On admissible estimation of a mean vector when the scale is unknown
OOn admissible estimation of a meanvector when the scale is unknown
Yuzo Maruyama , and William, E. Strawderman Graduate School of Business Administration, Kobe Universitye-mail: [email protected] Department of Statistics and Biostatistics, Rutgers Universitye-mail: [email protected]
Abstract:
We consider admissibility of generalized Bayes estimators ofthe mean of a multivariate normal distribution when the scale is unknownunder quadratic loss. The priors considered put the improper invariantprior on the scale while the prior on the mean has a hierarchical normalstructure conditional on the scale. This conditional hierarchical prior is es-sentially that of Maruyama and Strawderman (2021, Biometrika) (MS21)which is indexed by a hyperparameter a . In that paper a is chosen so thisconditional prior is proper which corresponds to a > −
1. This paper ex-tends MS21 by considering improper conditional priors with a in the closedinterval [ − , − a . The authors, inMaruyama and Strawderman (2017, JMVA), have earlier shown that suchconditional priors with a < − a = −
2, with admis-sibility holding for a ≥ − a < −
2. This boundarycorresponds exactly to that in the known scale case for these conditionalpriors, and which follows from Brown (1971, AOMS). As a notable benefitof this enlargement of the class of admissible generalized Bayes estimators,we give admissible and minimax estimators in all dimensions greater than2 as opposed to MS21 which required the dimension to be greater than 4.In one particularly interesting special case, we establish that the joint Steinprior for the unknown scale case leads to a minimax admissible estimatorin all dimensions greater than 2.
MSC2020 subject classifications:
Primary 62C15; secondary 62C20.
Keywords and phrases: admissibility, Bayes estimators, minimaxity.
1. Introduction
We consider admissibility of generalized Bayes estimators of the mean of a mul-tivariate normal distribution when the scale is unknown under quadratic loss.Specifically, we consider the model X ∼ N p ( θ, σ I ), S ∼ σ χ n where X and S are independent with densities f ( x | θ, η ) = η p/ (2 π ) p/ exp (cid:18) − η (cid:107) x − θ (cid:107) (cid:19) ,f ( s | η ) = η n/ s n/ − Γ( n/ n/ exp (cid:16) − ηs (cid:17) , (1.1) a r X i v : . [ m a t h . S T ] F e b . Maruyama and W. E. Strawderman/Admissibility with η = 1 /σ . The loss function is scaled quadratic loss L ( δ ; θ, η ) = η (cid:107) δ ( x, s ) − θ (cid:107) . (1.2)The estimators considered are generalized Bayes with respect to the generalizedpriors on ( θ, η ), π ∗ ( θ, η ) = 1 η × (cid:90) η p/ (2 π ) p/ g p/ exp (cid:18) − η g (cid:107) θ (cid:107) (cid:19) π ( g )d g, (1.3)with the following hierarchical structure θ | { g, η } ∼ N p (0 , ( g/η ) I ) , π ( g ) = 1( g + 1) a +2 (cid:18) gg + 1 (cid:19) b , η ∼ η . (1.4)As detailed in Section 2, the generalized Bayes estimator under this prior isgiven by δ ∗ = (cid:32) − (cid:82) ∞ ( g + 1) − p/ − (1 + (cid:107) x (cid:107) / { s ( g + 1) } ) − p/ − n/ − π ( g )d g (cid:82) ∞ ( g + 1) − p/ (1 + (cid:107) x (cid:107) / { s ( g + 1) } ) − p/ − n/ − π ( g )d g (cid:33) x (1.5)which is well-defined if (cid:90) ∞ π ( g )d g ( g + 1) p/ = (cid:90) ∞ (cid:18) gg + 1 (cid:19) b d g ( g + 1) p/ a +2 < ∞ . This is assured provided p/ a + 1 > b + 1 > − < a < n/ b > −
1, i.e., in cases where π ( g )is proper ( (cid:82) ∞ π ( g )d g < ∞ ), even though, since the prior on η is proportional to1 /η , the prior π ∗ ( θ, η ) is improper. This paper considers the more challengingproblem where π ( g ) is itself improper ( (cid:82) ∞ π ( g )d g = ∞ ). We establish admissi-bility in two such cases: CASE I max( − p/ − , − < a ≤ − b > − p ≥ , (1.7) CASE II a = − b ≥ p ≥ . (1.8)The main theorem of this paper is as follows. Theorem 1.1.
Under either assumption, (1.7) or (1.8) , the generalized Bayesestimator under π ∗ is admissible among all estimators. The authors, in Maruyama and Strawderman (2017), have earlier shown thatsuch conditional priors with a < − . Maruyama and W. E. Strawderman/Admissibility class of priors. It establishes the the boundary as a = −
2, with admissibilityholding for a ≥ − a < −
2. The ultimate admissibilityresult by this paper result of this paper was foreshadowed by Maruyama andStrawderman (2020) where admissibility of generalized Bayes estimators with a ≥ − { − ψ ( (cid:107) x (cid:107) /s ) } x .The estimator X , with a constant risk p , is minimax for all p . It is admissiblefor p = 1 , p ≥
3. Therefore we are mainlyinterested in proposing admissible minimax estimators for p ≥
3. When p ≥ − p/ − ξ ( p, n ) = − p − n + 2)2(2 p + n − . Hence we have a following result.
Theorem 1.2.
Assume p ≥ and n ≥ . Then the generalized Bayes estimatorunder π ∗ is minimax and admissible among the class of all estimators if − ≤ a ≤ ξ ( p, n ) and b ≥ . Figure 1 presents a summary of admissibility/inadmissibility and minimaxityresults for the unknown scale case for this class of priors.
Remark . Brown (1971) largely settled the issue of admissibility of general-ized Bayes estimators of µ in the known scale case, X ∼ N p ( µ, I p ) with no S ,under quadratic loss (cid:107) d − µ (cid:107) . Studies involving admissibility and minimaxityin the known scale case largely focused on priors with the hierarchical structure (cid:90) π ) p/ g p/ exp (cid:18) − g (cid:107) µ (cid:107) (cid:19) π ( g )d g (1.9)with π ( g ) given by (1.4) for a > − p/ − b > −
1. The key results underthe prior above in the known scale case are summarized as follows and in Figure1. The estimator is inadmissible if − p/ − < a < − b > − a ≥ − b > − − < a ≤ p/ − b = 0 (Strawderman, 1971), if − p/ − < a ≤ − b = 0 (Berger, 1976) and if − p/ − < a ≤ p/ − b > π ij ( θ, η ) converging to π ∗ of the form π ij ( θ, η ) = h i ( η ) η (cid:90) η p/ (2 π ) p/ g p/ exp (cid:18) − η g (cid:107) θ (cid:107) (cid:19) π ( g ) k j ( g )d g, (1.10) . Maruyama and W. E. Strawderman/Admissibility − p/ − − − p/ − Brown (1971)Brown (1971)Strawderman (1971)Berger (1976) inadmissibleadmissibleproper Bayesminimax known scale caseunknown scale case ξ ( p, n ) n/ MS2017MS2020MS2021Current paperLin and Tsai (1973)MS2005
Fig 1 . Ranges of a for admissibility/inadmissibility and minimaxity with the hierarchical structure θ | { g, η } ∼ N p (0 , ( g/η ) I ) , g ∼ π ( g ) k j ( g ) , η ∼ h i ( η ) η . (1.11)In (1.10) and (1.11), h i ( η ) and k j ( g ) are given as follows, h i ( η ) = ii + | log η | , (1.12) k j ( g ) = 1 − log( g + 1)log( g + 1 + j ) . (1.13)Properties of h i ( η ) and k j ( g ) will be provided in Lemmas C.1 and C.2. Inparticular, we emphasize that h i ( η ) /η and π ( g ) k j ( g ) are both proper by part2 of Lemma C.1 and part 3 of Lemma C.2, respectively. Also we note thateventually, for large i and j we set i = log(1 + j ) (1.14)which is crucial in the proof of admissibility.Before proceeding with the development of the main result, we make severalremarks relating the current paper to earlier developments. Remark . Maruyama and Strawderman (2021) considered the joint improperprior given by 1 η (cid:124)(cid:123)(cid:122)(cid:125) improper × (cid:90) η p/ (2 π ) p/ g p/ exp (cid:18) − η g (cid:107) θ (cid:107) (cid:19) π ( g )d g (cid:124) (cid:123)(cid:122) (cid:125) proper (1.15) . Maruyama and W. E. Strawderman/Admissibility where π ( g ) is proper, or a > −
1, and proposed a class of admissible generalizedBayes estimators for any dimension p . Since the Stein phenomenon occurs for p ≥
3, we are interested in proposing admissible minimax estimators for p ≥ − p/ − < a ≤ ξ ( p, n )and the admissibility region a > − n ≥ p > n/ ( n − p = 3 ,
4. Theorem 1.2 provides such estimators in this case for n ≥ θ conditionalon η , since the generalized prior on g is improper in these cases. Thus, not onlyhave we extended the class of admissible estimators in all dimensions, but weprovide admissible minimax estimators for dimensions p = 3 and 4, where nosuch estimators were previously known. Remark . In Theorem 1.2, two interesting cases seem deserving of attention.When a = − b = 0, the prior corresponds to the joint Stein prior1 η × η p/ (cid:8) η (cid:107) θ (cid:107) (cid:9) − p/ = (cid:107) θ (cid:107) − p . That this estimator is admissible and minimax follows from Theorem 1.2. Addi-tionally, Kubokawa (1991) established that this estimator dominates the Jamesand Stein (1961) estimator (cid:18) − ( p − / ( n + 2) (cid:107) x (cid:107) /s (cid:19) x. Another interesting case is a variant of the James-Stein of the simple form (cid:18) − ( p − / ( n + 2) (cid:107) x (cid:107) /s + ( p − / ( n + 2) + 1 (cid:19) x, which is generalized Bayes corresponding to a = − b = n/
2, that is, π ( g ) = (cid:18) gg + 1 (cid:19) n/ . This estimator is also admissible and minimax. See Section 4.1 of Maruyamaand Strawderman (2020) and Section 3 of Maruyama and Strawderman (2021)for details.
