SHRINKAGE ESTIMATORS IN THE SUM OF PROFILES MODEL WITH CERTAIN CONDITIONS ON DESIGN MATRICES Yoshihiko KONNO Chiba University, Chiba 263, Japan

Key Words: GMANOVA model; Shrinkage estimators, Stein problem, multivariate growth curve model, covariates

ABSTRACT

Shrinkage estimation method is considered for an extension of the growth curve model, i.e., the multivariate normal linear model with the expected value of the response matrix is represented as the sum of profile under conditions where the treatment design matrices are nested subspace and the argumented matrix of the profile matrices is of full rank.

1. INTRODUCTION

Since the growth curve model, equivalently the GMANOVA model was introduced by Potthoff and Roy [14], several approaches to extend this model have been proposed. Chinchilli and Elswick [4] considered a mixture of the MANOVA model and the GMANOVA model. On the other hand Verblyla and Venables [21] considered so called the sum of profile model which may arise in the cases of parallel profiles, concurrent profiles on longitudinal data from designed experiments. Later von Rosen [15] considered a form of the multivariate normal linear model with the nested subspace condition and gave explicit form of the maximum likelihood estimators for the regression coefficient matrices and covariance matrix. General review of these models have been given by [17, 11]. More general multivariate linear model which is called totally ordered multivariate linear model, was proposed in [1].

On the other hand, Bilodeau and Kariya [2] considered the so called Stein or shrinkage effect in the MANOVA model and gave general sufficient conditions for an estimator of the mean coefficient regression matrix to dominate the MLE under the risk matrix. Since then, there has been growing literature to discuss this effect in more general multivariate normal linear model. In [12, 19] some classes of estimators were proposed in the GMANOVA model. Later Kariya, Konno and Strawderman [9] revealed the intrinsic structure of the estimation problem in the GMANOVA model and demonstrated to derive the double shrinkage estimators. More recently, Konno, Kubokawa and Saleh [10] considered the Stein effect in the model of [4] and showed that the estimation problem can be decomposed into the MANOVA and the GMANOVA parts and that the shrinkage estimators which are the sum of the single shrinkage estimators for the MANOVA part and the double shrinkage estimators for the GMANOVA part were obtained.

The model we look at is summarized in the following:

Y : n × p = ∑_{i=1}^{m} A_{i}β_{i}C_{i} + ϵ,  (1)

where Y is an observation matrix, and A_{i} and C_{i} are known n × a_{i} and c_{i} × p matrices of the full rank, and β_{i} are unknown a_{i} × c_{i} matrices, i = 1, 2, …, m. We also assume that the rows of ϵ are independently p-variate normally distributed with a covariance matrix Σ. We also assume that the nested subspace condition, i.e.,

span(A_{1}) ⊇ span(A_{2}) ⊇ ⋯ ⊇ span(A_{m})  (2)

is hold, where span(·) denotes the column vector space, and that the conditions on the argumented matrix of the profile matrices is of full rank, i.e.,

the matrix [C′_{1}, C′_{2}, …, C′_{m}]′ is of full rank.  (3)

The condition (2) was first introduced in [15] as a nested subspace condition. Without the condition (3), the model (1) is regarded as a special case of the totally ordered multivariate model in [1] which is amenable to explicit maximum likelihood analysis. But we make a crucial assumption (3) to make the model more tractable. Fujikoshi and Satoh [6] considered the model (1) when m = 2 with conditions

span(C_{2}) ⊇ span(C_{1}) and the matrix [A_{1}, A_{2}] is of full rank

and gave the basic properties of MLE’s of parameters.

We consider the estimation problem of the regression coefficient matrices in the model (1) in a decision theoretic way and obtain the shrinkage estimators under the loss function

L(β̂, β) = tr { ∑_{i=1}^{m} A_{i}(β̂_{i} − β_{i})C_{i}}Σ^{−1}{ ∑_{i=1}^{m} A_{i}(β̂_{i} − β_{i})C_{i}.}′,  (4)

where β = (β_{1}, …, β_{m}) and β̂ = (β̂_{1}, …, β̂_{m}). This loss function can be regarded as a natural extension to those employed in [9, 10].

