, the estimator has to be unbiased and have a good fit. This can be done by implementing the balanced loss function in the estimation procedure of the competitive estimator. The balanced loss function has been suggested by Zellner [15], and its performance and applications to estimators have been discussed by Sanjari Farsipour and Asgharzadeh [16], JafariJozani et al. [17], and Selahattin and Issam [18].Therefore, we consider the random vector ZZto be normally distributed with an unknown mean vector θ\theta and covariance matrix σ2Iq{\sigma }^{2}{I}_{q}, where qqis the dimension of parameter space and Iq{I}_{q}is the q×qq\times qidentity matrix. As the main object of this paper is to propose a new estimator of θ\theta , we estimated the unknown parameter σ2{\sigma }^{2}by S2{S}^{2}(S2∼σ2χn2{S}^{2}\hspace{0.33em} \sim \hspace{0.33em}{\sigma }^{2}{\chi }_{n}^{2}). Then, we construct a new class of shrinkage estimators of θ\theta derived from the MLE. Specifically, the new class of shrinkage estimators is proposed by modifying the James-Stein estimator. We consider adding a term of the form γ(S2/‖Z‖2)2Z\gamma {\left({S}^{2}\text{/}\Vert Z{\Vert }^{2})}^{2}Zto the James-Stein estimator Tα(1)(Z,S2)=(1−αS2/‖Z‖2)Z{T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})=\left(1-\alpha {S}^{2}\hspace{-0.08em}\text{/}\hspace{-0.08em}\Vert Z{\Vert }^{2})Z, where α\alpha and γ\gamma are real constant parameters that both depend on the integer parameters nnand qq. We show that these estimators are minimax and dominating the James-Stein estimator for any values of nnand qq. The balanced loss function is implemented in the computation of the risk function to compare the efficiency of the proposed estimators over the James-Stein estimator.The rest of this paper is composed of the following sections: In Section 2, we establish the minimaxity of the estimators defined by Tα(1)(Z,S2)=(1−αS2/‖Z‖2)Z{T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})=\left(1-\alpha {S}^{2}\hspace{-0.08em}\text{/}\hspace{-0.08em}\Vert Z{\Vert }^{2})Z. Section 3 introduces the new shrinkage estimator class and its domination criterion over the James-Stein estimator. The efficiency of the new estimator classes is explored through simulation studies in Section 4. Then, we conclude our work in Section 5.2A class of minimax shrinkage estimatorsWe assume here the random variable ZZis following an MVN distribution with mean vector θ\theta and a covariance matrix σ2Iq{\sigma }^{2}{I}_{q}, where the parameters θ\theta and σ2{\sigma }^{2}are unknown. Thus, the term ‖Z‖2σ2\frac{\Vert Z{\Vert }^{2}}{{\sigma }^{2}}follows a non-central chi-square distribution with qqdegrees of freedom and non-centrality parameter λ=∥θ∥2σ2\lambda =\frac{{\parallel \theta \parallel }^{2}}{{\sigma }^{2}}. As the aim of this paper is to establish an effective estimator for the mean parameter θ\theta , we consider the statistic S2{S}^{2}(S2∼σ2χn2{S}^{2}\hspace{0.33em} \sim \hspace{0.33em}{\sigma }^{2}{\chi }_{n}^{2}) as an estimate of the unknown parameter σ2{\sigma }^{2}. Thus, for any estimator TTof θ\theta , the balanced squared error loss function is defined as follows: (1)Lω(T,θ)=ω‖T−T0‖2+(1−ω)‖T−θ‖2,0≤ω<1,{L}_{\omega }\left(T,\theta )=\omega \Vert T-{T}_{0}{\Vert }^{2}+\left(1-\omega )\Vert T-\theta {\Vert }^{2},\hspace{1em}0\le \omega \lt 1,where T0{T}_{0}is the target estimator of θ\theta , ω\omega is the weight given to the closeness between the estimators TTand T0{T}_{0}, and 1−ω1-\omega is the relative weight attributed to the accuracy of the estimator TT. The associated risk function to the Lω(T,θ){L}_{\omega }\left(T,\theta )function is defined as follows: Rω(T,θ)=E(Lω(T,θ))=ωE(‖T−T0‖2)+(1−ω)E(‖T−θ‖2).{R}_{\omega }\left(T,\theta )=E\left({L}_{\omega }\left(T,\theta ))=\omega E\left(\Vert T-{T}_{0}{\Vert }^{2})+\left(1-\omega )E\left(\Vert T-\theta {\Vert }^{2}).Benkhaled et al. [19] demonstrated that the MLE of θ\theta is Z≔T0Z:= {T}_{0}. Then, its risk function becomes (1−ω)qσ2\left(1-\omega )q{\sigma }^{2}. This finding shows the minimaxity and inadmissibility property of T0{T}_{0}for q≥3q\ge 3. Consequently, the minimaxity property is also achieved for any estimator that dominates the estimator T0{T}_{0}.Now, let consider the estimator (2)Tα(1)(Z,S2)=1−αS2‖Z‖2Z=Z−αS2‖Z‖2Z,{T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})=\left(1-\alpha \frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)Z=Z-\alpha \frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z,where α\alpha is a real constant parameter that can be related to the values of the parameters nnand qq.Proposition 2.1The associated risk function of the estimator Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})given in equation (2) based on the balanced loss function given in equation (1) is(3)Rω(Tα(1)(Z,S2),θ)=(1−ω)σ2q−2αnσ2(q−2)E1‖Z‖2+α2n(n+2)σ4E1‖Z‖2.{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )=\left(1-\omega ){\sigma }^{2}\left[q-2\alpha n{\sigma }^{2}\left(q-2)E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)\right]+{\alpha }^{2}n\left(n+2){\sigma }^{4}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right).ProofRω(Tα(1)(Z,S2),θ)=ωE−αS2‖Z‖2Z2+(1−ω)EZ−θ−αS2‖Z‖2Z2=α2E((S2)2)E1‖Z‖2+(1−ω)qσ2−2α(1−ω)EZ−θ,1‖Z‖2ZE(S2).\begin{array}{rcl}{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )& =& \omega E\left({\left\Vert ,-\alpha \frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z,\right\Vert }^{2}\right)+\left(1-\omega )E\left({\left\Vert ,Z-\theta -\alpha \frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z,\right\Vert }^{2}\right)\\ & =& {\alpha }^{2}E({\left({S}^{2})}^{2})E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)+\left(1-\omega )q{\sigma }^{2}-2\alpha \left(1-\omega )E\left(\left\langle Z-\theta ,\frac{1}{\Vert Z{\Vert }^{2}}Z\right\rangle \right)E\left({S}^{2}).\end{array}The last equality comes from the independence between two random variables S2{S}^{2}and ∥Z∥2{\parallel Z\parallel }^{2}.As, EZ−θ,1‖Z‖2Z=∑i=1qE(Zi−θi)1‖Z‖2Zi=∑i=1qEyi−θiσ1‖y‖2yi,\begin{array}{rcl}E\left(\left\langle Z-\theta ,\frac{1}{\Vert Z{\Vert }^{2}}Z\right\rangle \right)& =& \mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left[\left({Z}_{i}-{\theta }_{i})\frac{1}{\Vert Z{\Vert }^{2}}{Z}_{i}\right]=\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left[\left({y}_{i}-\frac{{\theta }_{i}}{\sigma }\right)\frac{1}{\Vert y{\Vert }^{2}}{y}_{i}\right],\end{array}where y=Zσ=(y1,…,yq)ty=\frac{Z}{\sigma }={({y}_{1},\ldots ,{y}_{q})}^{t}and for all i=1,…,qi=1,\ldots ,q, yi=Ziσ∼Nθiσ,1{y}_{i}=\frac{{Z}_{i}}{\sigma }\hspace{0.33em} \sim \hspace{0.33em}N\left(\frac{{\theta }_{i}}{\sigma },1\right). Then, based on Lemma 1 given in Stein [20], we get EZ−θ,1‖Z‖2Z=∑i=1qE∂∂yi1‖y‖2yi=∑i=1qE1‖y‖2−2yi2‖y‖4=(q−2)E1‖y‖2=(q−2)σ2E1‖Z‖2.\begin{array}{rcl}E\left(\left\langle Z-\theta ,\frac{1}{\Vert Z{\Vert }^{2}}Z\right\rangle \right)& =& \mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\frac{\partial }{\partial {y}_{i}}\frac{1}{\Vert y{\Vert }^{2}}{y}_{i}\right)=\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\frac{1}{\Vert y{\Vert }^{2}}-\frac{2{y}_{i}^{2}}{\Vert y{\Vert }^{4}}\right)\\ & =& \left(q-2)E\left(\frac{1}{\Vert y{\Vert }^{2}}\right)=\left(q-2){\sigma }^{2}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right).\end{array}Then, Rω(Tα(1)(Z,S2),θ)=α2E((S2)2)E1‖Z‖2+(1−ω)qσ2−2α(1−ω)EZ−θ,1‖Z‖2ZE(S2)=α2E((S2)2)E1‖Z‖2+(1−ω)qσ2−2α(1−ω)(q−2)σ2E1‖Z‖2E(S2)=(1−ω)σ2q−2αnσ2(q−2)E1‖Z‖2+α2n(n+2)σ4E1‖Z‖2.□\hspace{3em}\begin{array}{rcl}{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )& =& {\alpha }^{2}E({\left({S}^{2})}^{2})E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)+\left(1-\omega )q{\sigma }^{2}-2\alpha \left(1-\omega )E\left(\left\langle Z-\theta ,\frac{1}{\Vert Z{\Vert }^{2}}Z\right\rangle \right)E\left({S}^{2})\\ & =& {\alpha }^{2}E({\left({S}^{2})}^{2})E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)+\left(1-\omega )q{\sigma }^{2}-2\alpha \left(1-\omega )\left(q-2){\sigma }^{2}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)E\left({S}^{2})\\ & =& \left(1-\omega ){\sigma }^{2}\left[q-2\alpha n{\sigma }^{2}\left(q-2)E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)\right]+{\alpha }^{2}n\left(n+2){\sigma }^{4}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right).