Remark . The proof of admissibility in this paper is closely related to andgreatly influenced by those in Brown and Hwang (1982) (BH) and James andStein (1961) (JS). It is also similar to that in Maruyama and Strawderman(2021) (MS). It seems worthwhile to comment on some of the technical differ-ences. In MS the (conditional on η ) priors on θ were proper and only the prior on η was improper. Hence only a proper sequence of priors on the (inverse) scale, η ,was required. In both BH and JS the variance is assumed known (for the normalcase in BH) and hence the required sequence of proper priors is needed only on θ , whereas we need to find proper sequences on both θ , and the (inverse) scale, . Maruyama and W. E. Strawderman/Admissibility η . In this paper, since the (conditional on η ) priors on θ are also improper, thejoint improper prior is given by1 η (cid:124)(cid:123)(cid:122)(cid:125) improper × (cid:90) η p/ (2 π ) p/ g p/ exp (cid:18) − η g (cid:107) θ (cid:107) (cid:19) π ( g )d g (cid:124) (cid:123)(cid:122) (cid:125) improper (1.16)where π ( g ) is improper. As we mentioned, for the invariant prior 1 /η , we usethe sequence of proper priors, h i ( η ) /η , which was investigated in MS.Our sequence of proper priors on g , π ( g ) k j ( g ), with k j ( g ) given by (1.13),may be viewed as a modified version of that in BH and in JS. In particular thesequence in BH is k BH j ( g ) = ≤ g ≤ − log g/ log( j + 1) 1 < g < j + 10 g ≥ j + 1 . BH proposes a sufficient condition for admissibility of generalized Bayes esti-mator with two components, which they referred to as an “asymptotic flatness”condition and “growth” condition. Because our proof is similar in spirit to thatof BH, there are some connections in terms of these two conditions between BHand this paper. We focus here on the “growth” condition. In section 4, the con-dition for integrability (in order to invoke the dominated convergence theorem)in (4.6) and (4.8) sup i,j ∈ N (cid:40)(cid:90) η { h (cid:48) i ( η ) } d η (cid:90) π ( g ) k j ( g ) g + 1 d g (cid:41) < ∞ (1.17)sup i,j ∈ N (cid:26)(cid:90) h i ( η ) η d η (cid:90) ( g + 1) π ( g ) { k (cid:48) j ( g ) } d g (cid:27) < ∞ (1.18)for all i, j ∈ N may be regarded as a “growth condition”. Of the four integralsappearing in (1.17) and (1.18), only the integral (cid:90) ( g + 1) π ( g ) { k (cid:48) j ( g ) } d g < ∞ appeared in Brown and Hwang (1982). They bounded (d / d g ) k BH j ( g ) as (cid:12)(cid:12) (d / d g ) k BH j ( g ) (cid:12)(cid:12) ≤ g ≤ / ( g log 2) 1 < g ≤ / ( g log g ) g > , for all j ∈ N and concluded thatsup j ∈ N (cid:90) ( g + 1) π ( g ) { (d / d g ) k BH j ( g ) } d g < ∞ . . Maruyama and W. E. Strawderman/Admissibility However Lemmas C.1 gives (cid:90) h i ( η ) η d η = 2 i which does not imply the finiteness of (1.18).In this paper, we very carefully bound the integrals in (1.17) and (1.18) fromabove. By Lemmas C.1 and C.2 as well as π ( g ) ≤ (cid:90) η { h (cid:48) i ( η ) } d η (cid:90) π ( g ) k j ( g ) g + 1 d g < j ) i , (cid:90) h i ( η ) η d η (cid:90) ( g + 1) π ( g ) { k (cid:48) j ( g ) } d g < i log(1 + j ) . (1.19)With the choice i = log(1 + j ), the finiteness given by (1.17) and (1.18) follows.The organization of this paper is as follows. Section 2 is denoted to developingexpressions for Bayes estimators and the risk differences which are used to proveTheorem 1.1. Sections 3 and 4 are devoted to the proof of Theorem 1.1 forCASES I and II given by (1.7) and (1.8), respectively. many of the proofs oftechnical lemmas are given in Appendix.
2. The form of Bayes estimators and risk differences
In this section we develop expressions for Bayes estimators and the risk differ-ences which are used to prove Theorem 1.1. Let m ( ψ ( θ, η )) = (cid:90) (cid:90) ψ ( θ, η ) f ( x | θ, η ) f ( s | η )d θ d η. (2.1)Then, under the loss (1.2), the generalized Bayes estimator under the improper π ∗ ( θ, η ) is δ ∗ = m ( ηθπ ∗ ( θ, η )) m ( ηπ ∗ ( θ, η )) , (2.2)and the proper Bayes estimator under the proper π ij ( θ, η ) is δ ij = m ( ηθπ ij ( θ, η )) m ( ηπ ij ( θ, η )) . (2.3)The Bayes risk difference under π ij given by∆ ij = (cid:90) R p (cid:90) ∞ (cid:8) E (cid:0) η (cid:107) δ ∗ − θ (cid:107) (cid:1) − E (cid:0) η (cid:107) δ ij − θ (cid:107) (cid:1)(cid:9) π ij ( θ, η )d θ d η (2.4)may be re-expressed as∆ ij = (cid:90) R p (cid:90) ∞ (cid:107) δ ∗ − δ ij (cid:107) m ( ηπ ij ( θ, η ))d x d s. (2.5)The basic structure of the proof is standard, as in Brown and Hwang (1982),and is based on the method of Blyth (1951). The following form of Blyth’ssufficient condition shows that lim i,j →∞ ∆ ij = 0 implies admissibility. . Maruyama and W. E. Strawderman/Admissibility Lemma 2.1.
Suppose π ij ( θ, η ) is an increasing (in i and j ) sequence of properpriors, lim i,j →∞ π ij ( θ, η ) = π ( θ, η ) and π ij ( θ, η ) > for all θ and η . Then δ ∗ isadmissible if ∆ ij satisfies lim i,j →∞ ∆ ij = 0 . (2.6)Both Lemma C.1 and Lemma C.2 guarantee that π ij ( θ, η ) given by (1.10)satisfies the assumptions of Lemma 2.1 as follows. Lemma 2.2.
