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

  • Abdelkader Benkhaled , Abdenour Hamdaoui EMAIL logo , Waleed Almutiry , Mohammed Alshahrani and Mekki Terbeche
Published/Copyright: February 21, 2022
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Open Mathematics
From the journal Open Mathematics Volume 20 Issue 1

Abstract

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 estimator (MLE). However, the MLE becomes inefficient 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 estimators. The efficiency of the proposed estimators is explored through simulation studies.

Keywords: balanced loss function; James-Stein estimator; multivariate normal distribution; non-central chi-square distribution; shrinkage estimators
MSC 2010: 62J07; 62C20; 62H10

1 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 2 I q , 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 2 ∼ σ 2 χ n 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 2 / ‖ Z ‖ 2 ) 2 Z to the James-Stein estimator T α ( 1 ) ( Z , S 2 ) = ( 1 − α S 2 / ‖ Z ‖ 2 ) 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 α ( 1 ) ( Z , S 2 ) = ( 1 − α S 2 / ‖ Z ‖ 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.

2 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 2 I q , where the parameters θ and σ 2 are unknown. Thus, the term ‖ Z ‖ 2 σ 2 follows a non-central chi-square distribution with q degrees of freedom and non-centrality parameter λ = ∥ θ ∥ 2 σ 2 . As the aim of this paper is to establish an effective estimator for the mean parameter θ , we consider the statistic S 2 ( S 2 ∼ σ 2 χ n 2 ) 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:

(1) L ω ( T , θ ) = ω ‖ T − T 0 ‖ 2 + ( 1 − ω ) ‖ T − θ ‖ 2 , 0 ≤ ω < 1 ,

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 1 − ω is the relative weight attributed to the accuracy of the estimator T . The associated risk function to the L ω ( T , θ ) function is defined as follows:

R ω ( T , θ ) = E ( L ω ( T , θ ) ) = ω E ( ‖ T − T 0 ‖ 2 ) + ( 1 − ω ) E ( ‖ T − θ ‖ 2 ) .

Benkhaled et al. [19] demonstrated that the MLE of θ is Z ≔ T 0 . Then, its risk function becomes ( 1 − ω ) q σ 2 . 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

(2) T α ( 1 ) ( Z , S 2 ) = 1 − α S 2 ‖ Z ‖ 2 Z = Z − α S 2 ‖ Z ‖ 2 Z ,

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 α ( 1 ) ( Z , S 2 ) given in equation (2) based on the balanced loss function given in equation (1) is

(3) R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) = ( 1 − ω ) σ 2 q − 2 α n σ 2 ( q − 2 ) E 1 ‖ Z ‖ 2 + α 2 n ( n + 2 ) σ 4 E 1 ‖ Z ‖ 2 .

Proof

R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) = ω E − α S 2 ‖ Z ‖ 2 Z 2 + ( 1 − ω ) E Z − θ − α S 2 ‖ Z ‖ 2 Z 2 = α 2 E ( ( S 2 ) 2 ) E 1 ‖ Z ‖ 2 + ( 1 − ω ) q σ 2 − 2 α ( 1 − ω ) E Z − θ , 1 ‖ Z ‖ 2 Z E ( S 2 ) .

The last equality comes from the independence between two random variables S 2 and ∥ Z ∥ 2 .

As,

E Z − θ , 1 ‖ Z ‖ 2 Z = ∑ i = 1 q E ( Z i − θ i ) 1 ‖ Z ‖ 2 Z i = ∑ i = 1 q E y i − θ i σ 1 ‖ y ‖ 2 y i ,

where y = Z σ = ( y 1 , … , y q ) t and for all i = 1 , … , q , y i = Z i σ ∼ N θ i σ , 1 . Then, based on Lemma 1 given in Stein [20], we get

E Z − θ , 1 ‖ Z ‖ 2 Z = ∑ i = 1 q E ∂ ∂ y i 1 ‖ y ‖ 2 y i = ∑ i = 1 q E 1 ‖ y ‖ 2 − 2 y i 2 ‖ y ‖ 4 = ( q − 2 ) E 1 ‖ y ‖ 2 = ( q − 2 ) σ 2 E 1 ‖ Z ‖ 2 .