\hspace{7em}\square \end{array}From Proposition (2.1), the minimaxity and domination criterion of the estimator Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})to the MLE is achieved under the following condition: 0≤α≤2(1−ω)(q−2)n+2.0\le \alpha \le \frac{2\left(1-\omega )\left(q-2)}{n+2}.Thus, the risk function Rω(Tα(1)(Z,S2),θ){R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )is minimized at the optimal α\alpha value (α^\widehat{\alpha }) as follows: (4)α^=(1−ω)(q−2)n+2.\widehat{\alpha }=\frac{\left(1-\omega )\left(q-2)}{n+2}.Then, by considering α=α^\alpha =\widehat{\alpha }, we get the James-Stein estimator (5)TJS(Z,S2)=Tα^(1)(Z,S2)=1−α^S2‖Z‖2Z.{T}_{JS}\left(Z,{S}^{2})={T}_{\widehat{\alpha }}^{\left(1)}\left(Z,{S}^{2})=\left(1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)Z.From Proposition 2.1, the risk function of TJS(Z,S2){T}_{JS}\left(Z,{S}^{2})is expressed as follows: (6)Rω(TJS(Z,S2),θ)=(1−ω)qσ2−(q−2)2(1−ω)2nn+2σ2E1‖Z‖2.{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )=\left(1-\omega )q{\sigma }^{2}-{\left(q-2)}^{2}{\left(1-\omega )}^{2}\frac{n}{n+2}{\sigma }^{2}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right).Based on equation (5), the positive part of James-Stein estimator can be defined as follows: (7)TJS+(Z,S2)=1−α^S2‖Z‖2+Z=1−α^S2‖Z‖2ZIα^S2‖Z‖2≤1,{T}_{JS}^{+}\left(Z,{S}^{2})={\left(1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{+}Z=\left(1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)Z{{\rm{I}}}_{\widehat{\alpha }\tfrac{{S}^{2}}{\Vert Z{\Vert }^{2}}\le 1},where 1−α^S2‖Z‖2+=max0,1−α^S2‖Z‖2{\left(1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{+}=\max \left(0,1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right), and its risk function associated with Lω{L}_{\omega }is shown in the following formula: (8)Rω(TJS+(Z,S2),θ)=Rω(TJS(Z,S2),θ)+E‖Z‖2−α^2S4‖Z‖2+2(1−ω)σ2(q−2)α^S2‖Z‖2−qσ2Iα^S2‖Z‖2≥1,{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )={R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+E\left[\left(\Vert Z{\Vert }^{2}-{\widehat{\alpha }}^{2}\frac{{S}^{4}}{\Vert Z{\Vert }^{2}}+2\left(1-\omega ){\sigma }^{2}\left(q-2)\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}-q{\sigma }^{2}\right){I}_{\widehat{\alpha }\tfrac{{S}^{2}}{\Vert Z{\Vert }^{2}}\ge 1}\right],where Iα^S2‖Z‖2≥1{I}_{\widehat{\alpha }\tfrac{{S}^{2}}{\Vert Z{\Vert }^{2}}\ge 1}represents the indicating function of the set α^S2‖Z‖2≥1\left(\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\ge 1\right). Both equations (6) and (8) show that Rω(TJS(Z,S2),θ){R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )and Rω(TJS+(Z,S2),θ){R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )are less than (1−ω)qσ2=Rω(Z,θ)\left(1-\omega )q{\sigma }^{2}={R}_{\omega }\left(Z,\theta ), which proves the domination and minimaxity of both estimators TJS{T}_{JS}and TJS+{T}_{JS}^{+}over the MLE.3The improved shrinkage estimators of the James-Stein estimatorIn this section, we construct a class of shrinkage estimators that has the domination property over the James-Stein estimator TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}). This class of estimators is a modified version of TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}). Specifically, we extend TJS(Z,S2){T}_{JS}\left(Z,{S}^{2})given in equation (5) by adding the term γ(S2/‖Z‖2)2Z\gamma {\left({S}^{2}\text{/}\Vert Z{\Vert }^{2})}^{2}Z, where γ\gamma behaves like α\alpha in equation (2). These new estimators are then investigated regarding their superiority to the James-Stein estimator TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}). The modified version of the James-Stein estimator is shown in the following formula: (9)Tγ,JS(2)(Z,S2)=TJS(Z,S2)+γS2‖Z‖22Z=Z−(1−ω)(q−2)n+2S2‖Z‖2Z+γS2‖Z‖22Z.{T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2})={T}_{JS}\left(Z,{S}^{2})+\gamma {\left(\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{2}Z=Z-\frac{\left(1-\omega )\left(q-2)}{n+2}\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z+\gamma {\left(\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{2}Z.Proposition 3.1The associated risk function of the estimator Tγ,JS(2)(Z,S2){T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2})given in equation (9) based on the balanced loss function given in equation (1) is(10)Rω(Tγ,JS(2)(Z,S2),θ)=Rω(TJS(Z,S2),θ)+2γn(n+2)(1−ω)σ2(q−4)−(q−2)(n+4)n+2E1‖y‖4+γ2n(n+2)(n+4)(n+6)σ2E1‖y‖6,{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )={R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+2\gamma n\left(n+2)\left(1-\omega ){\sigma }^{2}\left[\left(q-4)-\frac{\left(q-2)\left(n+4)}{n+2}\right]E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)+{\gamma }^{2}n\left(n+2)\left(n+4)\left(n+6){\sigma }^{2}E\left(\frac{1}{\Vert y{\Vert }^{6}}\right),where y=Zσ=(y1,…,yq)ty=\frac{Z}{\sigma }={({y}_{1},\ldots ,{y}_{q})}^{t}and yi=Ziσ∼Nθiσ,1{y}_{i}=\frac{{Z}_{i}}{\sigma }\hspace{0.33em} \sim \hspace{0.33em}N\left(\frac{{\theta }_{i}}{\sigma },1\right)for i=1,…,qi=1,\ldots ,q.ProofRω(Tγ,JS(2)(Z,S2),θ)=ωETJS(Z,S2)+γ(S2‖Z‖2)2Z−Z2+(1−ω)ETJS(Z,S2)+γ(S2‖Z‖2)2Z−θ2\begin{array}{rcl}{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )& =& \omega E\left({\left\Vert ,{T}_{JS}\left(Z,{S}^{2})+\gamma {\left(\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{2}Z-Z,\right\Vert }^{2}\right)+\left(1-\omega )E\left({\left\Vert ,{T}_{JS}\left(Z,{S}^{2})+\gamma {\left(\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{2}Z-\theta ,\right\Vert }^{2}\right)\end{array}=ωE∥TJS(Z,S2)−Z∥2+γ2(S2)4(‖Z‖2)3+2ωTJS(Z,S2)−Z,γ(S2)2(‖Z‖2)2Z+(1−ω)E∥TJS(Z,S2)−θ∥2+γ2(S2)4(‖Z‖2)3+2TJS(Z,S2)−θ,γ(S2)2(‖Z‖2)2Z=Rω(TJS(Z,S2),θ)+γ2E((S2)4)E1(‖Z‖2)3−2γω(1−ω)(q−2)n+2E((S2)3)E1(‖Z‖2)2+2(1−ω)EZ−θ−(1−ω)(q−2)n+2S2‖Z‖2Z,γ(S2)2(‖Z‖2)2Z,\begin{array}{rcl}& =& \omega E\left({\parallel {T}_{JS}\left(Z,{S}^{2})-Z\parallel }^{2}+{\gamma }^{2}\frac{{\left({S}^{2})}^{4}}{{\left(\Vert Z{\Vert }^{2})}^{3}}\right)+2\omega \left(\left\langle {T}_{JS}\left(Z,{S}^{2})-Z,\gamma \frac{{\left({S}^{2})}^{2}}{{\left(\Vert Z{\Vert }^{2})}^{2}}Z\right\rangle \right)\\ & & +\left(1-\omega )E\left({\parallel {T}_{JS}\left(Z,{S}^{2})-\theta \parallel }^{2}+{\gamma }^{2}\frac{{\left({S}^{2})}^{4}}{{\left(\Vert Z{\Vert }^{2})}^{3}}+2\left\langle {T}_{JS}\left(Z,{S}^{2})-\theta ,\gamma \frac{{\left({S}^{2})}^{2}}{{\left(\Vert Z{\Vert }^{2})}^{2}}Z\right\rangle \right)\\ & =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}E({\left({S}^{2})}^{4})E\left(\frac{1}{{\left(\Vert Z{\Vert }^{2})}^{3}}\right)-2\gamma \omega \frac{\left(1-\omega )\left(q-2)}{n+2}E({\left({S}^{2})}^{3})E\left(\frac{1}{{\left(\Vert Z{\Vert }^{2})}^{2}}\right)\\ & & +2\left(1-\omega )E\left(\left\langle Z-\theta -\frac{\left(1-\omega )\left(q-2)}{n+2}\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z,\gamma \frac{{\left({S}^{2})}^{2}}{{\left(\Vert Z{\Vert }^{2})}^{2}}Z\right\rangle \right),\end{array}where the last equality is obtained as a result of the independence between the two random variables S2{S}^{2}and ‖Z‖2\Vert Z{\Vert }^{2}. Thus, Rω(Tγ,JS(2)(Z,S2),θ)=Rω(TJS(Z,S2),θ)+γ2E((σ2χn2)4)E1‖Z‖6−2γω(1−ω)(q−2)n+2E((σ2χn2)3)E1‖Z‖4+2γ(1−ω)E((σ2χn2)2)∑i=1qE(Zi−θi)Zi‖Z‖4−2γ(1−ω)2(q−2)n+2E((σ2χn2)3)E1‖Z‖4.\begin{array}{rcl}{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )& =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}E({\left({\sigma }^{2}{\chi }_{n}^{2})}^{4})E\left(\frac{1}{\Vert Z{\Vert }^{6}}\right)-2\gamma \omega \frac{\left(1-\omega )\left(q-2)}{n+2}E({\left({\sigma }^{2}{\chi }_{n}^{2})}^{3})E\left(\frac{1}{\Vert Z{\Vert }^{4}}\right)\\ & & +2\gamma \left(1-\omega )E({\left({\sigma }^{2}{\chi }_{n}^{2})}^{2})\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\left({Z}_{i}-{\theta }_{i})\frac{{Z}_{i}}{\Vert Z{\Vert }^{4}}\right)-2\gamma \frac{{\left(1-\omega )}^{2}\left(q-2)}{n+2}E({\left({\sigma }^{2}{\chi }_{n}^{2})}^{3})E\left(\frac{1}{\Vert Z{\Vert }^{4}}\right).