The prior π ij ( θ, η ) given by (1.10) is increasing in i and j and in-tegrable for all fixed i and j . Further lim i,j →∞ π ij ( θ, η ) = π ∗ ( θ, η ) and π ij ( θ, η ) > for all θ and η . Now we rewrite δ ∗ , δ ij and the integrand of (2.5). Using the identity, (cid:107) x − θ (cid:107) + (cid:107) θ (cid:107) g = g + 1 g (cid:13)(cid:13)(cid:13)(cid:13) θ − gg + 1 x (cid:13)(cid:13)(cid:13)(cid:13) + (cid:107) x (cid:107) g + 1 , we have m ( ηπ ij ) = (cid:90) (cid:90) { ηπ ij ( θ, η ) } f ( x | θ, η ) f ( s | η )d θ d η (2.7)= (cid:90) (cid:90) (cid:90) η p/ (2 π ) p/ exp (cid:18) − η (cid:107) x − θ (cid:107) (cid:19) f ( s | η ) × η p/ (2 π ) p/ g p/ exp (cid:18) − η g (cid:107) θ (cid:107) (cid:19) h i ( η ) π ( g ) k j ( g )d θ d g d η = (cid:90) (cid:90) η p/ f ( s | η )(2 π ) p/ ( g + 1) p/ exp (cid:18) − η (cid:107) x (cid:107) g + 1) (cid:19) h i ( η ) π ( g ) k j ( g )d g d η = q ( p, n ) s n/ − (cid:90) (cid:90) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η, where w = (cid:107) x (cid:107) /s , F ( g, η ; w, s ) = η p/ n/ ( g + 1) p/ exp (cid:26) − ηs (cid:18) wg + 1 + 1 (cid:19)(cid:27) , (2.8)and q ( p, n ) = 1(2 π ) p/ Γ( n/ n/ . Similarly we have m ( ηθπ ij ) = (cid:90) (cid:90) { ηθπ ij ( θ, η ) } f ( x | θ, η ) f ( s | η )d θ d η = q ( p, n ) s n/ − (cid:90) (cid:90) gxg + 1 F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η. (2.9)By (2.7) and (2.9), the Bayes estimator under π ij is δ ij = m ( θηπ ij ) m ( ηπ ij ) = (cid:18) − φ ij ( w, s ) w (cid:19) x, (2.10) . Maruyama and W. E. Strawderman/Admissibility where φ ij ( w, s ) = w (cid:82)(cid:82) ( g + 1) − F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η (cid:82)(cid:82) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η . (2.11)With h i ≡ k j ≡ φ ∗ ( w, s ) = w (cid:82)(cid:82) ( g + 1) − F ( g, η ; w, s ) π ( g )d g d η (cid:82)(cid:82) F ( g, η ; w, s ) π ( g )d g d η (2.12)and our target generalized Bayes estimator given by δ ∗ = (cid:18) − φ ∗ ( w, s ) w (cid:19) x. (2.13)Note that (cid:90) F ( g, η ; w, s )d η = Γ( p/ n/ g + 1) p/ (cid:18) s { w/ ( g + 1) } (cid:19) p/ n/ which implies φ ∗ ( w, s ) w = (cid:82) ∞ ( g + 1) − p/ − { w/ ( g + 1) } − p/ − n/ − π ( g )d g (cid:82) ∞ ( g + 1) − p/ { w/ ( g + 1) } − p/ − n/ − π ( g )d g . (2.14)In the following, however, we keep (2.12) not (2.14) as the expression of φ ∗ ( w, s ).By (2.7), (2.10) and (2.13), (cid:107) δ ∗ − δ ij (cid:107) m ( ηπ ij ), in ∆ ij , is (cid:107) δ ∗ − δ ij (cid:107) m ( ηπ ij ) = (cid:107) x (cid:107) (cid:18) φ ∗ ( w, s ) w − φ ij ( w, s ) w (cid:19) m ( ηπ ij )= q ( p, n ) (cid:107) x (cid:107) s n/ − A ( π ; i, j ) , (2.15)where A ( π ; i, j ) (2.16)= (cid:32) (cid:82)(cid:82) ( g + 1) − F π d g d η (cid:82)(cid:82) F π d g d η − (cid:82)(cid:82) ( g + 1) − F h i πk j d g d η (cid:82)(cid:82) F h i πk j d g d η (cid:33) (cid:90) (cid:90) F h i πk j d g d η. In Sections 3 and 4 respectively, we will complete the proof of Theorem 1.1,for CASE I and CASE II by the dominated convergence theorem. We do so byshowing the integrand in ∆ ij = q ( p, n ) (cid:82)(cid:82) (cid:107) x (cid:107) s n/ − A ( π ; i, j )d x d s is boundedby an integrable function. It follows, since the integrand approaches 0, that∆ ij → . Maruyama and W. E. Strawderman/Admissibility
3. CASE I
This section is devoted to the proof of Theorem 1.1 for CASE I given by (1.7).Applying the inequality (cid:32) k (cid:88) i =1 a i (cid:33) ≤ k k (cid:88) i =1 a i , (3.1)to (2.16), we have A ( π ; i, j ) (3.2) ≤ (cid:40)(cid:18) (cid:82)(cid:82) ψ ( g ) F π d g d η (cid:82)(cid:82) F π d g d η − (cid:82)(cid:82) ψ ( g ) F h i π d g d η (cid:82)(cid:82) F h i π d g d η (cid:19) (cid:90) (cid:90) F h i πk j d g d η (cid:32) (cid:82)(cid:82) ψ ( g ) F h i π d g d η (cid:82)(cid:82) F h i π d g d η − (cid:82)(cid:82) ψ ( g ) F h i πk j d g d η (cid:82)(cid:82) F h i π d g d η (cid:33) (cid:90) (cid:90) F h i πk j d g d η (cid:32) (cid:82)(cid:82) ψ ( g ) F h i πk j d g d η (cid:82)(cid:82) F h i π d g d η − (cid:82)(cid:82) ψ ( g ) F h i πk j d g d η (cid:82)(cid:82) F h i πk j d g d η (cid:33) (cid:90) (cid:90) F h i πk j d g d η ≤ {A ( ψ ; π ; i, j ) + A ( ψ ; π ; i, j ) + A ( ψ ; π ; i, j ) } where ψ ( g ) = 1 / ( g + 1) and A ( ψ ) = (cid:26)(cid:90) (cid:90) ψ ( g ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:82)(cid:82) F π d g d η − h i (cid:82)(cid:82) F h i π d g d η (cid:12)(cid:12)(cid:12)(cid:12) F π d g d η (cid:27) (cid:90) (cid:90) F h i π d g d η, A ( ψ ) = (cid:18)(cid:90) (cid:90) ψ ( g ) F h i π (1 − k j )d g d η (cid:19) (cid:82)(cid:82) F h i π d g d η , A ( ψ ) = (cid:0)(cid:82)(cid:82) ψ ( g ) F h i πk j d g d η (cid:1) ( (cid:82)(cid:82) F h i π d g d η ) (cid:82)(cid:82) F h i πk j d g d η (cid:18)(cid:90) (cid:90) F h i π d g d η − (cid:90) (cid:90) F h i πk j d g d η (cid:19) . Since A i for i = 1 , , ψ will also appear in Section 4 (more preciselyin Lemma 4.2), we summarize useful properties in the following Lemma. Lemma 3.1.
Let ψ ( g ) = 1( g + 1) α for α = 1 or . (3.3) Let m satisfy a + m + 2 α > . Also let (cid:15) satisfy < (cid:15) <
12 min ( p/ a + 1 , a + m + 2 α, . Then there exist positive constants, ˜ A , ˜ A , and ˜ A , independent of i and j , . Maruyama and W. E. Strawderman/Admissibility such that (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m A ( ψ ; π ; i, j )d x d s ≤ ˜ A B ( a + m + 2 α, b + 1) , (3.4) (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m A (cid:96) ( ψ ; π ; i, j )d x d s ≤ i { log(1 + j ) } ˜ A (cid:96) (cid:15) B ( a + m + 2 α − (cid:15), b + 1) for (cid:96) = 2 , . (3.5) Proof.
See Appendix A.We set m = 0 and α = 1 in Lemma 3.1. With the choice (cid:15) = 14 min ( p/ a + 1 , a + 2 , , i = O ( { log(1 + j ) } ) , (3.6)we have sup j ∈ N i = O ( { log(1+ j ) } ) 3 (cid:88) (cid:96) =1 (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − A (cid:96) (1 / ( g + 1); π ; i, j )d x d s < ∞ . (3.7)By the dominated convergence theorem, we have a following result. Theorem 3.1.
For p ≥ , the generalized Bayes estimator under π ∗ with max( − p/ − , − < a ≤ − and b > − is admissible among the class of all estimators.
4. CASE II
This section is devoted to the proof of Theorem 1.1 for CASE II given by (1.8), a = − b ≥ p ≥
3. We need the condition b ≥ π (0) < ∞ , which isrequired in the following lemma. Lemma 4.1.
Assume a = − and b ≥ . Then ( n/ φ ij ( w, s ) (4.1)= ( p/ − − w (cid:82)(cid:82) ( g + 1) − ηF ( g, η ; w, s ) h i ( η ) h (cid:48) i ( η ) π ( g ) k j ( g )d g d η (cid:82)(cid:82) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η − (cid:82)(cid:82) ( g + 1 + w ) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g ) k (cid:48) j ( g )d g d η (cid:82)(cid:82) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η − ϕ ij ( w, s ) , where ϕ ij ( w, s ) = b (cid:82)(cid:82) { ( g + 1 + w ) /g ( g + 1) } F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η (cid:82)(cid:82) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η b > w ) (cid:82) F (0 , η ; w, s ) h i ( η )d η (cid:82)(cid:82) F ( g, η ; w, s ) h i ( η ) k j ( g )d g d η b = 0 . . Maruyama and W. E. Strawderman/Admissibility Proof.