Then,

R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) = α 2 E ( ( S 2 ) 2 ) E 1 ‖ Z ‖ 2 + ( 1 − ω ) q σ 2 − 2 α ( 1 − ω ) E Z − θ , 1 ‖ Z ‖ 2 Z E ( S 2 ) = α 2 E ( ( S 2 ) 2 ) E 1 ‖ Z ‖ 2 + ( 1 − ω ) q σ 2 − 2 α ( 1 − ω ) ( q − 2 ) σ 2 E 1 ‖ Z ‖ 2 E ( S 2 ) = ( 1 − ω ) σ 2 q − 2 α n σ 2 ( q − 2 ) E 1 ‖ Z ‖ 2 + α 2 n ( n + 2 ) σ 4 E 1 ‖ Z ‖ 2 . □

From Proposition (2.1), the minimaxity and domination criterion of the estimator T α ( 1 ) ( Z , S 2 ) to the MLE is achieved under the following condition:

0 ≤ α ≤ 2 ( 1 − ω ) ( q − 2 ) n + 2 .

Thus, the risk function R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) is minimized at the optimal α value ( α ^ ) as follows:

(4) α ^ = ( 1 − ω ) ( q − 2 ) n + 2 .

Then, by considering α = α ^ , we get the James-Stein estimator

(5) T J S ( Z , S 2 ) = T α ^ ( 1 ) ( Z , S 2 ) = 1 − α ^ S 2 ‖ Z ‖ 2 Z .

From Proposition 2.1, the risk function of T J S ( Z , S 2 ) is expressed as follows:

(6) R ω ( T J S ( Z , S 2 ) , θ ) = ( 1 − ω ) q σ 2 − ( q − 2 ) 2 ( 1 − ω ) 2 n n + 2 σ 2 E 1 ‖ Z ‖ 2 .

Based on equation (5), the positive part of James-Stein estimator can be defined as follows:

(7) T J S + ( Z , S 2 ) = 1 − α ^ S 2 ‖ Z ‖ 2 + Z = 1 − α ^ S 2 ‖ Z ‖ 2 Z I α ^ S 2 ‖ Z ‖ 2 ≤ 1 ,

where 1 − α ^ S 2 ‖ Z ‖ 2 + = max 0 , 1 − α ^ S 2 ‖ Z ‖ 2 , and its risk function associated with L ω is shown in the following formula:

(8) R ω ( T J S + ( Z , S 2 ) , θ ) = R ω ( T J S ( Z , S 2 ) , θ ) + E ‖ Z ‖ 2 − α ^ 2 S 4 ‖ Z ‖ 2 + 2 ( 1 − ω ) σ 2 ( q − 2 ) α ^ S 2 ‖ Z ‖ 2 − q σ 2 I α ^ S 2 ‖ Z ‖ 2 ≥ 1 ,

where I α ^ S 2 ‖ Z ‖ 2 ≥ 1 represents the indicating function of the set α ^ S 2 ‖ Z ‖ 2 ≥ 1 . Both equations (6) and (8) show that R ω ( T J S ( Z , S 2 ) , θ ) and R ω ( T J S + ( Z , S 2 ) , θ ) are less than ( 1 − ω ) q σ 2 = R ω ( Z , θ ) , which proves the domination and minimaxity of both estimators T J S and T J S + over the MLE.

3 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 J S ( Z , S 2 ) . This class of estimators is a modified version of T J S ( Z , S 2 ) . Specifically, we extend T J S ( Z , S 2 ) given in equation (5) by adding the term γ ( S 2 / ‖ Z ‖ 2 ) 2 Z , where γ behaves like α in equation (2). These new estimators are then investigated regarding their superiority to the James-Stein estimator T J S ( Z , S 2 ) . The modified version of the James-Stein estimator is shown in the following formula:

(9) T γ , J S ( 2 ) ( Z , S 2 ) = T J S ( Z , S 2 ) + γ S 2 ‖ Z ‖ 2 2 Z = Z − ( 1 − ω ) ( q − 2 ) n + 2 S 2 ‖ Z ‖ 2 Z + γ S 2 ‖ Z ‖ 2 2 Z .

Proposition 3.1

The associated risk function of the estimator T γ , J S ( 2 ) ( Z , S 2 ) given in equation (9) based on the balanced loss function given in equation (1) is

(10) R ω ( T γ , J S ( 2 ) ( Z , S 2 ) , θ ) = R ω ( T J S ( Z , S 2 ) , θ ) + 2 γ n ( n + 2 ) ( 1 − ω ) σ 2 ( q − 4 ) − ( q − 2 ) ( n + 4 ) n + 2 E 1 ‖ y ‖ 4 + γ 2 n ( n + 2 ) ( n + 4 ) ( n + 6 ) σ 2 E 1 ‖ y ‖ 6 ,

where y = Z σ = ( y 1 , … , y q ) t and y i = Z i σ ∼ N θ i σ , 1 for i = 1 , … , q .