\end{array}Then, by making the transformation y=Zσ=(y1,…,yq)ty=\frac{Z}{\sigma }={({y}_{1},\ldots ,{y}_{q})}^{t}, where yi=Ziσ∼Nθiσ,1{y}_{i}=\frac{{Z}_{i}}{\sigma }\hspace{0.33em} \sim \hspace{0.33em}N\left(\frac{{\theta }_{i}}{\sigma },1\right)for i=1,…,qi=1,\ldots ,q, and using Lemma 1 given in Stein [20], we get ∑i=1qE(Zi−θi)Zi‖Z‖4=1σ2∑i=1qE(yi−θiσ)yi‖y‖4=1σ2∑i=1qE∂∂yiyi‖y‖4=1σ2∑i=1qE1‖y‖4−4yi2‖y‖6=1σ2(q−4)E1‖y‖4.\begin{array}{rcl}\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\left({Z}_{i}-{\theta }_{i})\frac{{Z}_{i}}{\Vert Z{\Vert }^{4}}\right)& =& \frac{1}{{\sigma }^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\left({y}_{i}-\frac{{\theta }_{i}}{\sigma }\right)\frac{{y}_{i}}{\Vert y{\Vert }^{4}}\right)=\frac{1}{{\sigma }^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\frac{\partial }{\partial {y}_{i}}\frac{{y}_{i}}{\Vert y{\Vert }^{4}}\right)\\ & =& \frac{1}{{\sigma }^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\frac{1}{\Vert y{\Vert }^{4}}-4\frac{{y}_{i}^{2}}{\Vert y{\Vert }^{6}}\right)=\frac{1}{{\sigma }^{2}}\left(q-4)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right).\end{array}Thus, Rω(Tγ,JS(2)(Z,S2),θ)=Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)E1‖y‖6−2γω(1−ω)(q−2)σ2n(n+4)E1‖y‖4+2γ(1−ω)σ2n(n+2)(q−4)E1‖y‖4−2γ(1−ω)2(q−2)σ2n(n+4)E1‖y‖4=Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)E1‖y‖6+2γn(n+2)(1−ω)σ2(q−4)−(q−2)(n+4)n+2E1‖y‖4.□\hspace{1.25em}\begin{array}{rcl}{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )& =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}n\left(n+2)\left(n+4)\left(n+6)E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)\\ & & -2\gamma \omega \left(1-\omega )\left(q-2){\sigma }^{2}n\left(n+4)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)+2\gamma \left(1-\omega ){\sigma }^{2}n\left(n+2)\left(q-4)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\\ & & -2\gamma {\left(1-\omega )}^{2}\left(q-2){\sigma }^{2}n\left(n+4)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\\ & =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}n\left(n+2)\left(n+4)\left(n+6)E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)\\ & & +2\gamma n\left(n+2)\left(1-\omega ){\sigma }^{2}\left[\left(q-4)-\frac{\left(q-2)\left(n+4)}{n+2}\right]E\left(\frac{1}{\Vert y{\Vert }^{4}}\right).\hspace{10em}\square \end{array}Theorem 3.1Under the balanced loss function Lω{L}_{\omega }, the estimator Tγ,JS(2)(Z,S2){T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2})with q>6q\gt 6andγ=2(1−ω)(q−6)(n+4)(n+6),\gamma =\frac{2\left(1-\omega )\left(q-6)}{\left(n+4)\left(n+6)},dominates the James-Stein estimator TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}).ProofAccording to Proposition 3.1, we have Rω(Tγ,JS(2)(Z,S2),θ)=Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)E1‖y‖6E1‖y‖4E1‖y‖4+2γσ2(1−ω)n(n+2)(q−4)−(q−2)(n+4)n+2E1‖y‖4\begin{array}{rcl}{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )& =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}n\left(n+2)\left(n+4)\left(n+6)\frac{E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)}{E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)}E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\\ & & +2\gamma {\sigma }^{2}\left(1-\omega )n\left(n+2)\left[\left(q-4)-\frac{\left(q-2)\left(n+4)}{n+2}\right]E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\end{array}≤Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)E1‖y‖6E1‖y‖4E1‖y‖4+2γσ2(1−ω)n(n+2)[(q−4)−(q−2)]E1‖y‖4.\begin{array}{rcl}& \le & {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}n\left(n+2)\left(n+4)\left(n+6)\frac{E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)}{E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)}E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\\ & & +2\gamma {\sigma }^{2}\left(1-\omega )n\left(n+2){[}\left(q-4)-\left(q-2)]E\left(\frac{1}{\Vert y{\Vert }^{4}}\right).\end{array}Following Lemma 2 given in the study by Benkhaled et al. [19], we obtain E1‖y‖6E1‖y‖4=E(‖y‖−6)E(‖y‖−4)≤2−4+22Γq2−4+1Γq−42=1q−6.\frac{E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)}{E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)}=\frac{E(\Vert y{\Vert }^{-6})}{E(\Vert y{\Vert }^{-4})}\le {2}^{\tfrac{-4+2}{2}}\frac{\Gamma \left(\frac{q}{2}-4+1\right)}{\Gamma \left(\frac{q-4}{2}\right)}=\frac{1}{q-6}.Then, (11)Rω(Tγ,JS(2)(Z,S2),θ)≤Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)(q−6)E1‖y‖4−4γσ2(1−ω)n(n+2)E1‖y‖4.{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )\le {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}\frac{n\left(n+2)\left(n+4)\left(n+6)}{\left(q-6)}E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)-4\gamma {\sigma }^{2}\left(1-\omega )n\left(n+2)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right).The right side of the aforementioned inequality is minimized at the optimal value of γ\gamma as follows: (12)γ^=2(1−ω)(q−6)(n+4)(n+6).\widehat{\gamma }=\frac{2\left(1-\omega )\left(q-6)}{\left(n+4)\left(n+6)}.Then, by replacing γ\gamma by γ^\widehat{\gamma }in equation (11), we obtain Rω(Tγ^,JS(2)(Z,S2),θ)≤Rω(TJS(Z,S2),θ)−4σ2(1−ω)2n(n+2)(q−6)(n+4)(n+6)≤Rω(TJS(Z,S2),θ).□\hspace{2.35em}{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}),\theta )\le {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )-4{\sigma }^{2}\frac{{\left(1-\omega )}^{2}n\left(n+2)\left(q-6)}{\left(n+4)\left(n+6)}\le {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta ).\hspace{5em}\square 4Simulation resultsWe conduct here a simulation study for comparing the efficiency of the proposed estimators Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})and Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2})to the estimators TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}), TJS+(Z,S2){T}_{JS}^{+}\left(Z,{S}^{2})and the MLE. We consider here α=(1−ω)(q−2)2(n+2)\alpha =\frac{\left(1-\omega )\left(q-2)}{2\left(n+2)}in the estimator Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}). This comparison is done based on the risk ratio of these estimators to the MLE. Thus, the risk ratios of these estimators are denoted as follows: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}. We consider here all estimators to be functions of λ=‖θ‖2σ2\lambda =\frac{\Vert \theta {\Vert }^{2}}{{\sigma }^{2}}.Figures 1, 2, 3, 4, 5, show the curve of the risk ratios for simulated values of λ\lambda in the interval (1,30)\left(1,30)and for relatively low and high values of nn, qq, and ω\omega . The risk ratio of the MLE is represented by the horizontal line at the value of one. The gap between the curves of estimators indicates the gain magnitude of the estimator. We observed that the curves of all risk ratios for the different sets of nn, qq, and ω\omega values are entirely located below 1, which indicate the domination of these estimators to the MLE ZZ. Consequently, these estimators are considered minimax.Figure 1Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=6n=6, q=8q=8, and ω=0.1\omega =0.1.Figure 2Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=20n=20, q=8q=8, and ω=0.1\omega =0.1.Figure 3Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=6n=6, q=8q=8, and ω=0.5\omega =0.5.Figure 4Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=20n=20, q=12q=12, and ω=0.1\omega =0.1.Figure 5Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=20n=20, q=12q=12, and ω=0.5\omega =0.5.Among these estimators, the positive-part James-Stein estimator (TJS+{T}_{JS}^{+}) was the more efficient estimator for values of λ\lambda less than approximately 10. It means that TJS+{T}_{JS}^{+}dominates all the considered estimators. Also, we