See Appendix C.2.With h i ≡ k j ≡ n/ φ ∗ ( w, s ) = ( p/ − − ϕ ∗ ( w, s ) , (4.2)where ϕ ∗ ( w, s ) = b (cid:82)(cid:82) { ( g + 1 + w ) /g ( g + 1) } F ( g, η ; w, s ) π ( g )d g d η (cid:82)(cid:82) F ( g, η ; w, s ) π ( g )d g d η b > w ) (cid:82) F (0 , η ; w, s )d η (cid:82)(cid:82) F ( g, η ; w, s )d g d η b = 0 . (4.3)By (2.15), (4.1) and (4.2), (cid:107) δ ∗ − δ ij (cid:107) m ( ηπ ij ), in ∆ ij , is rewritten as (cid:107) δ ∗ − δ ij (cid:107) m ( ηπ ij ) = (cid:107) x (cid:107) (cid:18) φ ∗ ( w, s ) w − φ ij ( w, s ) w (cid:19) m ( ηπ ij ) (4.4)= q ( p, n ) (cid:107) x (cid:107) s n/ − ( n/ w (cid:26) ϕ ∗ ( w, s ) − ϕ ij ( w, s ) − w (cid:82)(cid:82) ( g + 1) − ηF ( g, η ; w, s ) h i ( η ) h (cid:48) i ( η ) π ( g ) k j ( g )d g d η (cid:82)(cid:82) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η − (cid:82)(cid:82) ( g + 1 + w ) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g ) k (cid:48) j ( g )d g d η (cid:82)(cid:82) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η (cid:27) × (cid:90) (cid:90) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η. Applying the inequality (3.1) to (4.4), we have (cid:107) δ ∗ − δ ij (cid:107) m ( ηπ ij ) ≤ q ( p, n )( n/ (cid:107) x (cid:107) s n/ − { B ( π ; i, j ) + 4 C ( π ; i, j ) + D ( π ; i, j ) } , (4.5)where B ( π ; i, j ) = { (cid:82)(cid:82) ( g + 1) − ηF ( g, η ; w, s ) h i ( η ) h (cid:48) i ( η ) π ( g ) k j ( g )d g d η } (cid:82)(cid:82) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η , C ( π ; i, j ) = { (cid:82)(cid:82) ( g + 1 + w ) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g ) k (cid:48) j ( g )d g d η } w (cid:82)(cid:82) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η , D ( π ; i, j ) = { ϕ ∗ ( w, s ) − ϕ ij ( w, s ) } w (cid:90) (cid:90) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η. For B , by Cauchy-Schwarz inequality, we have B ( π ; i, j ) ≤ (cid:90) (cid:90) η ( g + 1) F ( g, η ; w, s ) { h (cid:48) i ( η ) } π ( g ) k j ( g )d g d η. . Maruyama and W. E. Strawderman/Admissibility By Lemma C.4, the integral for B is (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − B ( π ; i, j )d x d s ≤ q (0) (cid:90) η { h (cid:48) i ( η ) } d η (cid:90) π ( g ) k j ( g ) g + 1 d g, (4.6)where q ( m ) = 2 p/ n/ π p/ Γ( p/ − m )Γ( n/ m )Γ( p/ . (4.7)For C , again by Cauchy-Schwarz inequality, we have C ( π ; i, j ) ≤ w (cid:90) ( g + 1 + w ) F ( g, η ; w, s ) h i ( η ) π ( g ) { k (cid:48) j ( g ) } d g d η ≤ (cid:90) (cid:18) ( g + 1) w + 1 (cid:19) F ( g, η ; w, s ) h i ( η ) π ( g ) { k (cid:48) j ( g ) } d g d η. By Lemma C.4, the integral for C is (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − C ( π ; i, j )d x d s ≤ { q (0) + q (2) } (cid:90) h i ( η ) η d η (cid:90) ( g + 1) π ( g ) { k (cid:48) j ( g ) } d g. (4.8)By Lemmas C.1 and C.2 as well as π ( g ) ≤ (cid:90) η { h (cid:48) i ( η ) } d η (cid:90) π ( g ) k j ( g ) g + 1 d g < j ) i , (cid:90) h i ( η ) η d η (cid:90) ( g + 1) π ( g ) { k (cid:48) j ( g ) } d g < i log(1 + j ) . (4.9)For D , using Lemma 3.1, we have a following lemma. Lemma 4.2.
There exist positive constants ˜ D and ˜ D all independent of i and j such that (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − D ( π ; i, j )d x d s ≤ ˜ D + i { log(1 + j ) } ˜ D . (4.10) Proof.
See Appendix B.By (4.6), (4.8), (4.9) and Lemma 4.2, with the choice i = log(1 + j )we havesup j ∈ N i =log(1+ j ) (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − { B ( π ; i, j ) + 4 C ( π ; i, j ) + D ( π ; i, j ) } d x d s < ∞ (4.11)and by the dominated convergence theorem, we have a following result. . Maruyama and W. E. Strawderman/Admissibility Theorem 4.1.
For p ≥ , the generalized Bayes estimator under π ∗ with a = − and b ≥ is admissible among the class of all estimators. Appendix A: Proof of Lemma 3.1
A.1. Proof for A Recall A ( ψ ) = (cid:26)(cid:90) (cid:90) ψ ( g ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:82)(cid:82) F π d g d η − h i (cid:82)(cid:82) F h i π d g d η (cid:12)(cid:12)(cid:12)(cid:12) F π d g d η (cid:27) (cid:90) (cid:90) F h i π d g d η. By Cauchy-Schwarz inequality, we have A ( ψ ) ≤ (cid:90) (cid:90) ψ ( g ) F π d g d η (cid:90) (cid:90) (cid:18) (cid:82)(cid:82) F π d g d η − h i (cid:82)(cid:82) F h i π d g d η (cid:19) F π d g d η × (cid:90) (cid:90) F h i π d g d η. (A.1)Note (cid:18) (cid:82)(cid:82) F π d g d η − h i (cid:82)(cid:82) F h i π d g d η (cid:19) (A.2)= (cid:113)(cid:82)(cid:82) F π d g d η − h i (cid:113)(cid:82)(cid:82) F h i π d g d η (cid:113)(cid:82)(cid:82) F π d g d η + h i (cid:113)(cid:82)(cid:82) F h i π d g d η = 1 (cid:82)(cid:82) F π d g d η (cid:82)(cid:82) F h i π d g d η − h i (cid:113)(cid:82)(cid:82) F π d g d η (cid:113)(cid:82)(cid:82) F h i π d g d η (cid:113)(cid:82)(cid:82) F h i π d g d η (cid:113)(cid:82)(cid:82) F π d g d η + h i ≤ (cid:82)(cid:82) F π d g d η (cid:82)(cid:82) F h i π d g d η − h i (cid:113)(cid:82)(cid:82) F π d g d η (cid:113)(cid:82)(cid:82) F h i π d g d η , where the inequality follows from the fact 0 ≤ h i ≤ . Maruyama and W. E. Strawderman/Admissibility Further we have (cid:90) (cid:90)
F π − h i (cid:113)(cid:82)(cid:82) F π d g d η (cid:113)(cid:82)(cid:82) F h i π d g d η d g d η = 2 (cid:90) (cid:90) F π d g d η − (cid:113)(cid:82)(cid:82) F π d g d η (cid:113)(cid:82)(cid:82) F h i π d g d η (cid:90) (cid:90) F h i π d g d η = 2 (cid:90) (cid:90) F π d g d η (cid:32) − (cid:115) ( (cid:82)(cid:82) F h i π d g d η ) (cid:82)(cid:82) F π d g d η (cid:82)(cid:82) F h i π d g d η (cid:33) ≤ (cid:90) (cid:90) F π d g d η (cid:18) − ( (cid:82)(cid:82) F h i π d g d η ) (cid:82)(cid:82) F π d g d η (cid:82)(cid:82) F h i π d g d η (cid:19) , (A.3)where the inequality follows from the fact( (cid:82)(cid:82) F h i π d g d η ) (cid:82)(cid:82) F π d g d η (cid:82)(cid:82) F h i π d g d η ∈ (0 , , which is shown by Cauchy-Schwarz inequality. By (A.1), (A.2) and (A.3), wehave A ( ψ ) ≤ (cid:90) (cid:90) F ψ ( g ) π ( g )d g d η (cid:18) − ( (cid:82)(cid:82) F h i π d g d η ) (cid:82)(cid:82) F π d g d η (cid:82)(cid:82) F h i π d g d η (cid:19) . (A.4)By Lemma C.7, we have (cid:90) (cid:90) F h li π d g d η = (1 − z ) p/ a +1 s p/ n/ × (cid:90) (cid:90) t p/ a (1 − t ) b (1 − zt ) p/ a + b +2 v p/ n/ exp (cid:18) − v − zt ) (cid:19) h li ( v/s )d t d v, where l = 0 , ,