Proof

R ω ( T γ , J S ( 2 ) ( Z , S 2 ) , θ ) = ω E T J S ( Z , S 2 ) + γ ( S 2 ‖ Z ‖ 2 ) 2 Z − Z 2 + ( 1 − ω ) E T J S ( Z , S 2 ) + γ ( S 2 ‖ Z ‖ 2 ) 2 Z − θ 2

= ω E ∥ T J S ( Z , S 2 ) − Z ∥ 2 + γ 2 ( S 2 ) 4 ( ‖ Z ‖ 2 ) 3 + 2 ω T J S ( Z , S 2 ) − Z , γ ( S 2 ) 2 ( ‖ Z ‖ 2 ) 2 Z + ( 1 − ω ) E ∥ T J S ( Z , S 2 ) − θ ∥ 2 + γ 2 ( S 2 ) 4 ( ‖ Z ‖ 2 ) 3 + 2 T J S ( Z , S 2 ) − θ , γ ( S 2 ) 2 ( ‖ Z ‖ 2 ) 2 Z = R ω ( T J S ( Z , S 2 ) , θ ) + γ 2 E ( ( S 2 ) 4 ) E 1 ( ‖ Z ‖ 2 ) 3 − 2 γ ω ( 1 − ω ) ( q − 2 ) n + 2 E ( ( S 2 ) 3 ) E 1 ( ‖ Z ‖ 2 ) 2 + 2 ( 1 − ω ) E Z − θ − ( 1 − ω ) ( q − 2 ) n + 2 S 2 ‖ Z ‖ 2 Z , γ ( S 2 ) 2 ( ‖ Z ‖ 2 ) 2 Z ,

where the last equality is obtained as a result of the independence between the two random variables S 2 and ‖ Z ‖ 2 . Thus,

R ω ( T γ , J S ( 2 ) ( Z , S 2 ) , θ ) = R ω ( T J S ( Z , S 2 ) , θ ) + γ 2 E ( ( σ 2 χ n 2 ) 4 ) E 1 ‖ Z ‖ 6 − 2 γ ω ( 1 − ω ) ( q − 2 ) n + 2 E ( ( σ 2 χ n 2 ) 3 ) E 1 ‖ Z ‖ 4 + 2 γ ( 1 − ω ) E ( ( σ 2 χ n 2 ) 2 ) ∑ i = 1 q E ( Z i − θ i ) Z i ‖ Z ‖ 4 − 2 γ ( 1 − ω ) 2 ( q − 2 ) n + 2 E ( ( σ 2 χ n 2 ) 3 ) E 1 ‖ Z ‖ 4 .

Then, by making the transformation y = Z σ = ( y 1 , … , y q ) t , where y i = Z i σ ∼ N θ i σ , 1 for i = 1 , … , q , and using Lemma 1 given in Stein [20], we get

∑ i = 1 q E ( Z i − θ i ) Z i ‖ Z ‖ 4 = 1 σ 2 ∑ i = 1 q E ( y i − θ i σ ) y i ‖ y ‖ 4 = 1 σ 2 ∑ i = 1 q E ∂ ∂ y i y i ‖ y ‖ 4 = 1 σ 2 ∑ i = 1 q E 1 ‖ y ‖ 4 − 4 y i 2 ‖ y ‖ 6 = 1 σ 2 ( q − 4 ) E 1 ‖ y ‖ 4 .

Thus,

R ω ( T γ , J S ( 2 ) ( Z , S 2 ) , θ ) = R ω ( T J S ( Z , S 2 ) , θ ) + γ 2 σ 2 n ( n + 2 ) ( n + 4 ) ( n + 6 ) E 1 ‖ y ‖ 6 − 2 γ ω ( 1 − ω ) ( q − 2 ) σ 2 n ( n + 4 ) E 1 ‖ y ‖ 4 + 2 γ ( 1 − ω ) σ 2 n ( n + 2 ) ( q − 4 ) E 1 ‖ y ‖ 4 − 2 γ ( 1 − ω ) 2 ( q − 2 ) σ 2 n ( n + 4 ) E 1 ‖ y ‖ 4 = R ω ( T J S ( Z , S 2 ) , θ ) + γ 2 σ 2 n ( n + 2 ) ( n + 4 ) ( n + 6 ) E 1 ‖ y ‖ 6 + 2 γ n ( n + 2 ) ( 1 − ω ) σ 2 ( q − 4 ) − ( q − 2 ) ( n + 4 ) n + 2 E 1 ‖ y ‖ 4 . □