note that the estimator Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2})dominates the James-Stein estimator TJS{T}_{JS}for the various values of nn, qq, and ω\omega . We also observe a larger gain of the estimator Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2})for low values of ω\omega . The gain of the estimators Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}), TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}), and TJS+(Z,S2){T}_{JS}^{+}\left(Z,{S}^{2})was very similar in a specific period of λ\lambda values, which depend on the combination of the values of the nnand qq. To study this similarity, we conduct simulation studies for all combinations of the selected values of nnand qqfor different sets of values of λ\lambda and ω\omega .Tables 1, 2, 3, 4 show the results of the risk ratios of the estimators Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}), Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}), and TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}). Each cell of the tables represents the risk ratio of these estimators in order. We observe a strong relationship between the gain of the risk ratios and the values of ω\omega and λ\lambda . The gain of all risk ratios was large with small values of λ\lambda and ω\omega and tended to vanish with the increase of λ\lambda and ω\omega values. Also, the difference in the gain of risk ratios was observed in small values of λ\lambda . This difference indicated the domination of an estimator to another. Thus, the estimator Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2})dominated both estimators Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})and TJS(Z,S2){T}_{JS}\left(Z,{S}^{2})for small values of λ\lambda . However, as the values of λ\lambda and ω\omega increased, the difference in the gain of these estimators became negligible (i.e., no improvement of the proposed estimators over the James-Stein estimator). The other parameters nnand qqhave also an influence on the gain of the estimators. The gain of the estimators was large for large values of nnand qqunder fixed values of ω\omega . Specifically, the increase of qqhad significant influence on the gain than the increase of nnvalues. This means that having large values of nn, qq, and λ\lambda with value of ω\omega close to zero leads to a larger gain of the estimators, which leads to a significant improvement. Thus, we conclude that the improvement of the considered estimators is clearly affected by the values of the parameters nn, qq, ω\omega , and λ\lambda .Table 1Values of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, and Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}at various values of λ\lambda and ω\omega , and n=6n=6, and q=8q=8λ\lambda ω\omega 0.00.10.20.70.91.24180.63620.67260.70900.89090.96360.51500.56350.61200.85450.95150.48240.53710.59110.85160.95125.00190.75010.77510.80010.92500.97500.66680.70010.73340.90000.96670.65020.68670.72280.89850.966510.43110.83260.84940.86610.94980.98330.77690.79920.82150.93300.97770.76940.79320.81670.93240.977620.00000.89620.90660.91700.96890.98960.86160.87550.88930.95850.98650.85890.87330.88760.95820.9861Table 2Values of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ),\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )},and Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}at various values of λ\lambda and ω\omega , and n=6,q=12n=6,q=12λ\lambda ω\omega 0.00.10.20.70.91.24180.57580.61820.66060.87270.95760.43430.49090.54750.83030.94340.40490.46710.52860.82760.94315.00190.67380.70640.73900.90210.96740.56510.60860.65210.86950.95650.54680.59380.64040.86790.956310.43110.75850.78270.80680.92760.97580.67810.71020.74240.90340.96780.66780.70200.73590.90250.967720.00000.83630.85270.86900.95090.98360.78170.80360.82540.93450.97820.77700.79980.82240.93410.9781Table 3Values of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ),\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )},and Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}at various values of λ\lambda and ω\omega and n=20,q=8n=20,q=8λ\lambda ω\omega 0.00.10.20.70.91.24180.55910.60320.64730.86770.95590.41210.47090.52970.82360.94120.37530.44110.50620.82030.94085.00190.69710.72740.75770.90910.96970.59610.63650.67690.87880.95960.57630.62040.66420.87700.959410.43110.79710.81740.83770.93910.97970.72950.75660.78360.91880.97290.72030.74910.77770.91800.972920.00000.87420.88680.89940.96230.98740.83230.84900.86580.94970.98320.82880.84630.86360.94940.9832Table 4Values of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ),\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )},and Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}at various values of λ\lambda and ω\omega , and n=20n=20and q=12q=12λ\lambda ω\omega 0.00.10.20.70.91.24180.48580.53720.58860.84570.94860.31440.38290.45150.79430.93140.28540.35950.43300.79170.93115.00190.60460.64420.68370.88140.96050.47280.52550.57830.84180.94730.45400.51030.56620.84020.947110.43110.70730.73660.76590.91220.97070.60980.64880.68780.88290.96100.59880.63990.68080.88190.960920.00000.80160.82140.84130.94050.98010.73540.76190.78840.92060.97350.73030.75770.78510.92020.97355ConclusionIn this paper, we constructed a new class of shrinkage estimator that dominate the James-Stein estimator for the estimation of the mean θ\theta of the MVN distribution Z∼Nq(θ,σ2Iq)Z\hspace{0.33em} \sim \hspace{0.33em}{N}_{q}\left(\theta ,{\sigma }^{2}{I}_{q}), where σ2{\sigma }^{2}is unknown. We implemented the balanced square function in the form of the risk function of the estimators for the purpose of comparing the efficiency of two estimators. We started establishing a class of the minimaxity property for the estimator defined by Tα(1)(Z,S2)=(1−αS2/‖Z‖2)Z{T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})=\left(1-\alpha {S}^{2}\hspace{-0.08em}\text{/}\hspace{-0.08em}\Vert Z{\Vert }^{2})Z. We found then the minimum risk of this class that resulted in the James-Stein estimator. Then, we constructed a new class of shrinkage estimator that is a modified version of the James-Stein estimator. Mainly, a term γ(S2/‖Z‖2)2Z\gamma {\left({S}^{2}\text{/}\Vert Z{\Vert }^{2})}^{2}Zwas added to the James-Stein estimator. The efficiency of the constructed estimator was explored by simulation studies under various values of the model parameters, and it has been shown that the constructed estimators beat the James-Stein estimator under the balanced loss function.An extension of this work is to implement the similar procedures of this paper in the Bayesian framework and explore possible shrinkage estimators for the mean parameters of the MVN distribution, such as the ridge estimators. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function

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de Gruyter
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© 2022 Abdelkader Benkhaled et al., published by De Gruyter
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2391-5455
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2391-5455
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10.1515/math-2022-0008
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Abstract