2. For (A.4), by Lemma 2.8 of Maruyama and Strawderman(2021), there exists a positive constant q ( a, b ), independent of i , x and s , suchthat 1 − ( (cid:82)(cid:82) F h i π d g d η ) (cid:82)(cid:82) F π d g d η (cid:82)(cid:82) F h i π d g d η ≤ q ( a, b )(1 + | log s | ) . (A.5)By (A.4), (A.5) and Lemma C.5, we have (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m A ( ψ ; π ; i, j )d x d s ≤ q ( a, b ) (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ψ ( g ) F π ( (cid:107) x (cid:107) /s ) m (1 + | log s | ) d g d η ≤ q ( m ) q ( a, b ) B ( a + m + 2 α, b + 1) , where q ( m ) is given by (4.7). This completes the proof. . Maruyama and W. E. Strawderman/Admissibility A.2. Proof for A Recall A ( ψ ) = (cid:18)(cid:90) (cid:90) ψ ( g ) F h i π (1 − k j )d g d η (cid:19) (cid:82)(cid:82) F h i π d g d η . By Cauchy-Schwarz inequality, we have A ( ψ ) ≤ (cid:90) (cid:90) ψ (1 − k j ) F h i π d g d η. (A.6)Note 1 − k j = (1 + k j )(1 − k j ) ≤ − k j ) = 2 log( g + 1)log( g + 1 + j ) , log( g + 1) ≤ ( g + 1) (cid:15) (cid:15) , for (cid:15) > , (A.7)which gives 1 − k j ≤ g + 1)log(1 + j ) ≤ g ) (cid:15) (cid:15) log(1 + j ) . (A.8)By (A.6) and (A.8), A ( ψ ) ≤ (cid:15) { log(1 + j ) } (cid:90) (cid:90) F h i ψ ( g + 1) (cid:15) π d g d η, where the integral in the right-hand side is integrable when p/ a + 1 + 2 α − (cid:15) = ( p/ a + 1 − (cid:15) ) + 2 α > A isbounded as follows: (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m A ( ψ ; π ; i, j )d x d s ≤ i { log(1 + j ) } q ( m ) (cid:15) (cid:90) ( g + 1) − m +2 (cid:15) +1 ψ ( g ) π ( g )d g = i { log(1 + j ) } q ( m ) (cid:15) B ( a + m + 2 α − (cid:15), b + 1) , which completes the proof. . Maruyama and W. E. Strawderman/Admissibility A.3. Proof for A Recall A ( ψ ) = (cid:0)(cid:82)(cid:82) ψ ( g ) F h i πk j d g d η (cid:1) ( (cid:82)(cid:82) F h i π d g d η ) (cid:82)(cid:82) F h i πk j d g d η (cid:18)(cid:90) (cid:90) F h i π d g d η − (cid:90) (cid:90) F h i πk j d g d η (cid:19) . By Cauchy-Schwarz inequality, we have (cid:18)(cid:90) (cid:90) ψ ( g ) F h i πk j d g d η (cid:19) ≤ (cid:90) (cid:90) F h i πk j d g d η (cid:90) (cid:90) ψ ( g ) F h i πk j d g d η ≤ (cid:90) (cid:90) F h i πk j d g d η (cid:90) (cid:90) ψ ( g ) F h i π d g d η, (A.9)where the second inequality follows from k j ≤
1. Hence we have (cid:18)(cid:90) (cid:90)
F h i π d g d η − (cid:90) (cid:90) F h i πk j d g d η (cid:19) ≤ (cid:90) (cid:90) F h i π d g d η (cid:90) (cid:90) (1 − k j ) F h i π d g d η ≤ (cid:15) { log(1 + j ) } (cid:90) (cid:90) F h i π d g d η (cid:90) (cid:90) ( g + 1) (cid:15) F h i π d g d η, where the second inequality follows from (A.8). Note the integral (cid:82) ( g +1) (cid:15) F π d g is integrable when p/ a + 1 − (cid:15) > q such that (cid:82)(cid:82) ( g + 1) (cid:15) F h i π d g d η (cid:82)(cid:82) F h i π d g d η ≤ q ( w + 1) (cid:15) (A.10)which implies that A ( ψ ) ≤ q ( w + 1) (cid:15) (cid:15) { log(1 + j ) } (cid:90) (cid:90) ψ ( g ) F h i πk j d g d η. (A.11)Note, for 0 < (cid:15) <
1, we have( w + 1) (cid:15) ≤ w (cid:15) + 1 . . Maruyama and W. E. Strawderman/Admissibility Then, by Lemma C.4 as well as Part 2 of Lemma C.1, we have (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m A ( ψ ; π ; i, j )d x d s ≤ i { log(1 + j ) } q (cid:15) (cid:18) q ( m − (cid:15) ) (cid:90) ( g + 1) − m +2 (cid:15) +1 ψ ( g ) π ( g )d g + q ( m ) (cid:90) ( g + 1) − m +1 ψ ( g ) π ( g )d g (cid:19) ≤ i { log(1 + j ) } q { q ( m − (cid:15) ) + q ( m ) } (cid:15) B ( a + m + 2 α − (cid:15), b + 1) , which completes the proof. Appendix B: Proof of Lemma 4.2
B.1. D for b = 0 Recall π ( g ) ≡ a = − b = 0 and D ( π ; i, j ) = (cid:18) w + 1 w (cid:19) (cid:32) (cid:82) F (0 , η ; w, s )d η (cid:82)(cid:82) F ( g, η ; w, s )d g d η − (cid:82) F (0 , η ; w, s ) h i d η (cid:82)(cid:82) F ( g, η ; w, s ) h i k j d g d η (cid:33) × (cid:90) (cid:90) F ( g, η ; w, s ) h i k j d g d η. By (3.1), we have D ( π ; i, j ) (B.1) ≤ (cid:0) /w (cid:1) (cid:40)(cid:18) (cid:82) F (0 , η ; w, s )d η (cid:82)(cid:82) F ( g, η ; w, s )d g d η − (cid:82) F (0 , η ; w, s ) h i d η (cid:82)(cid:82) F ( g, η ; w, s ) h i d g d η (cid:19) + (cid:32) (cid:82) F (0 , η ; w, s ) h i d η (cid:82)(cid:82) F ( g, η ; w, s ) h i d g d η − (cid:82) F (0 , η ; w, s ) h i d η (cid:82)(cid:82) F ( g, η ; w, s ) h i k j d g d η (cid:33) × (cid:90) (cid:90) F ( g, η ; w, s ) h i k j d g d η. In (B.1), we have (cid:18) (cid:82) F (0 , η ; w, s )d η (cid:82)(cid:82) F ( g, η ; w, s )d g d η − (cid:82) F (0 , η ; w, s ) h i d η (cid:82)(cid:82) F ( g, η ; w, s ) h i d g d η (cid:19) (B.2) ≤ (cid:18)(cid:90) F (0 , η ; w, s ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:82)(cid:82) F d g d η − h i (cid:82)(cid:82) F h i d g d η (cid:12)(cid:12)(cid:12)(cid:12) d η (cid:19) . Maruyama and W. E. Strawderman/Admissibility and (cid:32) (cid:82) F (0 , η ; w, s ) h i d η (cid:82)(cid:82) F ( g, η ; w, s ) h i d g d η − (cid:82) F (0 , η ; w, s ) h i d η (cid:82)(cid:82) F ( g, η ; w, s ) h i k j d g d η (cid:33) (B.3)= (cid:32) (cid:82) F (0 , η ; w, s ) h i d η (cid:82)(cid:82) F ( g, η ; w, s ) h i (1 − k j )d g d η (cid:82)(cid:82) F ( g, η ; w, s ) h i d g d η (cid:82)(cid:82) F ( g, η ; w, s ) h i k j d g d η (cid:33) . Applying the following inequality to (B.2), F (0 , η ; w, s ) = η p/ n/ exp (cid:16) − ηs w + 1) (cid:17) (B.4)= η p/ n/ exp (cid:16) − ηs w + 1) (cid:17) (cid:82) ( g + 1) − p/ − d g (cid:82) ( g + 1) − p/ − d g ≤ η p/ n/ exp( − ηs ( w + 1) / (cid:82) ( g + 1) − p/ − d g (cid:90) g + 1) p/ exp (cid:18) ηs w gg + 1 (cid:19) d g = (cid:82) ( g + 1) − F ( g, η ; w, s )d g (cid:82) ( g + 1) − p/ − d g . and the following inequality to (B.3), F (0 , η ; w, s ) ≤ (cid:82) ( g + 1) − F ( g, η ; w, s ) π ( g ) k j ( g )d g (cid:82) ( g + 1) − p/ − k j ( g )d g , (B.5)we have D ( π ; i, j ) ≤ /w ) { (cid:82) ( g + 1) − p/ − k ( g )d g } (cid:8) A (( g + 1) − ) + A (( g + 1) − ) (cid:9) . In Lemma 3.1, we set α = 2, m = 0 or 2, (cid:15) = 1 / a = − b = 0.There exist positive constants ˜ D and ˜ D , both independent of i and j , suchthat (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − D ( π ; i, j )d x d s ≤ ˜ D + i { log(1 + j ) } ˜ D . B.2. D for b > When b >
0, we have D ( π ; i, j )= b w (cid:82)(cid:82) (cid:18) g + wg ( g + 1) (cid:19) F π d g d η (cid:82)(cid:82) F π d g d η − (cid:82)(cid:82) (cid:18) g + wg ( g + 1) (cid:19) F h i πk j d g d η (cid:82)(cid:82) F h i πk j d g d η × (cid:90) (cid:90) F h i πk j d g d η. . Maruyama and W. E. Strawderman/Admissibility By (3.1) and Lemma B.1 below, we have D ( π ; i, j ) ≤ b (cid:26) A (1 /g ) w + A (1 / { g ( g + 1) } ) + A (1 /g ) w + A (1 / { g ( g + 1) } ) + A (1 /g ) w + A (1 / { g ( g + 1) } ) (cid:27) ≤ b r ∗ (cid:26) A (1 / ( g + 1)) w + A (1 / ( g + 1) ) + A (1 / ( g + 1)) w + A (1 / ( g + 1) ) + A (1 / ( g + 1)) w + A (1 / ( g + 1) ) (cid:27) , where r ∗ = max( r , r , r ) defined in Lemma B.1 below.In Lemma 3.1, we set α = 2, m = 0 or 2, (cid:15) = 1 / a = − b >
0. Then there exist positive constants ˜ D and ˜ D both independent of i and j such that (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − D ( π ; i, j )d x d s ≤ ˜ D + i { log(1 + j ) } ˜ D . Lemma B.1. (cid:90)
F π d gg ≤ r (cid:90) F π d gg + 1 , (cid:90) F π d gg ( g + 1) ≤ r (cid:90) F π d g ( g + 1) , (cid:90) F π (1 − k j )d gg ≤ r (cid:90) F π (1 − k j )d gg + 1 , (cid:90) F π (1 − k j )d gg ( g + 1) ≤ r (cid:90) F π (1 − k j )d g ( g + 1) , (cid:90) F πk j d gg ≤ r (cid:90) F πk j d gg + 1 , (cid:90) F πk j d gg ( g + 1) ≤ r (cid:90) F πk j d g ( g + 1) , where r = (cid:82) { ( g + 1) /g } π ( g )( g + 1) − p/ − d g (cid:82) π ( g )( g + 1) − p/ − d g ,r = max j (cid:82) { ( g + 1) /g } π ( g ) { − k j ( g ) } ( g + 1) − p/ − d g (cid:82) π ( g ) { − k j ( g ) } ( g + 1) − p/ − d g ,r = max j (cid:82) { ( g + 1) /g } π ( g ) k j ( g )( g + 1) − p/ − d g (cid:82) π ( g ) k j ( g )( g + 1) − p/ − d g . Proof.