Theorem 3.1

Under the balanced loss function L ω , the estimator T γ , J S ( 2 ) ( Z , S 2 ) with q > 6 and

γ = 2 ( 1 − ω ) ( q − 6 ) ( n + 4 ) ( n + 6 ) ,

dominates the James-Stein estimator T J S ( Z , S 2 ) .

Proof

According to Proposition 3.1, we have

R ω ( T γ , J S ( 2 ) ( Z , S 2 ) , θ ) = R ω ( T J S ( Z , S 2 ) , θ ) + γ 2 σ 2 n ( n + 2 ) ( n + 4 ) ( n + 6 ) E 1 ‖ y ‖ 6 E 1 ‖ y ‖ 4 E 1 ‖ y ‖ 4 + 2 γ σ 2 ( 1 − ω ) n ( n + 2 ) ( q − 4 ) − ( q − 2 ) ( n + 4 ) n + 2 E 1 ‖ y ‖ 4

≤ R ω ( T J S ( Z , S 2 ) , θ ) + γ 2 σ 2 n ( n + 2 ) ( n + 4 ) ( n + 6 ) E 1 ‖ y ‖ 6 E 1 ‖ y ‖ 4 E 1 ‖ y ‖ 4 + 2 γ σ 2 ( 1 − ω ) n ( n + 2 ) [ ( q − 4 ) − ( q − 2 ) ] E 1 ‖ y ‖ 4 .

Following Lemma 2 given in the study by Benkhaled et al. [19], we obtain

E 1 ‖ y ‖ 6 E 1 ‖ y ‖ 4 = E ( ‖ y ‖ − 6 ) E ( ‖ y ‖ − 4 ) ≤ 2 − 4 + 2 2 Γ q 2 − 4 + 1 Γ q − 4 2 = 1 q − 6 .

Then,

(11) R ω ( T γ , J S ( 2 ) ( Z , S 2 ) , θ ) ≤ R ω ( T J S ( Z , S 2 ) , θ ) + γ 2 σ 2 n ( n + 2 ) ( n + 4 ) ( n + 6 ) ( q − 6 ) E 1 ‖ y ‖ 4 − 4 γ σ 2 ( 1 − ω ) n ( n + 2 ) E 1 ‖ y ‖ 4 .

The right side of the aforementioned inequality is minimized at the optimal value of γ as follows:

(12) γ ^ = 2 ( 1 − ω ) ( q − 6 ) ( n + 4 ) ( n + 6 ) .

Then, by replacing γ by γ ^ in equation (11), we obtain

R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) , θ ) ≤ R ω ( T J S ( Z , S 2 ) , θ ) − 4 σ 2 ( 1 − ω ) 2 n ( n + 2 ) ( q − 6 ) ( n + 4 ) ( n + 6 ) ≤ R ω ( T J S ( Z , S 2 ) , θ ) . □

4 Simulation results

We conduct here a simulation study for comparing the efficiency of the proposed estimators T α ( 1 ) ( Z , S 2 ) and T γ ^ , J S ( 2 ) ( Z , S 2 ) to the estimators T J S ( Z , S 2 ) , T J S + ( Z , S 2 ) and the MLE. We consider here α = ( 1 − ω ) ( q − 2 ) 2 ( n + 2 ) in the estimator T α ( 1 ) ( 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: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) . We consider here all estimators to be functions of λ = ‖ θ ‖ 2 σ 2 .

Figures 1, 2, 3, 4, 5, show the curve of the risk ratios for simulated values of λ in the interval ( 1 , 30 ) and for relatively low and high values of n , q , and ω . 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 n , q , and ω values are entirely located below 1, which indicate the domination of these estimators to the MLE Z . Consequently, these estimators are considered minimax.

Figure 1 Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )} , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} for n = 6 n=6 , q = 8 q=8 , and ω = 0.1 \omega =0.1 .
Figure 1

Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) for n = 6 , q = 8 , and ω = 0.1 .