1IntroductionEstimating the mean parameters is one of the most often encountered difficulties in multivariate statistical analysis. Various studies have dealt with this issue in the context of MVN distribution. When the dimensionality of the parameter space is greater than three, the efficiency of the MLE approach is not fulfilled. There are certain limitations to this approach, which have been shown by Stein [1] and James and Stein [2].A common strategy for enhancing the MLE is the shrinkage estimation approach, which reduces the components of the MLE to zero. The shrinkage estimation approach has been used for enhancing different estimators, such as ordinary least squares estimator [3], and preliminary test and Stein-type shrinkage ridge estimators in robust regression [4]. In the context of enhancing the mean of the MVN distribution, Khursheed [5] studied the domination and admissibility properties of the MLE of a family of shrinkage estimators. Baranchik [6] and Shinozaki [7] also studied the minimaxity of some shrinkage estimators. In addition, several studies have examined the minimaxity and domination properties for various shrinkage estimators under the Bayesian framework, including Efron and Morris [8,9], Berger and Strawderman [10], Benkhaled and Hamdaoui [11], Hamdaoui et al. [12,13], and Zinodiny et al. [14]. Most of these studies have used the quadratic loss function to compute the risk function.This paper introduces a new class of shrinkage estimators that dominate the James-Stein estimator and the MLE. In order to get a competitive estimator, the estimator has to be unbiased and have a good fit. This can be done by implementing the balanced loss function in the estimation procedure of the competitive estimator. The balanced loss function has been suggested by Zellner [15], and its performance and applications to estimators have been discussed by Sanjari Farsipour and Asgharzadeh [16], JafariJozani et al. [17], and Selahattin and Issam [18].Therefore, we consider the random vector ZZto be normally distributed with an unknown mean vector θ\theta and covariance matrix σ2Iq{\sigma }^{2}{I}_{q}, where qqis the dimension of parameter space and Iq{I}_{q}is the q×qq\times qidentity matrix. As the main object of this paper is to propose a new estimator of θ\theta , we estimated the unknown parameter σ2{\sigma }^{2}by S2{S}^{2}(S2∼σ2χn2{S}^{2}\hspace{0.33em} \sim \hspace{0.33em}{\sigma }^{2}{\chi }_{n}^{2}). Then, we construct a new class of shrinkage estimators of θ\theta derived from the MLE. Specifically, the new class of shrinkage estimators is proposed by modifying the James-Stein estimator. We consider adding a term of the form γ(S2/‖Z‖2)2Z\gamma {\left({S}^{2}\text{/}\Vert Z{\Vert }^{2})}^{2}Zto the James-Stein estimator Tα(1)(Z,S2)=(1−αS2/‖Z‖2)Z{T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})=\left(1-\alpha {S}^{2}\hspace{-0.08em}\text{/}\hspace{-0.08em}\Vert Z{\Vert }^{2})Z, where α\alpha and γ\gamma are real constant parameters that both depend on the integer parameters nnand qq. We show that these estimators are minimax and dominating the James-Stein estimator for any values of nnand qq. The balanced loss function is implemented in the computation of the risk function to compare the efficiency of the proposed estimators over the James-Stein estimator.The rest of this paper is composed of the following sections: In Section 2, we establish the minimaxity of the estimators defined by Tα(1)(Z,S2)=(1−αS2/‖Z‖2)Z{T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})=\left(1-\alpha {S}^{2}\hspace{-0.08em}\text{/}\hspace{-0.08em}\Vert Z{\Vert }^{2})Z. Section 3 introduces the new shrinkage estimator class and its domination criterion over the James-Stein estimator. The efficiency of the new estimator classes is explored through simulation studies in Section 4. Then, we conclude our work in Section 5.2A class of minimax shrinkage estimatorsWe assume here the random variable ZZis following an MVN distribution with mean vector θ\theta and a covariance matrix σ2Iq{\sigma }^{2}{I}_{q}, where the parameters θ\theta and σ2{\sigma }^{2}are unknown. Thus, the term ‖Z‖2σ2\frac{\Vert Z{\Vert }^{2}}{{\sigma }^{2}}follows a non-central chi-square distribution with qqdegrees of freedom and non-centrality parameter λ=∥θ∥2σ2\lambda =\frac{{\parallel \theta \parallel }^{2}}{{\sigma }^{2}}. As the aim of this paper is to establish an effective estimator for the mean parameter θ\theta , we consider the statistic S2{S}^{2}(S2∼σ2χn2{S}^{2}\hspace{0.33em} \sim \hspace{0.33em}{\sigma }^{2}{\chi }_{n}^{2}) as an estimate of the unknown parameter σ2{\sigma }^{2}. Thus, for any estimator TTof θ\theta , the balanced squared error loss function is defined as follows: (1)Lω(T,θ)=ω‖T−T0‖2+(1−ω)‖T−θ‖2,0≤ω<1,{L}_{\omega }\left(T,\theta )=\omega \Vert T-{T}_{0}{\Vert }^{2}+\left(1-\omega )\Vert T-\theta {\Vert }^{2},\hspace{1em}0\le \omega \lt 1,where T0{T}_{0}is the target estimator of θ\theta , ω\omega is the weight given to the closeness between the estimators TTand T0{T}_{0}, and 1−ω1-\omega is the relative weight attributed to the accuracy of the estimator TT. The associated risk function to the Lω(T,θ){L}_{\omega }\left(T,\theta )function is defined as follows: Rω(T,θ)=E(Lω(T,θ))=ωE(‖T−T0‖2)+(1−ω)E(‖T−θ‖2).{R}_{\omega }\left(T,\theta )=E\left({L}_{\omega }\left(T,\theta ))=\omega E\left(\Vert T-{T}_{0}{\Vert }^{2})+\left(1-\omega )E\left(\Vert T-\theta {\Vert }^{2}).Benkhaled et al. [19] demonstrated that the MLE of θ\theta is Z≔T0Z:= {T}_{0}. Then, its risk function becomes (1−ω)qσ2\left(1-\omega )q{\sigma }^{2}. This finding shows the minimaxity and inadmissibility property of T0{T}_{0}for q≥3q\ge 3. Consequently, the minimaxity property is also achieved for any estimator that dominates the estimator T0{T}_{0}.Now, let consider the estimator (2)Tα(1)(Z,S2)=1−αS2‖Z‖2Z=Z−αS2‖Z‖2Z,{T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})=\left(1-\alpha \frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)Z=Z-\alpha \frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z,where α\alpha is a real constant parameter that can be related to the values of the parameters nnand qq.Proposition 2.1The associated risk function of the estimator Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})given in equation (2) based on the balanced loss function given in equation (1) is(3)Rω(Tα(1)(Z,S2),θ)=(1−ω)σ2q−2αnσ2(q−2)E1‖Z‖2+α2n(n+2)σ4E1‖Z‖2.{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )=\left(1-\omega ){\sigma }^{2}\left[q-2\alpha n{\sigma }^{2}\left(q-2)E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)\right]+{\alpha }^{2}n\left(n+2){\sigma }^{4}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right).ProofRω(Tα(1)(Z,S2),θ)=ωE−αS2‖Z‖2Z2+(1−ω)EZ−θ−αS2‖Z‖2Z2=α2E((S2)2)E1‖Z‖2+(1−ω)qσ2−2α(1−ω)EZ−θ,1‖Z‖2ZE(S2).\begin{array}{rcl}{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )& =& \omega E\left({\left\Vert ,-\alpha \frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z,\right\Vert }^{2}\right)+\left(1-\omega )E\left({\left\Vert ,Z-\theta -\alpha \frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z,\right\Vert }^{2}\right)\\ & =& {\alpha }^{2}E({\left({S}^{2})}^{2})E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)+\left(1-\omega )q{\sigma }^{2}-2\alpha \left(1-\omega )E\left(\left\langle Z-\theta ,\frac{1}{\Vert Z{\Vert }^{2}}Z\right\rangle \right)E\left({S}^{2}).\end{array}The last equality comes from the independence between two random variables S2{S}^{2}and ∥Z∥2{\parallel Z\parallel }^{2}.As, EZ−θ,1‖Z‖2Z=∑i=1qE(Zi−θi)1‖Z‖2Zi=∑i=1qEyi−θiσ1‖y‖2yi,\begin{array}{rcl}E\left(\left\langle Z-\theta ,\frac{1}{\Vert Z{\Vert }^{2}}Z\right\rangle \right)& =& \mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left[\left({Z}_{i}-{\theta }_{i})\frac{1}{\Vert Z{\Vert }^{2}}{Z}_{i}\right]=\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left[\left({y}_{i}-\frac{{\theta }_{i}}{\sigma }\right)\frac{1}{\Vert y{\Vert }^{2}}{y}_{i}\right],\end{array}where y=Zσ=(y1,…,yq)ty=\frac{Z}{\sigma }={({y}_{1},\ldots ,{y}_{q})}^{t}and for all i=1,…,qi=1,\ldots ,q, yi=Ziσ∼Nθiσ,1{y}_{i}=\frac{{Z}_{i}}{\sigma }\hspace{0.33em} \sim \hspace{0.33em}N\left(\frac{{\theta }_{i}}{\sigma },1\right). Then, based on Lemma 1 given in Stein [20], we get EZ−θ,1‖Z‖2Z=∑i=1qE∂∂yi1‖y‖2yi=∑i=1qE1‖y‖2−2yi2‖y‖4=(q−2)E1‖y‖2=(q−2)σ2E1‖Z‖2.\begin{array}{rcl}E\left(\left\langle Z-\theta ,\frac{1}{\Vert Z{\Vert }^{2}}Z\right\rangle \right)& =& \mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\frac{\partial }{\partial {y}_{i}}\frac{1}{\Vert y{\Vert }^{2}}{y}_{i}\right)=\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\frac{1}{\Vert y{\Vert }^{2}}-\frac{2{y}_{i}^{2}}{\Vert y{\Vert }^{4}}\right)\\ & =& \left(q-2)E\left(\frac{1}{\Vert y{\Vert }^{2}}\right)=\left(q-2){\sigma }^{2}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right).