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

: One of the most common challenges in multivariate statistical analysis is estimating the mean parameters. A well - known approach of estimating the mean parameters is the maximum likelihood esti mator ( MLE ) . However, the MLE becomes ine ﬃ cient in the case of having large - dimensional parameter space. A popular estimator that tackles this issue is the James - Stein estimator. Therefore, we aim to use the shrinkage method based on the balanced loss function to construct estimators for the mean parameters of the multivariate normal ( MVN ) distribution that dominates both the MLE and James - Stein estimators. Two classes of shrinkage estimators have been established that generalized the James - Stein estimator. We study their domination and minimaxity properties to the MLE and their performances to the James - Stein estima -tors. The e ﬃ ciency of the proposed estimators is explored through simulation studies.


Introduction
Estimating 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 Z to be normally distributed with an unknown mean vector θ and covariance matrix σ I q 2 , where q is the dimension of parameter space and I q is the q q × identity matrix. As the main object of this paper is to propose a new estimator of θ, we estimated the unknown parameter σ 2 by S 2 (S σ χ n 2 2 2 ). Then, we construct a new class of shrinkage estimators of θ 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 S Z Z where α and γ are real constant parameters that both depend on the integer parameters n and q. We show that these estimators are minimax and dominating the James-Stein estimator for any values of n and q. 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 T Z S αS Z Z , 1 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.

A class of minimax shrinkage estimators
We assume here the random variable Z is following an MVN distribution with mean vector θ and a covariance matrix σ I q 2 , where the parameters θ and σ 2 are unknown. Thus, the term Z σ 2 2 ‖ ‖ follows a non-central chisquare distribution with q degrees of freedom and non-centrality parameter λ θ σ As the aim of this paper is to establish an effective estimator for the mean parameter θ, we consider the statistic S 2 (S σ χ as an estimate of the unknown parameter σ 2 . Thus, for any estimator T of θ, the balanced squared error loss function is defined as follows: where T 0 is the target estimator of θ, ω is the weight given to the closeness between the estimators T and T 0 , and ω 1 − is the relative weight attributed to the accuracy of the estimator T . The associated risk function to the L T θ , ω ( ) function is defined as follows: Benkhaled et al. [19] demonstrated that the MLE of θ is Z T 0 ≔ . Then, its risk function becomes ω qσ 1 . This finding shows the minimaxity and inadmissibility property of T 0 for q 3 ≥ . Consequently, the minimaxity property is also achieved for any estimator that dominates the estimator T 0 . Now, let consider the estimator where α is a real constant parameter that can be related to the values of the parameters n and q.
Proposition 2.1. The associated risk function of the estimator T Z S , given in equation (2) based on the balanced loss function given in equation (1) is Proof.
The last equality comes from the independence between two random variables S 2 and Z 2 ∥ ∥ . As, From Proposition (2.1), the minimaxity and domination criterion of the estimator T Z S , to the MLE is achieved under the following condition: is minimized at the optimal α value (α ) as follows: Then, by considering α α  = , we get the James-Stein estimator From Proposition 2.1, the risk function of T Z S , JS 2 ( ) is expressed as follows: Based on equation (5), the positive part of James-Stein estimator can be defined as follows: ‖ ‖ , and its risk function associated with L ω is shown in the following formula: , represents the indicating function of the set α 1 . Both equations (6) and (8) show that , which proves the domination and minimaxity of both estimators T JS and T JS + over the MLE.

The improved shrinkage estimators of the James-Stein estimator
In this section, we construct a class of shrinkage estimators that has the domination property over the James-Stein estimator T Z S , ). The modified version of the James-Stein estimator is shown in the following formula: Proposition 3.1. The associated risk function of the estimator T Z S , given in equation (9) based on the balanced loss function given in equation (1) is where y y y , , where the last equality is obtained as a result of the independence between the two random variables S 2 and Z 2 ‖ ‖ . Thus, Then, by making the transformation y y y , , The right side of the aforementioned inequality is minimized at the optimal value of γ as follows: γ ω q n n Then, by replacing γ by γ  in equation (11), we obtain

Simulation results
We conduct here a simulation study for comparing the efficiency of the proposed estimators T Z S , Specifically, the increase of q had significant influence on the gain than the increase of n values. This means that having large values of n, q, and λ with value of ω 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 n, q, ω, and λ.

Conclusion
In this paper, we constructed a new class of shrinkage estimator that dominate the James-Stein estimator for the estimation of the mean θ of the MVN distribution Z N θ σ Ĩ , q q 2 ( ), where σ 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 T Z S αS Z Z , 1 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.