Figure 2 Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )} , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} for n = 20 n=20 , q = 8 q=8 , and ω = 0.1 \omega =0.1 .
Figure 2

Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) for n = 20 , q = 8 , and ω = 0.1 .

Figure 3 Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )} , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} for n = 6 n=6 , q = 8 q=8 , and ω = 0.5 \omega =0.5 .
Figure 3

Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) for n = 6 , q = 8 , and ω = 0.5 .

Figure 4 Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )} , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} for n = 20 n=20 , q = 12 q=12 , and ω = 0.1 \omega =0.1 .
Figure 4

Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) for n = 20 , q = 12 , and ω = 0.1 .

Figure 5 Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\alpha }^{\left(1)}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{\widehat{\gamma },JS}^{\left(2)}\left(Z,{S}^{2}))}{{R}_{\omega }\left(Z,\theta )} , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) \frac{{R}_{\omega }\left({T}_{JS}^{+}\left(Z,{S}^{2}),\theta )}{{R}_{\omega }\left(Z,\theta )} for n = 20 n=20 , q = 12 q=12 , and ω = 0.5 \omega =0.5 .
Figure 5

Curves of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) , and R ω ( T J S + ( Z , S 2 ) , θ ) R ω ( Z , θ ) for n = 20 , q = 12 , and ω = 0.5 .

Among these estimators, the positive-part James-Stein estimator ( T J S + ) was the more efficient estimator for values of λ less than approximately 10. It means that T J S + dominates all the considered estimators. Also, we note that the estimator T γ ^ , J S ( 2 ) ( Z , S 2 ) dominates the James-Stein estimator T J S for the various values of n , q , and ω . We also observe a larger gain of the estimator T γ ^ , J S ( 2 ) ( Z , S 2 ) for low values of ω . The gain of the estimators T γ ^ , J S ( 2 ) ( Z , S 2 ) , T J S ( Z , S 2 ) , and T J S + ( Z , S 2 ) was very similar in a specific period of λ values, which depend on the combination of the values of the n and q . To study this similarity, we conduct simulation studies for all combinations of the selected values of n and q for different sets of values of λ and ω .

Tables 1, 2, 3, 4 show the results of the risk ratios of the estimators T α ( 1 ) ( Z , S 2 ) , T γ ^ , J S ( 2 ) ( Z , S 2 ) , and T J S ( 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 and λ . The gain of all risk ratios was large with small values of λ and ω and tended to vanish with the increase of λ and ω values. Also, the difference in the gain of risk ratios was observed in small values of λ . This difference indicated the domination of an estimator to another. Thus, the estimator T γ ^ , J S ( 2 ) ( Z , S 2 ) dominated both estimators T α ( 1 ) ( Z , S 2 ) and T J S ( Z , S 2 ) for small values of λ . However, as the values of λ and ω 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 n and q have also an influence on the gain of the estimators. The gain of the estimators was large for large values of n and q under fixed values of ω . 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 .

Table 1

Values of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , and R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) at various values of λ and ω , and n = 6 , and q = 8

λ ω
0.0 0.1 0.2 0.7 0.9
1.2418 0.6362 0.6726 0.7090 0.8909 0.9636
0.5150 0.5635 0.6120 0.8545 0.9515
0.4824 0.5371 0.5911 0.8516 0.9512
5.0019 0.7501 0.7751 0.8001 0.9250 0.9750
0.6668 0.7001 0.7334 0.9000 0.9667
0.6502 0.6867 0.7228 0.8985 0.9665
10.4311 0.8326 0.8494 0.8661 0.9498 0.9833
0.7769 0.7992 0.8215 0.9330 0.9777
0.7694 0.7932 0.8167 0.9324 0.9776
20.0000 0.8962 0.9066 0.9170 0.9689 0.9896
0.8616 0.8755 0.8893 0.9585 0.9865
0.8589 0.8733 0.8876 0.9582 0.9861
Table 2

Values of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , and R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) at various values of λ and ω , and n = 6 , q = 12

λ ω
0.0 0.1 0.2 0.7 0.9
1.2418 0.5758 0.6182 0.6606 0.8727 0.9576
0.4343 0.4909 0.5475 0.8303 0.9434
0.4049 0.4671 0.5286 0.8276 0.9431
5.0019 0.6738 0.7064 0.7390 0.9021 0.9674
0.5651 0.6086 0.6521 0.8695 0.9565
0.5468 0.5938 0.6404 0.8679 0.9563
10.4311 0.7585 0.7827 0.8068 0.9276 0.9758
0.6781 0.7102 0.7424 0.9034 0.9678
0.6678 0.7020 0.7359 0.9025 0.9677
20.0000 0.8363 0.8527 0.8690 0.9509 0.9836
0.7817 0.8036 0.8254 0.9345 0.9782
0.7770 0.7998 0.8224 0.9341 0.9781
Table 3