\end{array}Then, Rω(Tα(1)(Z,S2),θ)=α2E((S2)2)E1‖Z‖2+(1−ω)qσ2−2α(1−ω)EZ−θ,1‖Z‖2ZE(S2)=α2E((S2)2)E1‖Z‖2+(1−ω)qσ2−2α(1−ω)(q−2)σ2E1‖Z‖2E(S2)=(1−ω)σ2q−2αnσ2(q−2)E1‖Z‖2+α2n(n+2)σ4E1‖Z‖2.□\hspace{3em}\begin{array}{rcl}{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )& =& {\alpha }^{2}E({\left({S}^{2})}^{2})E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)+\left(1-\omega )q{\sigma }^{2}-2\alpha \left(1-\omega )E\left(\left\langle Z-\theta ,\frac{1}{\Vert Z{\Vert }^{2}}Z\right\rangle \right)E\left({S}^{2})\\ & =& {\alpha }^{2}E({\left({S}^{2})}^{2})E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)+\left(1-\omega )q{\sigma }^{2}-2\alpha \left(1-\omega )\left(q-2){\sigma }^{2}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)E\left({S}^{2})\\ & =& \left(1-\omega ){\sigma }^{2}\left[q-2\alpha n{\sigma }^{2}\left(q-2)E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right)\right]+{\alpha }^{2}n\left(n+2){\sigma }^{4}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right).\hspace{7em}\square \end{array}From Proposition (2.1), the minimaxity and domination criterion of the estimator Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})to the MLE is achieved under the following condition: 0≤α≤2(1−ω)(q−2)n+2.0\le \alpha \le \frac{2\left(1-\omega )\left(q-2)}{n+2}.Thus, the risk function Rω(Tα(1)(Z,S2),θ){R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )is minimized at the optimal α\alpha value (α^\widehat{\alpha }) as follows: (4)α^=(1−ω)(q−2)n+2.\widehat{\alpha }=\frac{\left(1-\omega )\left(q-2)}{n+2}.Then, by considering α=α^\alpha =\widehat{\alpha }, we get the James-Stein estimator (5)TJS(Z,S2)=Tα^(1)(Z,S2)=1−α^S2‖Z‖2Z.{T}_{JS}\left(Z,{S}^{2})={T}_{\widehat{\alpha }}^{\left(1)}\left(Z,{S}^{2})=\left(1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)Z.From Proposition 2.1, the risk function of TJS(Z,S2){T}_{JS}\left(Z,{S}^{2})is expressed as follows: (6)Rω(TJS(Z,S2),θ)=(1−ω)qσ2−(q−2)2(1−ω)2nn+2σ2E1‖Z‖2.{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )=\left(1-\omega )q{\sigma }^{2}-{\left(q-2)}^{2}{\left(1-\omega )}^{2}\frac{n}{n+2}{\sigma }^{2}E\left(\frac{1}{\Vert Z{\Vert }^{2}}\right).Based on equation (5), the positive part of James-Stein estimator can be defined as follows: (7)TJS+(Z,S2)=1−α^S2‖Z‖2+Z=1−α^S2‖Z‖2ZIα^S2‖Z‖2≤1,{T}_{JS}^{+}\left(Z,{S}^{2})={\left(1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{+}Z=\left(1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)Z{{\rm{I}}}_{\widehat{\alpha }\tfrac{{S}^{2}}{\Vert Z{\Vert }^{2}}\le 1},where 1−α^S2‖Z‖2+=max0,1−α^S2‖Z‖2{\left(1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{+}=\max \left(0,1-\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right), and its risk function associated with Lω{L}_{\omega }is shown in the following formula: (8)Rω(TJS+(Z,S2),θ)=Rω(TJS(Z,S2),θ)+E‖Z‖2−α^2S4‖Z‖2+2(1−ω)σ2(q−2)α^S2‖Z‖2−qσ2Iα^S2‖Z‖2≥1,{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )={R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+E\left[\left(\Vert Z{\Vert }^{2}-{\widehat{\alpha }}^{2}\frac{{S}^{4}}{\Vert Z{\Vert }^{2}}+2\left(1-\omega ){\sigma }^{2}\left(q-2)\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}-q{\sigma }^{2}\right){I}_{\widehat{\alpha }\tfrac{{S}^{2}}{\Vert Z{\Vert }^{2}}\ge 1}\right],where Iα^S2‖Z‖2≥1{I}_{\widehat{\alpha }\tfrac{{S}^{2}}{\Vert Z{\Vert }^{2}}\ge 1}represents the indicating function of the set α^S2‖Z‖2≥1\left(\widehat{\alpha }\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\ge 1\right). Both equations (6) and (8) show that Rω(TJS(Z,S2),θ){R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )and Rω(TJS+(Z,S2),θ){R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )are less than (1−ω)qσ2=Rω(Z,θ)\left(1-\omega )q{\sigma }^{2}={R}_{\omega }\left(Z,\theta ), which proves the domination and minimaxity of both estimators TJS{T}_{JS}and TJS+{T}_{JS}^{+}over the MLE.3The improved shrinkage estimators of the James-Stein estimatorIn this section, we construct a class of shrinkage estimators that has the domination property over the James-Stein estimator TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}). This class of estimators is a modified version of TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}). Specifically, we extend TJS(Z,S2){T}_{JS}\left(Z,{S}^{2})given in equation (5) by adding the term γ(S2/‖Z‖2)2Z\gamma {\left({S}^{2}\text{/}\Vert Z{\Vert }^{2})}^{2}Z, where γ\gamma behaves like α\alpha in equation (2). These new estimators are then investigated regarding their superiority to the James-Stein estimator TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}). The modified version of the James-Stein estimator is shown in the following formula: (9)Tγ,JS(2)(Z,S2)=TJS(Z,S2)+γS2‖Z‖22Z=Z−(1−ω)(q−2)n+2S2‖Z‖2Z+γS2‖Z‖22Z.{T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2})={T}_{JS}\left(Z,{S}^{2})+\gamma {\left(\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{2}Z=Z-\frac{\left(1-\omega )\left(q-2)}{n+2}\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z+\gamma {\left(\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{2}Z.Proposition 3.1The associated risk function of the estimator Tγ,JS(2)(Z,S2){T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2})given in equation (9) based on the balanced loss function given in equation (1) is(10)Rω(Tγ,JS(2)(Z,S2),θ)=Rω(TJS(Z,S2),θ)+2γn(n+2)(1−ω)σ2(q−4)−(q−2)(n+4)n+2E1‖y‖4+γ2n(n+2)(n+4)(n+6)σ2E1‖y‖6,{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )={R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+2\gamma n\left(n+2)\left(1-\omega ){\sigma }^{2}\left[\left(q-4)-\frac{\left(q-2)\left(n+4)}{n+2}\right]E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)+{\gamma }^{2}n\left(n+2)\left(n+4)\left(n+6){\sigma }^{2}E\left(\frac{1}{\Vert y{\Vert }^{6}}\right),where y=Zσ=(y1,…,yq)ty=\frac{Z}{\sigma }={({y}_{1},\ldots ,{y}_{q})}^{t}and yi=Ziσ∼Nθiσ,1{y}_{i}=\frac{{Z}_{i}}{\sigma }\hspace{0.33em} \sim \hspace{0.33em}N\left(\frac{{\theta }_{i}}{\sigma },1\right)for i=1,…,qi=1,\ldots ,q.ProofRω(Tγ,JS(2)(Z,S2),θ)=ωETJS(Z,S2)+γ(S2‖Z‖2)2Z−Z2+(1−ω)ETJS(Z,S2)+γ(S2‖Z‖2)2Z−θ2\begin{array}{rcl}{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )& =& \omega E\left({\left\Vert ,{T}_{JS}\left(Z,{S}^{2})+\gamma {\left(\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{2}Z-Z,\right\Vert }^{2}\right)+\left(1-\omega )E\left({\left\Vert ,{T}_{JS}\left(Z,{S}^{2})+\gamma {\left(\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}\right)}^{2}Z-\theta ,\right\Vert }^{2}\right)\end{array}=ωE∥TJS(Z,S2)−Z∥2+γ2(S2)4(‖Z‖2)3+2ωTJS(Z,S2)−Z,γ(S2)2(‖Z‖2)2Z+(1−ω)E∥TJS(Z,S2)−θ∥2+γ2(S2)4(‖Z‖2)3+2TJS(Z,S2)−θ,γ(S2)2(‖Z‖2)2Z=Rω(TJS(Z,S2),θ)+γ2E((S2)4)E1(‖Z‖2)3−2γω(1−ω)(q−2)n+2E((S2)3)E1(‖Z‖2)2+2(1−ω)EZ−θ−(1−ω)(q−2)n+2S2‖Z‖2Z,γ(S2)2(‖Z‖2)2Z,\begin{array}{rcl}& =& \omega E\left({\parallel {T}_{JS}\left(Z,{S}^{2})-Z\parallel }^{2}+{\gamma }^{2}\frac{{\left({S}^{2})}^{4}}{{\left(\Vert Z{\Vert }^{2})}^{3}}\right)+2\omega \left(\left\langle {T}_{JS}\left(Z,{S}^{2})-Z,\gamma \frac{{\left({S}^{2})}^{2}}{{\left(\Vert Z{\Vert }^{2})}^{2}}Z\right\rangle \right)\\ & & +\left(1-\omega )E\left({\parallel {T}_{JS}\left(Z,{S}^{2})-\theta \parallel }^{2}+{\gamma }^{2}\frac{{\left({S}^{2})}^{4}}{{\left(\Vert Z{\Vert }^{2})}^{3}}+2\left\langle {T}_{JS}\left(Z,{S}^{2})-\theta ,\gamma \frac{{\left({S}^{2})}^{2}}{{\left(\Vert Z{\Vert }^{2})}^{2}}Z\right\rangle \right)\\ & =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}E({\left({S}^{2})}^{4})E\left(\frac{1}{{\left(\Vert Z{\Vert }^{2})}^{3}}\right)-2\gamma \omega \frac{\left(1-\omega )\left(q-2)}{n+2}E({\left({S}^{2})}^{3})E\left(\frac{1}{{\left(\Vert Z{\Vert }^{2})}^{2}}\right)\\ & & +2\left(1-\omega )E\left(\left\langle Z-\theta -\frac{\left(1-\omega )\left(q-2)}{n+2}\frac{{S}^{2}}{\Vert Z{\Vert }^{2}}Z,\gamma \frac{{\left({S}^{2})}^{2}}{{\left(\Vert Z{\Vert }^{2})}^{2}}Z\right\rangle \right),\end{array}where the last equality is obtained as a result of the independence between the two random variables S2{S}^{2}and ‖Z‖2\Vert Z{\Vert }^{2}. Thus, Rω(Tγ,JS(2)(Z,S2),θ)=Rω(TJS(Z,S2),θ)+γ2E((σ2χn2)4)E1‖Z‖6−2γω(1−ω)(q−2)n+2E((σ2χn2)3)E1‖Z‖4+2γ(1−ω)E((σ2χn2)2)∑i=1qE(Zi−θi)Zi‖Z‖4−2γ(1−ω)2(q−2)n+2E((σ2χn2)3)E1‖Z‖4.