We only prove two inequalities in the first line. By the covariance in-equality, we have (cid:90) g π ( g )( g + 1) p/ exp (cid:18) − ηsg + 1 w (cid:19) d g = (cid:90) g + 1 g π ( g )( g + 1) p/ exp (cid:18) − ηsg + 1 w (cid:19) d g ≤ (cid:82) { ( g + 1) /g } π ( g )( g + 1) − p/ − d g (cid:82) π ( g )( g + 1) − p/ − d g (cid:90) g + 1 π ( g )( g + 1) p/ exp (cid:18) − ηsg + 1 w (cid:19) d g. . Maruyama and W. E. Strawderman/Admissibility where the inequality follows from the fact ( g +1) /g is decreasing and exp( − ηsw/ { g +1) } ) is increasing. Similarly we have (cid:90) g ( g + 1) π ( g )( g + 1) p/ exp (cid:18) − ηsg + 1 w (cid:19) d g ≤ (cid:82) { ( g + 1) /g } π ( g )( g + 1) − p/ − d g (cid:82) π ( g )( g + 1) − p/ − d g (cid:90) g + 1) π ( g )( g + 1) p/ exp (cid:18) − ηsg + 1 w (cid:19) d g. Since ( g + 1) /g is decreasing and g + 1 is increasing, (cid:82) { ( g + 1) /g } π ( g )( g + 1) − p/ − d g (cid:82) π ( g )( g + 1) − p/ − d g = (cid:82) { ( g + 1) /g } ( g + 1) π ( g )( g + 1) − p/ − d g (cid:82) ( g + 1) π ( g )( g + 1) − p/ − d g< (cid:82) { ( g + 1) /g } π ( g )( g + 1) − p/ − d g (cid:82) π ( g )( g + 1) − p/ − d g = r , which completes the proof of two inequalities in the first line. Appendix C: Lemmas and Proofs
C.1. the sequence
Lemma C.1.
Let h i ( η ) = ii + | log η | . (C.1) h i ( η ) is increasing in i and lim i →∞ h i ( η ) = 1 for all η > . (cid:90) ∞ η − h i ( η )d η = 2 i .3. (cid:90) ∞ η { h (cid:48) i ( η ) } d η ≤ i .Proof. [Part 1] This part is straightforward given the form of h i ( η ).[Part 2] The results follow from the integrals, (cid:90) ∞ η − h i ( η )d η = (cid:90) i d ηη { i + log(1 /η ) } + (cid:90) ∞ i d ηη { i + log η } = (cid:20) i i + log(1 /η ) (cid:21) + (cid:20) − i i + log η (cid:21) ∞ = 2 i. (C.2)[Part 3] Note h (cid:48) i ( η ) = iη { i + log(1 /η ) } < η < , − iη { i + log η } η ≥ . . Maruyama and W. E. Strawderman/Admissibility Hence { h (cid:48) i ( η ) } = i η { i + log(1 /η ) } < η < − i η { i + log η } η ≥ ≤ η { i + log(1 /η ) } < η < , η { i + log η } η ≥ . Then (cid:90) ∞ η { h (cid:48) i ( η ) } d η ≤ (cid:90) d ηη { i + log(1 /η ) } + (cid:90) ∞ d ηη { i + log η } = 2 i . Lemma C.2.
Let k j ( g ) = 1 − log( g + 1)log( g + 1 + j ) . (C.3) k j ( g ) is increasing in j for fixed g , and decreasing in g for fixed j . Further lim j →∞ k j ( g ) = 1 for fixed g ≥ .2. For fixed j ≥ , k j ( g ) ≤ (1 + j ) log(1 + j )( g + 1 + j ) log( g + 1 + j ) .
3. Let π ( g ) = ( g + 1) − a − { g/ ( g + 1) } b for a ≥ − and b > − . Then (cid:90) ∞ π ( g ) k j ( g )d g ≤ b + 1 + max(1 , − b )(1 + j ) . (cid:90) ∞ k j ( g ) g + 1 d g ≤ j ) .
5. For g ≥ , k (cid:48) j ( g ) = − j/ ( g + 1) + k j ( g )( g + 1 + j ) log( g + 1 + j ) (cid:90) ∞ ( g + 1) { k (cid:48) j ( g ) } d g ≤ j ) . . Maruyama and W. E. Strawderman/Admissibility Proof. [Part 1] This part is straightforward given the form of k j ( g ).[Part 2] The function k j ( g ) is rewritten as k j ( g ) = jζ ( j/ ( g + 1 + j ))( g + 1 + j ) log( g + 1 + j ) , where ζ ( x ) = − log(1 − x ) /x = 1 + (cid:80) ∞ l =1 x l / ( l + 1) which is increasing in x .Hence k j ( g ) ≤ jζ ( j/ (1 + j ))( g + 1 + j ) log( g + 1 + j ) = (1 + j ) log(1 + j )( g + 1 + j ) log( g + 1 + j ) . [Part 3] By Part 2, (cid:90) ∞ k j ( g )d g ≤ (cid:90) ∞ (1 + j ) { log(1 + j ) } ( g + 1 + j ) { log( g + 1 + j ) } d g ≤ (cid:90) ∞ (1 + j ) ( g + 1 + j ) d g = 1 + j. Under the condition, we have π ( g ) ≤ { g/ ( g +1) } b . When b ≥
0, we have π ( g ) ≤ (cid:90) ∞ π ( g ) k j ( g )d g ≤ (cid:90) ∞ k j ( g )d g ≤ j. When − < b < (cid:18) gg + 1 (cid:19) b ≤ (cid:0) g b (cid:1) I (0 , ( g ) + 2 − b I (1 , ∞ ) ( g ) ≤ g b I (0 , ( g ) + 2 − b I (0 , ∞ ) ( g ) . Note (cid:90) π ( g ) k j ( g )d g ≤ (cid:90) g b d g = 1 b + 1 . Then the result follows.[Part 4] Note k j ≤ (cid:90) j k j ( g ) g + 1 d g ≤ (cid:90) j g + 1 d g = log(1 + j ) . Also, by Part 2, (cid:90) ∞ j k j ( g )( g + 1) d g ≤ (cid:90) ∞ j (1 + j ) { log(1 + j ) } ( g + 1)( g + 1 + j ) { log( g + 1 + j ) } d g ≤ (cid:90) ∞ j { log(1 + j ) } ( g + 1 + j ) { log( g + 1 + j ) } d g = { log(1 + j ) } log(1 + 2 j ) ≤ log(1 + j ) . . Maruyama and W. E. Strawderman/Admissibility Then the result follows.[Part 5] The derivative is k (cid:48) j ( g ) = − g + 1) log( g + 1 + j ) + log( g + 1)( g + 1 + j ) { log( g + 1 + j ) } . (C.4)Then log( g + 1 + j ) k (cid:48) j ( g ) = − g + 1 + log( g + 1)( g + 1 + j ) log( g + 1 + j )= − g + 1 + 1 g + 1 + j { − k j ( g ) } = − j ( g + 1)( g + 1 + j ) − k j ( g ) g + 1 + j . [Part 6] By Part 5, we have( g + 1) { k (cid:48) j ( g ) } ≤ (cid:32) j ( g + 1)( g + 1 + j ) { log( g + 1 + j ) } + ( g + 1) k j ( g )( g + 1 + j ) { log( g + 1 + j ) } (cid:33) . Then we have (cid:90) j j d g ( g + 1)( g + 1 + j ) { log( g + 1 + j ) } ≤ { log(1 + j ) } (cid:90) j d gg + 1= 1log(1 + j )and (cid:90) ∞ j j d g ( g + 1)( g + 1 + j ) { log( g + 1 + j ) } ≤ (cid:90) ∞ j d g ( g + 1) { log( g + 1) } = 14 log(1 + j ) . Further, by 0 ≤ k j ≤ (cid:90) ∞ ( g + 1) k j ( g ) d g ( g + 1 + j ) { log( g + 1 + j ) } ≤ (cid:90) ∞ d g ( g + 1 + j ) { log( g + 1 + j ) } = 1log(1 + j ) . Then the result follows.