Values of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , and R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) at various values of λ and ω and n = 20 , q = 8

λ ω
0.0 0.1 0.2 0.7 0.9
1.2418 0.5591 0.6032 0.6473 0.8677 0.9559
0.4121 0.4709 0.5297 0.8236 0.9412
0.3753 0.4411 0.5062 0.8203 0.9408
5.0019 0.6971 0.7274 0.7577 0.9091 0.9697
0.5961 0.6365 0.6769 0.8788 0.9596
0.5763 0.6204 0.6642 0.8770 0.9594
10.4311 0.7971 0.8174 0.8377 0.9391 0.9797
0.7295 0.7566 0.7836 0.9188 0.9729
0.7203 0.7491 0.7777 0.9180 0.9729
20.0000 0.8742 0.8868 0.8994 0.9623 0.9874
0.8323 0.8490 0.8658 0.9497 0.9832
0.8288 0.8463 0.8636 0.9494 0.9832
Table 4

Values of the risk ratios: R ω ( T α ( 1 ) ( Z , S 2 ) , θ ) R ω ( Z , θ ) , R ω ( T J S ( Z , S 2 ) , θ ) R ω ( Z , θ ) , and R ω ( T γ ^ , J S ( 2 ) ( Z , S 2 ) ) R ω ( Z , θ ) at various values of λ and ω , and n = 20 and q = 12

λ ω
0.0 0.1 0.2 0.7 0.9
1.2418 0.4858 0.5372 0.5886 0.8457 0.9486
0.3144 0.3829 0.4515 0.7943 0.9314
0.2854 0.3595 0.4330 0.7917 0.9311
5.0019 0.6046 0.6442 0.6837 0.8814 0.9605
0.4728 0.5255 0.5783 0.8418 0.9473
0.4540 0.5103 0.5662 0.8402 0.9471
10.4311 0.7073 0.7366 0.7659 0.9122 0.9707
0.6098 0.6488 0.6878 0.8829 0.9610
0.5988 0.6399 0.6808 0.8819 0.9609
20.0000 0.8016 0.8214 0.8413 0.9405 0.9801
0.7354 0.7619 0.7884 0.9206 0.9735
0.7303 0.7577 0.7851 0.9202 0.9735

5 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 q ( θ , σ 2 I q ) , 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 α ( 1 ) ( Z , S 2 ) = ( 1 − α S 2 / ‖ Z ‖ 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 ( S 2 / ‖ Z ‖ 2 ) 2 Z was 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.

Acknowledgements

The authors are very grateful to the editor and the referees for their valuable suggestions and advice that enhance the whole paper.

  1. Funding information: This research received no external funding.

  2. Conflict of interest: The authors state no conflict of interest.

References

[1] C. Stein, Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, in: Proceedings of the 3th Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, Vol. 3.1, University of California Press, Berkeley, 1956, pp. 197–206. 10.1525/9780520313880-018Search in Google Scholar

[2] W. James and C. Stein, Estimation with quadratic loss, in: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, Vol. 4.1, University of California Press, Berkeley, 1961, pp. 361–379. 10.1007/978-1-4612-0919-5_30Search in Google Scholar

[3] O. Nimet and K. Selahattin, Risk performance of some shrinkage estimators, Comm. Statist. Simulation Comput. 50 (2021), no. 2, 323–342. 10.1080/03610918.2018.1554116Search in Google Scholar

[4] M. Norouzirad and M. Arashi, Preliminary test and Stein-type shrinkage ridge estimators in robust regression, Statist. Papers 60 (2019), no. 6, 1849–1882. 10.1007/s00362-017-0899-3Search in Google Scholar

[5] A. Khursheed, A family of admissible minimax estimators of a mean of a multivariate normal distribution, Ann. Statist. 1 (1973), 517–525. 10.1214/aos/1176342417Search in Google Scholar

[6] A. J. Baranchik, Inadmissibility of maximum likelihood estimators in some multiple regression problems with three or more independents variables, Ann. Statist. 1 (1973), 312–321. 10.1214/aos/1176342368Search in Google Scholar