\begin{array}{rcl}{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )& =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}E({\left({\sigma }^{2}{\chi }_{n}^{2})}^{4})E\left(\frac{1}{\Vert Z{\Vert }^{6}}\right)-2\gamma \omega \frac{\left(1-\omega )\left(q-2)}{n+2}E({\left({\sigma }^{2}{\chi }_{n}^{2})}^{3})E\left(\frac{1}{\Vert Z{\Vert }^{4}}\right)\\ & & +2\gamma \left(1-\omega )E({\left({\sigma }^{2}{\chi }_{n}^{2})}^{2})\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\left({Z}_{i}-{\theta }_{i})\frac{{Z}_{i}}{\Vert Z{\Vert }^{4}}\right)-2\gamma \frac{{\left(1-\omega )}^{2}\left(q-2)}{n+2}E({\left({\sigma }^{2}{\chi }_{n}^{2})}^{3})E\left(\frac{1}{\Vert Z{\Vert }^{4}}\right).\end{array}Then, by making the transformation y=Zσ=(y1,…,yq)ty=\frac{Z}{\sigma }={({y}_{1},\ldots ,{y}_{q})}^{t}, where yi=Ziσ∼Nθiσ,1{y}_{i}=\frac{{Z}_{i}}{\sigma }\hspace{0.33em} \sim \hspace{0.33em}N\left(\frac{{\theta }_{i}}{\sigma },1\right)for i=1,…,qi=1,\ldots ,q, and using Lemma 1 given in Stein [20], we get ∑i=1qE(Zi−θi)Zi‖Z‖4=1σ2∑i=1qE(yi−θiσ)yi‖y‖4=1σ2∑i=1qE∂∂yiyi‖y‖4=1σ2∑i=1qE1‖y‖4−4yi2‖y‖6=1σ2(q−4)E1‖y‖4.\begin{array}{rcl}\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\left({Z}_{i}-{\theta }_{i})\frac{{Z}_{i}}{\Vert Z{\Vert }^{4}}\right)& =& \frac{1}{{\sigma }^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\left({y}_{i}-\frac{{\theta }_{i}}{\sigma }\right)\frac{{y}_{i}}{\Vert y{\Vert }^{4}}\right)=\frac{1}{{\sigma }^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\frac{\partial }{\partial {y}_{i}}\frac{{y}_{i}}{\Vert y{\Vert }^{4}}\right)\\ & =& \frac{1}{{\sigma }^{2}}\mathop{\displaystyle \sum }\limits_{i=1}^{q}E\left(\frac{1}{\Vert y{\Vert }^{4}}-4\frac{{y}_{i}^{2}}{\Vert y{\Vert }^{6}}\right)=\frac{1}{{\sigma }^{2}}\left(q-4)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right).\end{array}Thus, Rω(Tγ,JS(2)(Z,S2),θ)=Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)E1‖y‖6−2γω(1−ω)(q−2)σ2n(n+4)E1‖y‖4+2γ(1−ω)σ2n(n+2)(q−4)E1‖y‖4−2γ(1−ω)2(q−2)σ2n(n+4)E1‖y‖4=Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)E1‖y‖6+2γn(n+2)(1−ω)σ2(q−4)−(q−2)(n+4)n+2E1‖y‖4.□\hspace{1.25em}\begin{array}{rcl}{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )& =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}n\left(n+2)\left(n+4)\left(n+6)E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)\\ & & -2\gamma \omega \left(1-\omega )\left(q-2){\sigma }^{2}n\left(n+4)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)+2\gamma \left(1-\omega ){\sigma }^{2}n\left(n+2)\left(q-4)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\\ & & -2\gamma {\left(1-\omega )}^{2}\left(q-2){\sigma }^{2}n\left(n+4)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\\ & =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}n\left(n+2)\left(n+4)\left(n+6)E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)\\ & & +2\gamma n\left(n+2)\left(1-\omega ){\sigma }^{2}\left[\left(q-4)-\frac{\left(q-2)\left(n+4)}{n+2}\right]E\left(\frac{1}{\Vert y{\Vert }^{4}}\right).\hspace{10em}\square \end{array}Theorem 3.1Under the balanced loss function Lω{L}_{\omega }, the estimator Tγ,JS(2)(Z,S2){T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2})with q>6q\gt 6andγ=2(1−ω)(q−6)(n+4)(n+6),\gamma =\frac{2\left(1-\omega )\left(q-6)}{\left(n+4)\left(n+6)},dominates the James-Stein estimator TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}).ProofAccording to Proposition 3.1, we have Rω(Tγ,JS(2)(Z,S2),θ)=Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)E1‖y‖6E1‖y‖4E1‖y‖4+2γσ2(1−ω)n(n+2)(q−4)−(q−2)(n+4)n+2E1‖y‖4\begin{array}{rcl}{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )& =& {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}n\left(n+2)\left(n+4)\left(n+6)\frac{E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)}{E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)}E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\\ & & +2\gamma {\sigma }^{2}\left(1-\omega )n\left(n+2)\left[\left(q-4)-\frac{\left(q-2)\left(n+4)}{n+2}\right]E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\end{array}≤Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)E1‖y‖6E1‖y‖4E1‖y‖4+2γσ2(1−ω)n(n+2)[(q−4)−(q−2)]E1‖y‖4.\begin{array}{rcl}& \le & {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}n\left(n+2)\left(n+4)\left(n+6)\frac{E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)}{E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)}E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)\\ & & +2\gamma {\sigma }^{2}\left(1-\omega )n\left(n+2){[}\left(q-4)-\left(q-2)]E\left(\frac{1}{\Vert y{\Vert }^{4}}\right).\end{array}Following Lemma 2 given in the study by Benkhaled et al. [19], we obtain E1‖y‖6E1‖y‖4=E(‖y‖−6)E(‖y‖−4)≤2−4+22Γq2−4+1Γq−42=1q−6.\frac{E\left(\frac{1}{\Vert y{\Vert }^{6}}\right)}{E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)}=\frac{E(\Vert y{\Vert }^{-6})}{E(\Vert y{\Vert }^{-4})}\le {2}^{\tfrac{-4+2}{2}}\frac{\Gamma \left(\frac{q}{2}-4+1\right)}{\Gamma \left(\frac{q-4}{2}\right)}=\frac{1}{q-6}.Then, (11)Rω(Tγ,JS(2)(Z,S2),θ)≤Rω(TJS(Z,S2),θ)+γ2σ2n(n+2)(n+4)(n+6)(q−6)E1‖y‖4−4γσ2(1−ω)n(n+2)E1‖y‖4.{R}_{\omega }\left({T}_{\gamma ,JS}^{\left(2)}\left(Z,{S}^{2}),\theta )\le {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )+{\gamma }^{2}{\sigma }^{2}\frac{n\left(n+2)\left(n+4)\left(n+6)}{\left(q-6)}E\left(\frac{1}{\Vert y{\Vert }^{4}}\right)-4\gamma {\sigma }^{2}\left(1-\omega )n\left(n+2)E\left(\frac{1}{\Vert y{\Vert }^{4}}\right).The right side of the aforementioned inequality is minimized at the optimal value of γ\gamma as follows: (12)γ^=2(1−ω)(q−6)(n+4)(n+6).\widehat{\gamma }=\frac{2\left(1-\omega )\left(q-6)}{\left(n+4)\left(n+6)}.Then, by replacing γ\gamma by γ^\widehat{\gamma }in equation (11), we obtain Rω(Tγ^,JS(2)(Z,S2),θ)≤Rω(TJS(Z,S2),θ)−4σ2(1−ω)2n(n+2)(q−6)(n+4)(n+6)≤Rω(TJS(Z,S2),θ).□\hspace{2.35em}{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}),\theta )\le {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )-4{\sigma }^{2}\frac{{\left(1-\omega )}^{2}n\left(n+2)\left(q-6)}{\left(n+4)\left(n+6)}\le {R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta ).\hspace{5em}\square 4Simulation resultsWe conduct here a simulation study for comparing the efficiency of the proposed estimators Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})and Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2})to the estimators TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}), TJS+(Z,S2){T}_{JS}^{+}\left(Z,{S}^{2})and the MLE. We consider here α=(1−ω)(q−2)2(n+2)\alpha =\frac{\left(1-\omega )\left(q-2)}{2\left(n+2)}in the estimator Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}). This comparison is done based on the risk ratio of these estimators to the MLE. Thus, the risk ratios of these estimators are denoted as follows: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}. We consider here all estimators to be functions of λ=‖θ‖2σ2\lambda =\frac{\Vert \theta {\Vert }^{2}}{{\sigma }^{2}}.Figures 1, 2, 3, 4, 5, show the curve of the risk ratios for simulated values of λ\lambda in the interval (1,30)\left(1,30)and for relatively low and high values of nn, qq, and ω\omega . The risk ratio of the MLE is represented by the horizontal line at the value of one. The gap between the curves of estimators indicates the gain magnitude of the estimator. We observed that the curves of all risk ratios for the different sets of nn, qq, and ω\omega values are entirely located below 1, which indicate the domination of these