C.2. Proof of Lemma 4.1
Lemma 4.1 follows from the following result. . Maruyama and W. E. Strawderman/Admissibility Lemma C.3.
Assume a = − and b ≥ . Then ( n/ w (cid:90) (cid:90) ( g + 1) − F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η = ( p/ − (cid:90) (cid:90) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η − w (cid:90) (cid:90) ηg + 1 F ( g, η ; w, s ) h i ( η ) h (cid:48) i ( η ) π ( g ) k j ( g )d g d η − (cid:90) (cid:90) ( g + 1 + w ) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g ) k (cid:48) j ( g )d g d η − b (cid:90) (cid:90) g + 1 + wg ( g + 1) F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η b > , (cid:90) (1 + w ) F (0 , η ; w, s ) h i ( η )d η b = 0 . Proof.
By change of variables u = η { w/ ( g +1) } with d η/ d u = 1 / { w/ ( g +1) } , we have (cid:90) (cid:90) ( g + 1) − F ( g, η ; w, s ) h i ( η ) π ( g ) k j ( g )d g d η = (cid:90) (cid:90) (cid:18) wg + 1 (cid:19) − p/ − n/ − u p/ n/ ( g + 1) p/ exp (cid:16) − su (cid:17) × h i (cid:18) u w/ ( g + 1) (cid:19) π ( g ) k j ( g )d g d u = (cid:90) ( g + 1) n/ ( g + 1 + w ) p/ n/ ζ i ( w/ ( g + 1) , s ) (cid:18) gg + 1 (cid:19) b k j ( g )d g, where ζ i ( v, s ) = (cid:90) u p/ n/ exp (cid:16) − su (cid:17) h i (cid:18) u v (cid:19) d u. Note w ( g + 1) n/ ( g + 1 + w ) p/ n/ = w (cid:18) g + 1 g + 1 + w (cid:19) n/ ( g + 1 + w ) − p/ − = w (cid:18) − wg + 1 + w (cid:19) n/ ( g + 1 + w ) − p/ − = dd g (cid:40) n/ (cid:18) − wg + 1 + w (cid:19) n/ (cid:41) ( g + 1 + w ) − p/ . . Maruyama and W. E. Strawderman/Admissibility Then an integration by parts gives w (cid:90) ( g + 1) n/ ( g + 1 + w ) p/ n/ ζ i ( w/ ( g + 1) , s ) (cid:18) gg + 1 (cid:19) b k j ( g )d g (C.5)= 1 n/ (cid:40)(cid:34)(cid:18) − wg + 1 + w (cid:19) n/ ζ i ( w/ ( g + 1) , s )( g + 1 + w ) p/ − (cid:18) gg + 1 (cid:19) b k j ( g ) (cid:35) ∞ + ( p/ − (cid:90) ∞ (cid:18) − wg + 1 + w (cid:19) n/ ζ i ( w/ ( g + 1) , s )( g + 1 + w ) p/ (cid:18) gg + 1 (cid:19) b k j ( g )d g − (cid:90) ∞ (cid:18) − wg + 1 + w (cid:19) n/ ζ (cid:48) i ( w/ ( g + 1) , s )( g + 1 + w ) p/ − (cid:26) − w ( g + 1) (cid:27) (cid:18) gg + 1 (cid:19) b k j ( g )d g − (cid:90) ∞ (cid:18) − wg + 1 + w (cid:19) n/ ζ i ( w/ ( g + 1) , s )( g + 1 + w ) p/ − (cid:40) dd g (cid:18) gg + 1 (cid:19) b (cid:41) k j ( g )d g − (cid:90) ∞ (cid:18) − wg + 1 + w (cid:19) n/ ζ i ( w/ ( g + 1) , s )( g + 1 + w ) p/ − (cid:18) gg + 1 (cid:19) b k j ( g ) k (cid:48) j ( g )d g (cid:41) , where π (0) = 1 for b = 0, π (0) = 0 for b >
0, anddd g (cid:18) gg + 1 (cid:19) b = b (cid:18) gg + 1 (cid:19) b − g + 1) = bg ( g + 1) (cid:18) gg + 1 (cid:19) b , for b >
0. Further we have ζ (cid:48) i ( v, s ) = ∂∂v ζ i ( v, s )= − v + 1) (cid:90) u p/ n/ exp (cid:16) − su (cid:17) h i (cid:18) u v (cid:19) h (cid:48) i (cid:18) u v (cid:19) d u = − v + 1) p/ n/ (cid:90) η p/ n/ exp (cid:18) − s (1 + v ) η (cid:19) h i ( η ) h (cid:48) i ( η )d η. In (C.5), note (cid:18) − wg + 1 + w (cid:19) n/ g + 1 + w ) p/ (cid:18) gg + 1 (cid:19) b = (cid:18) wg + 1 (cid:19) − p/ − n/ − π ( g )( g + 1) p/ . Change of variables η = u/ { w/ ( g +1) } with d u/ d η = 1+ w/ ( g +1) completesthe proof. . Maruyama and W. E. Strawderman/Admissibility C.3. Lemmas for Lemma 3.1
Lemma C.4.
Assume − n/ < m < p/ and that both H ( η ) /η and Π( g ) / ( g +1) m − are integrable. Then (cid:90) (cid:90) (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m F ( g, η ; (cid:107) x (cid:107) /s, s ) H ( η )Π( g )d g d η d x d s = q ( m ) (cid:90) H ( η ) η d η (cid:90) Π( g )( g + 1) m − d g, where q ( m ) = 2 p/ n/ π p/ Γ( p/ − m )Γ( n/ m )Γ( p/ . Proof. (cid:90) (cid:90) (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m F ( g, η ; (cid:107) x (cid:107) /s, s ) H ( η )Π( g )d g d η d x d s = (cid:90) (cid:90) (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m η p/ n/ ( g + 1) p/ exp (cid:18) − sη (cid:18) (cid:107) x (cid:107) /s g + 1 (cid:19)(cid:19) H ( η )Π( g )d g d η d x d s = (cid:90) (cid:90) (cid:90) (cid:90) ( g + 1) p/ s p/ (cid:107) y (cid:107) s ( g + 1) s n/ − (cid:107) y (cid:107) m (1 + g ) m × η p/ n/ ( g + 1) p/ exp (cid:16) − sη (cid:0) (cid:107) y (cid:107) + 1 (cid:1)(cid:17) H ( η )Π( g )d g d η d y d s = Γ( p/ n/ − p/ − n/ − (cid:90) (cid:107) y (cid:107) − m ) d y (1 + (cid:107) y (cid:107) ) p/ n/ (cid:90) H ( η ) η d η (cid:90) Π( g )d g ( g + 1) m − = Γ( p/ n/ − p/ − n/ − π p/ Γ( p/ − m )Γ( n/ m )Γ( p/ p/ n/ (cid:90) H ( η ) η d η (cid:90) Π( g )d g ( g + 1) m − , where the second equality follows from change of variables y i = x i / ( √ g √ s )with Jacobian | ∂x/∂y | = (1 + g ) p/ s p/ as well as w/ (1 + g ) = (cid:107) y (cid:107) and (cid:107) x (cid:107) =(1 + g ) s (cid:107) y (cid:107) , and the last equality follows from Lemma C.6. Lemma C.5.
Assume − n/ < m < p/ and Π( g ) / ( g + 1) m − is integrable.Then (cid:90) (cid:90) (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m F ( g, η ; (cid:107) x (cid:107) /s, s )Π( g )(1 + | log s | ) d g d η d x d s = 2 q ( m ) (cid:90) Π( g )d g ( g + 1) m − . . Maruyama and W. E. Strawderman/Admissibility Proof. (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m F ( g, η ; (cid:107) x (cid:107) /s, s )Π( g )(1 + | log s | ) d g d η d x d s = (cid:90) (cid:90) (cid:90) (cid:90) (cid:107) x (cid:107) s n/ − ( (cid:107) x (cid:107) /s ) m η p/ n/ ( g + 1) p/ exp (cid:18) − ηs (cid:26) (cid:107) x (cid:107) /sg + 1 + 1 (cid:27)(cid:19) Π( g )d g d η d x d s (1 + | log s | ) = (cid:90) (cid:90) (cid:90) (cid:90) (1 + g ) p/ s p/ s n/ − (cid:107) y (cid:107) (1 + g ) s { (1 + g ) (cid:107) y (cid:107) } m × η p/ n/ ( g + 1) p/ exp (cid:16) − ηs (cid:8) (cid:107) y (cid:107) + 1 (cid:9)(cid:17) Π( g )(1 + | log s | ) d g d η d y d s = Γ( p/ n/ − p/ − n/ − (cid:90) d ss (1 + | log s | ) (cid:90) (cid:107) y (cid:107) − m ) d y (1 + (cid:107) y (cid:107) ) p/ n/ (cid:90) Π( g )d g ( g + 1) m − = 2 q ( m ) (cid:90) Π( g )d g ( g + 1) m − . Lemma C.6.