[7] N. Shinozaki, A note on estimating the mean vector of a multivariate normal distribution with general quadratic loss function, Keio Engrg. Rep. 27 (1974), 105–112. Search in Google Scholar

[8] B. Efron and C. N. Morris, Stein’s estimation rule and its competitors: An empirical Bayes approach, J. Amer. Statist. Assoc. 68 (1973), 117–130. 10.1080/01621459.1973.10481350Search in Google Scholar

[9] B. Efron and C. N. Morris, Data analysis using Stein’s estimator and its generalizations, J. Amer. Statist. Assoc. 70 (1975), 311–319. 10.1080/01621459.1975.10479864Search in Google Scholar

[10] J. O. Berger and W. E. Strawderman, Choice of hierarchical priors: Admissibility in estimation of normal means, Ann. Statist. 24 (1996), 931–951. 10.1214/aos/1032526950Search in Google Scholar

[11] A. Benkhaled and A. Hamdaoui, General classes of shrinkage estimators for the multivariate normal mean with unknown variance: Minimaxity and limit of risk ratios, Kragujevac J. Math. 46 (2019), no. 2, 193–213. 10.46793/KgJMat2202.193BSearch in Google Scholar

[12] A. Hamdaoui, A. Benkhaled, and M. Terbeche, Baranchick-type estimators of a multivariate normal mean under the general quadratic loss function, J. Sib. Fed. Univ. Math. Phys. 13 (2020), no. 5, 608–621. 10.17516/1997-1397-2020-13-5-608-621Search in Google Scholar

[13] A. Hamdaoui, A. Benkhaled, and M. Mezouar, Minimaxity and limits of risk ratios of shrinkage estimators of a multivariate normal mean in the Bayesian case, Stat. Optim. Inf. Comput. 8 (2020), no. 2, 507–520. 10.19139/soic-2310-5070-735Search in Google Scholar

[14] S. Zinodiny, S. Rezaei, and S. Nadarajah, Bayes minimax estimation of the mean matrix of matrix-variate normal distribution under balanced loss function, Statist. Probab. Lett. 125 (2017), 110–120, http://doi.org/10.1016/j.spl.2017.02.003. 10.1016/j.spl.2017.02.003Search in Google Scholar

[15] A. Zellner, Bayesian and non-Bayesian estimation using balanced loss functions, in: J. O. Berger, S. S. Gupta (eds.), Statistical Decision Theory and Related Topics, Vol. 8, Springer, New York, 1994, pp. 337–390. 10.1007/978-1-4612-2618-5_28Search in Google Scholar

[16] N. Sanjari Farsipour and A. Asgharzadeh, Estimation of a normal mean relative to balanced loss functions, Statist. Papers 45 (2004), 279–286. 10.1007/BF02777228Search in Google Scholar

[17] M. Jafari Jozani, A. Leblanc, and E. Marchand, On continuous distribution functions, minimax and best invariant estimators and integrated balanced loss functions, Canad. J. Statistist. 42 (2014), 470–486. 10.1002/cjs.11217Search in Google Scholar

[18] K. Selahattin and D. Issam, The optimal extended balanced loss function estimators, J. Comput. Appl. Math. 345 (2019), 86–98. 10.1016/j.cam.2018.06.021Search in Google Scholar

[19] A. Benkhaled, M. Terbeche, and A. Hamdaoui, Polynomials shrinkage estimators of a multivariate normal mean, Stat. Optim. Inf. Comput. (2021), https://doi.org/10.19139/soic-2310-5070-1095. 10.19139/soic-2310-5070-1095Search in Google Scholar

[20] C. Stein, Estimation of the mean of a multivariate normal distribution, Ann. Statist. 9 (1981), 1135–1151. 10.1214/aos/1176345632Search in Google Scholar
Received: 2021-05-12
Revised: 2021-10-05
Accepted: 2021-12-09
Published Online: 2022-02-21