estimators to the MLE ZZ. Consequently, these estimators are considered minimax.Figure 1Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=6n=6, q=8q=8, and ω=0.1\omega =0.1.Figure 2Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=20n=20, q=8q=8, and ω=0.1\omega =0.1.Figure 3Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=6n=6, q=8q=8, and ω=0.5\omega =0.5.Figure 4Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=20n=20, q=12q=12, and ω=0.1\omega =0.1.Figure 5Curves of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}, and Rω(TJS+(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}for n=20n=20, q=12q=12, and ω=0.5\omega =0.5.Among these estimators, the positive-part James-Stein estimator (TJS+{T}_{JS}^{+}) was the more efficient estimator for values of λ\lambda less than approximately 10. It means that TJS+{T}_{JS}^{+}dominates all the considered estimators. Also, we note that the estimator Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2})dominates the James-Stein estimator TJS{T}_{JS}for the various values of nn, qq, and ω\omega . We also observe a larger gain of the estimator Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2})for low values of ω\omega . The gain of the estimators Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}), TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}), and TJS+(Z,S2){T}_{JS}^{+}\left(Z,{S}^{2})was very similar in a specific period of λ\lambda values, which depend on the combination of the values of the nnand qq. To study this similarity, we conduct simulation studies for all combinations of the selected values of nnand qqfor different sets of values of λ\lambda and ω\omega .Tables 1, 2, 3, 4 show the results of the risk ratios of the estimators Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}), Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}), and TJS(Z,S2){T}_{JS}\left(Z,{S}^{2}). Each cell of the tables represents the risk ratio of these estimators in order. We observe a strong relationship between the gain of the risk ratios and the values of ω\omega and λ\lambda . The gain of all risk ratios was large with small values of λ\lambda and ω\omega and tended to vanish with the increase of λ\lambda and ω\omega values. Also, the difference in the gain of risk ratios was observed in small values of λ\lambda . This difference indicated the domination of an estimator to another. Thus, the estimator Tγ^,JS(2)(Z,S2){T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2})dominated both estimators Tα(1)(Z,S2){T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})and TJS(Z,S2){T}_{JS}\left(Z,{S}^{2})for small values of λ\lambda . However, as the values of λ\lambda and ω\omega increased, the difference in the gain of these estimators became negligible (i.e., no improvement of the proposed estimators over the James-Stein estimator). The other parameters nnand qqhave also an influence on the gain of the estimators. The gain of the estimators was large for large values of nnand qqunder fixed values of ω\omega . Specifically, the increase of qqhad significant influence on the gain than the increase of nnvalues. This means that having large values of nn, qq, and λ\lambda with value of ω\omega close to zero leads to a larger gain of the estimators, which leads to a significant improvement. Thus, we conclude that the improvement of the considered estimators is clearly affected by the values of the parameters nn, qq, ω\omega , and λ\lambda .Table 1Values of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, and Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}at various values of λ\lambda and ω\omega , and n=6n=6, and q=8q=8λ\lambda ω\omega 0.00.10.20.70.91.24180.63620.67260.70900.89090.96360.51500.56350.61200.85450.95150.48240.53710.59110.85160.95125.00190.75010.77510.80010.92500.97500.66680.70010.73340.90000.96670.65020.68670.72280.89850.966510.43110.83260.84940.86610.94980.98330.77690.79920.82150.93300.97770.76940.79320.81670.93240.977620.00000.89620.90660.91700.96890.98960.86160.87550.88930.95850.98650.85890.87330.88760.95820.9861Table 2Values of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ),\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )},and Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}at various values of λ\lambda and ω\omega , and n=6,q=12n=6,q=12λ\lambda ω\omega 0.00.10.20.70.91.24180.57580.61820.66060.87270.95760.43430.49090.54750.83030.94340.40490.46710.52860.82760.94315.00190.67380.70640.73900.90210.96740.56510.60860.65210.86950.95650.54680.59380.64040.86790.956310.43110.75850.78270.80680.92760.97580.67810.71020.74240.90340.96780.66780.70200.73590.90250.967720.00000.83630.85270.86900.95090.98360.78170.80360.82540.93450.97820.77700.79980.82240.93410.9781Table 3Values of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ),\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )},and Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}at various values of λ\lambda and ω\omega and n=20,q=8n=20,q=8λ\lambda ω\omega 0.00.10.20.70.91.24180.55910.60320.64730.86770.95590.41210.47090.52970.82360.94120.37530.44110.50620.82030.94085.00190.69710.72740.75770.90910.96970.59610.63650.67690.87880.95960.57630.62040.66420.87700.959410.43110.79710.81740.83770.93910.97970.72950.75660.78360.91880.97290.72030.74910.77770.91800.972920.00000.87420.88680.89940.96230.98740.83230.84900.86580.94970.98320.82880.84630.86360.94940.9832Table 4Values of the risk ratios: Rω(Tα(1)(Z,S2),θ)Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )}, Rω(TJS(Z,S2),θ)Rω(Z,θ),\frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )},and Rω(Tγ^,JS(2)(Z,S2))Rω(Z,θ)\frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )}at various values of λ\lambda and ω\omega , and n=20n=20and q=12q=12λ\lambda ω\omega 0.00.10.20.70.91.24180.48580.53720.58860.84570.94860.31440.38290.45150.79430.93140.28540.35950.43300.79170.93115.00190.60460.64420.68370.88140.96050.47280.52550.57830.84180.94730.45400.51030.56620.84020.947110.43110.70730.73660.76590.91220.97070.60980.64880.68780.88290.96100.59880.63990.68080.88190.960920.00000.80160.82140.84130.94050.98010.73540.76190.78840.92060.97350.73030.75770.78510.92020.97355ConclusionIn this paper, we constructed a new class of shrinkage estimator that dominate the James-Stein estimator for the estimation of the mean θ\theta of the MVN distribution Z∼Nq(θ,σ2Iq)Z\hspace{0.33em} \sim \hspace{0.33em}{N}_{q}\left(\theta ,{\sigma }^{2}{I}_{q}), where σ2{\sigma }^{2}is unknown. We implemented the balanced square function in the form of the risk function of the estimators for the purpose of comparing the efficiency of two estimators. We started establishing a class of the minimaxity property for the estimator defined by Tα(1)(Z,S2)=(1−αS2/‖Z‖2)Z{T}_{\alpha }^{\left(1)}\left(Z,{S}^{2})=\left(1-\alpha {S}^{2}\hspace{-0.08em}\text{/}\hspace{-0.08em}\Vert Z{\Vert }^{2})Z. We found then the minimum risk of this class that resulted in the James-Stein estimator. Then, we constructed a new class of shrinkage estimator that is a modified version of the James-Stein estimator. Mainly, a term γ(S2/‖Z‖2)2Z\gamma {\left({S}^{2}\text{/}\Vert Z{\Vert }^{2})}^{2}Zwas added to the James-Stein estimator. The efficiency of the constructed estimator was explored by simulation studies under various values of the model parameters, and it has been shown that the constructed estimators beat the James-Stein estimator under the balanced loss function.An extension of this work is to implement the similar procedures of this paper in the Bayesian framework and explore possible shrinkage estimators for the mean parameters of the MVN distribution, such as the ridge estimators.

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: balanced loss function; James-Stein estimator; multivariate normal distribution; non-central chi-square distribution; shrinkage estimators; 62J07; 62C20; 62H10

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