Let y ∈ R p . Assume β > α > − p/ . Then (cid:90) R p ( (cid:107) y (cid:107) ) α (1 + (cid:107) y (cid:107) ) − p/ − β d y = π p/ Γ( p/ α )Γ( β − α )Γ( p/ p/ β ) . (C.6) Proof. (cid:90) R p ( (cid:107) y (cid:107) ) α (1 + (cid:107) y (cid:107) ) − p/ − β d y = π p/ Γ( p/ (cid:90) ∞ u p/ − α (1 + u ) p/ β d u = π p/ Γ( p/
2) Be( p/ α, β − α )= π p/ Γ( p/ α )Γ( β − α )Γ( p/ p/ β ) . Lemma C.7.
Assume p/ a + 1 > and ≤ γ < p/ a + 1 .1. (cid:90) ∞ (cid:90) ∞ F h i ( η )( g + 1) γ π ( g )d g d η = (1 − z ) p/ − γ + a +1 s p/ n/ H ( z, s ; i ) , where z = w/ (1 + w ) and H ( z, s ; i ) (C.7)= (cid:90) (cid:90) ∞ t p/ − γ + a (1 − t ) b (1 − zt ) p/ − γ + a + b +2 v p/ n/ exp (cid:18) − v − zt ) (cid:19) h i ( v/s )d t d v.
2. The function { ( i + | log s | ) /i } H ( z, s ; i ) is bounded from above and below,where the lower and upper bounds are independent of i , z and s . . Maruyama and W. E. Strawderman/Admissibility
3. There exists a positive constant q , independent of i , z and s , such that (cid:82) ∞ (cid:82) ∞ F h i ( η )( g + 1) γ π ( g )d g d η (cid:82) ∞ (cid:82) ∞ F h i ( η ) π ( g )d g d η ≤ q ( w + 1) γ . (C.8) Proof. [Part 1] Note (cid:90) (cid:90) F ( g + 1) γ h i π d g d η = (cid:90) (cid:90) η p/ n/ ( g + 1) p/ exp (cid:26) − ηs (cid:18) wg + 1 + 1 (cid:19)(cid:27) h i ( η )( g + 1) γ − a − (cid:18) gg + 1 (cid:19) b d g d η. Apply the change of variables g = 1 − t (1 − z ) t where z = ww + 1with g + 1 = 1 − zt (1 − z ) t , wg + 1 = 1 + z/ (1 − z ) g + 1 = 11 − zt , (cid:12)(cid:12)(cid:12)(cid:12) d g d t (cid:12)(cid:12)(cid:12)(cid:12) = 1(1 − z ) t . Then= (cid:90) (cid:90) (cid:18) (1 − z ) t − zt (cid:19) p/ − γ + a +2 (cid:18) − t − zt (cid:19) b η p/ n/ (1 − z ) t exp (cid:18) − ηs − zt ) (cid:19) h i ( η )d t d η = (1 − z ) p/ − γ + a +1 (cid:90) (cid:90) t p/ − γ + a (1 − t ) b (1 − zt ) p/ − γ + a + b +2 η p/ n/ exp (cid:18) − ηs − zt ) (cid:19) h i ( η )d t d η = (1 − z ) p/ − γ + a +1 s p/ n/ (cid:90) (cid:90) t p/ − γ + a (1 − t ) b (1 − zt ) p/ − γ + a + b +2 v p/ n/ exp (cid:18) − v − zt ) (cid:19) h i ( v/s )d t d v. [Part 2] Let the probability density investigated in Maruyama and Strawder-man (2021) be f ( v | z ) = v ( p + n ) / ψ ( z ) (cid:90) t p/ a − γ (1 − t ) b (1 − zt ) p/ − γ + a + b +2 exp (cid:18) − v − zt ) (cid:19) d t, with normalizing constant ψ ( z ) given by ψ ( z ) = (cid:90) ∞ (cid:90) t p/ a − γ (1 − t ) b (1 − zt ) p/ − γ + a + b +2 v ( p + n ) / exp (cid:18) − v − zt ) (cid:19) d v d t. Then H ( z, s ; i ) = ψ ( z ) E (cid:2) h i ( V /s ) | z (cid:3) . By Lemma A.4 of Maruyama and Strawderman (2021), 0 < ψ ( z ) < ∞ for z ∈ [0 , (cid:18) i + | log s | i (cid:19) E (cid:2) h i ( V /s ) | z (cid:3) . . Maruyama and W. E. Strawderman/Admissibility (cid:104)(cid:104) lower bound (cid:105)(cid:105) Note i + | log s | i + | log v/s | ≥ i + | log s | i + | log s | + | log v | ≥
11 + | log v | . (C.9)By the Jensen inequality, we have (cid:18) i + | log s | i (cid:19) E (cid:2) h i ( V /s ) | z (cid:3) ≥ (cid:18)
11 + E [ | log V || z ] (cid:19) ≥ (cid:18)
11 + max z E [ | log V || z ] (cid:19) . (cid:104)(cid:104) < s <
1, upper bound (cid:105)(cid:105)
Assume 0 < s <
1. When v ≥ s , we have (cid:18) i + log(1 /s ) i + log v + log(1 /s ) (cid:19) = (cid:18) − log vi + log v + log(1 /s ) (cid:19) ≤ | log v | { i + log v + log(1 /s ) } ≤ | log v | . (C.10)When v < s , we have (cid:18) i + log(1 /s ) i + log( s/v ) (cid:19) ≤ (cid:18) i + log(1 /s ) i (cid:19) ≤ { /s ) } . Then (cid:18) i + log(1 /s ) i (cid:19) E (cid:2) h i ( V /s ) | z (cid:3) = (cid:90) s (cid:18) i + log(1 /s ) i + log( s/v ) (cid:19) f ( v | z )d v + (cid:90) ∞ s (cid:18) i + log(1 /s ) i + log( v/s ) (cid:19) f ( v | z )d v = (cid:90) s { /s ) } f ( v | z )d v + (cid:90) ∞ s { | log v | } f ( v | z )d v ≤ max z ∈ [0 , (cid:18) { /s ) } (cid:90) s f ( v | z )d v (cid:19) + 2 + 2 max z ∈ [0 , E [ | log V | | z ] . As in Lemma A.2 of Maruyama and Strawderman (2021), f ( v | z ) is regarded asa generalization of Gamma distribution. We have f (0 | z ) = 0 and f ( v | z ) growswith polynomial order around zero. Hencesup z ∈ [0 , ,s ∈ (0 , (cid:18) { /s ) } (cid:90) s f ( v | z )d v (cid:19) < ∞ . (cid:104)(cid:104) s >
1, upper bound (cid:105)(cid:105)
Assume s > v ≤ s . As in (C.10), we have (cid:18) i + log si + log( s/v ) (cid:19) = (cid:18) − log(1 /v ) i + log( s/v ) (cid:19) ≤ | log v | { i + log( s/v ) } ≤ | log v | . . Maruyama and W. E. Strawderman/Admissibility When v > s , (cid:18) i + log si + log( v/s ) (cid:19) ≤ (cid:18) i + log si (cid:19) ≤ (1 + log s ) . Then we have (cid:18) i + log si (cid:19) E (cid:2) h i ( V /s ) | z (cid:3) = (cid:90) s (cid:18) i + log si + log( s/v ) (cid:19) f ( v | z )d v + (cid:90) ∞ s (cid:18) i + log si + log( v/s ) (cid:19) f ( v | z )d v ≤ (cid:90) s (cid:0) | log v | (cid:1) f ( v | z )d v + (cid:90) ∞ s (1 + log s ) f ( v | z )d v ≤ z ∈ [0 , E [ | log V | | z ] + max z ∈ [0 , (cid:18) (1 + log s ) (cid:90) ∞ s f ( v | z )d v (cid:19) . As in Lemma A.2 of Maruyama and Strawderman (2021), f ( v | z ) is regardedas a generalization of Gamma distribution. We have f ( v | z ) is with exponentialdecay at infinity. Hencemax z ∈ [0 , ,s ∈ (1 , ∞ ) (cid:18) (1 + log s ) (cid:90) ∞ s f ( v | z )d v (cid:19) < ∞ . [Part 3] Part 3 follows from Parts 1 and 2. References
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