© 2022 Abdelkader Benkhaled et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2022-0008
Keywords for this article
balanced loss function; James-Stein estimator; multivariate normal distribution; non-central chi-square distribution; shrinkage estimators
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  1. Regular Articles
  2. A random von Neumann theorem for uniformly distributed sequences of partitions
  3. Note on structural properties of graphs
  4. Mean-field formulation for mean-variance asset-liability management with cash flow under an uncertain exit time
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  80. Stein-Weiss inequality for local mixed radial-angular Morrey spaces
  81. Eigenvalues of transition weight matrix for a family of weighted networks
  82. A modified Tikhonov regularization for unknown source in space fractional diffusion equation
  83. Modular forms of half-integral weight on Γ0(4) with few nonvanishing coefficients modulo ℓ
  84. Some estimates for commutators of bilinear pseudo-differential operators
  85. Extension of isometries in real Hilbert spaces
  86. Existence of positive periodic solutions for first-order nonlinear differential equations with multiple time-varying delays
  87. B-Fredholm elements in primitive C*-algebras
  88. Unique solvability for an inverse problem of a nonlinear parabolic PDE with nonlocal integral overdetermination condition
  89. An algebraic semigroup method for discovering maximal frequent itemsets
  90. Class-preserving Coleman automorphisms of some classes of finite groups
  91. Exponential stability of traveling waves for a nonlocal dispersal SIR model with delay
  92. Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses
  93. The transitivity of primary conjugacy in regular ω-semigroups
  94. Stability estimation of some Markov controlled processes
  95. On nonnil-coherent modules and nonnil-Noetherian modules
  96. N-Tuples of weighted noncommutative Orlicz space and some geometrical properties
  97. The dimension-free estimate for the truncated maximal operator
  98. A human error risk priority number calculation methodology using fuzzy and TOPSIS grey
  99. Compact mappings and s-mappings at subsets
  100. The structural properties of the Gompertz-two-parameter-Lindley distribution and associated inference
  101. A monotone iteration for a nonlinear Euler-Bernoulli beam equation with indefinite weight and Neumann boundary conditions
  102. Delta waves of the isentropic relativistic Euler system coupled with an advection equation for Chaplygin gas
  103. Multiplicity and minimality of periodic solutions to fourth-order super-quadratic difference systems
  104. On the reciprocal sum of the fourth power of Fibonacci numbers
  105. Averaging principle for two-time-scale stochastic differential equations with correlated noise
  106. Phragmén-Lindelöf alternative results and structural stability for Brinkman fluid in porous media in a semi-infinite cylinder
  107. Study on r-truncated degenerate Stirling numbers of the second kind
  108. On 7-valent symmetric graphs of order 2pq and 11-valent symmetric graphs of order 4pq
  109. Some new characterizations of finite p-nilpotent groups
  110. A Billingsley type theorem for Bowen topological entropy of nonautonomous dynamical systems
  111. F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface
  112. On modules related to McCoy modules
  113. On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems
  114. Solvability for a nonlocal dispersal model governed by time and space integrals
  115. Finite groups whose maximal subgroups of even order are MSN-groups
  116. Symmetric results of a Hénon-type elliptic system with coupled linear part
  117. On the connection between Sp-almost periodic functions defined on time scales and ℝ
  118. On a class of Harada rings
  119. On regular subgroup functors of finite groups
  120. Fast iterative solutions of Riccati and Lyapunov equations
  121. Weak measure expansivity of C2 dynamics
  122. Admissible congruences on type B semigroups
  123. Generalized fractional Hermite-Hadamard type inclusions for co-ordinated convex interval-valued functions
  124. Inverse eigenvalue problems for rank one perturbations of the Sturm-Liouville operator
  125. Data transmission mechanism of vehicle networking based on fuzzy comprehensive evaluation
  126. Dual uniformities in function spaces over uniform continuity
  127. Review Article
  128. On Hahn-Banach theorem and some of its applications
  129. Rapid Communication
  130. Discussion of foundation of mathematics and quantum theory
  131. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part II)
  132. A study of minimax shrinkage estimators dominating the James-Stein estimator under the balanced loss function
  133. Representations by degenerate Daehee polynomials
  134. Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme
  135. Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator
  136. Special Issue on Evolution Equations, Theory and Applications (Part II)
  137. Coupled measure of noncompactness and functional integral equations
  138. Existence results for neutral evolution equations with nonlocal conditions and delay via fractional operator
  139. Global weak solution of 3D-NSE with exponential damping
  140. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  141. Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells
  142. A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN
  143. Jensen-type inequalities for m-convex functions
  144. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part III)
  145. The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation
  146. Basic inequalities for statistical submanifolds in Golden-like statistical manifolds
  147. Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term
  148. Hopf bifurcation and Turing instability in a diffusive predator-prey model with hunting cooperation
  149. Efficient fixed-point iteration for generalized nonexpansive mappings and its stability in Banach spaces
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