In Section 2, we give a factorization of a natural loss function (4). Following the argument in [1, 15], we provide a factorization of the loss function (4) and a formula to decompose the likelihood function for (1) into a product of a simpler likelihood functions. In Section 3, we treat the estimation problem of (β_{1}, …, β_{m}) under the loss function (4) when the covariance matrix is known and demonstrate that, using the result in the previous section, we can decompose the problem into the sum of the usual MANOVA estimation problems from which we derive a Stein-type shrinkage estimator. In Section 4, we consider the case when the covariance matrix is unknown. Unlike the known case, the loss function (4) is intractable, so we only modifies the first component of expectation Y and give a generalization of double shrinkage estimators obtained in [9].

2. FACTORIZATION OF THE ESTIMATION PROBLEM

In order to make clear the intrinsic structure of the estimation problem of (β_{1}, …, β_{m}) in the model (1), we first decompose the loss function (4). To this end, we introduce the following notation: For i = 1, 2, …, m, let

Q_{i} : a_{i} × n = (A′_{i}A_{i})^{−1/2}A′_{i};
H_{i} : (n − a_{i}) × n  is a matrix such that H_{i}H′_{i} = I_{n−a_{i}} and H′_{i}H_{i} = I_{n} − Q′_{i}Q_{i};
H_{i}^{∗} : (a_{i} − a_{i+1}) × n  is a matrix such that H_{i}^{∗}(H_{i}^{∗})′ = I_{a_{i}−a_{i+1}} and
(H_{i}^{∗})′H_{i}^{∗} = Q′_{i}Q_{i} − Q′_{i+1}Q_{i+1} with Q_{m+1}^{∗} = 0;  (5)
C̃_{i} : c_{i} × (p − c_{i−1}^{+}) = C_{i}D_{0}D_{1}D_{2} × ⋯ × D_{i−1};

D_{i} : (p − c_{i−1}^{+}) × (p − c_{i}^{+}) is a matrix such that C̃_{i}D_{i} = 0, D′_{i}D_{i} = I_{p−c_{i}^{+}} for i ≥ 1

with  D_{0} = I_{p} and c_{i}^{+} = ∑_{j=1}^{i} c_{i} for i ≥ 1 and c_{0}^{+} = 0.

Furthermore put, for i = 1, 2, …, m,

X_{[i]} : a_{i} × c_{i} = Q_{i}Y D_{0}D_{1} × ⋯ × D_{i−1}C̃′_{i}(C̃_{i}C̃′_{i})^{−1};  (6)
X_{<i>} : a_{i} × (p − c_{i}^{+}) = Q_{i}Y D_{0}D_{1} × ⋯ × D_{i};  (7)
Z_{[i]} : (n − a_{i}) × c_{i} = H_{i}Y D_{0}D_{1} × ⋯ × D_{i−1}C̃′_{i}(C̃_{i}C̃′_{i})^{−1};  (8)
Z_{<i>} : (n − a_{i}) × (p − c_{i}^{+}) = H_{i}Y D_{0}D_{1} × ⋯ × D_{i};  (9)
Z_{<i>}^{∗} : (a_{i} − a_{i+1}) × (p − c_{i}^{+}) = H_{i}^{∗}Y D_{0}D_{1} × ⋯ × D_{i}  (10)

with a_{m+1} = 0 ( so Z_{<m>}^{∗} = X_{<m>});
μ_{[i]} : a_{i} × c_{i} = E[X_{[i]}]
= (A′_{i}A_{i})^{1/2}β_{i} + Q_{i}( ∑_{j=i+1}^{m} A_{j}β_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}C̃′_{i}(C̃_{i}C̃′_{i})^{−1}; (11)
μ_{<i>} : a_{i} × (p − c_{i}^{+}) = E[X_{<i>}]

= Q_{i} ( ∑_{j=i+1}^{m} A_{j}β_{j}C_{j}) D_{0}D_{1} × ⋯ × D_{i}

= Q_{i}A_{i+1}β_{i+1}C̃_{i+1} + Q_{i}( ∑_{j=i+2}^{m} A_{j}β_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i} with μ_{<m>} = 0; (12)
Σ_{[i]} : c_{i} × c_{i} = cov[X_{[i]}, X_{[i]}];
Σ_{[i>} : c_{i} × (p − c_{i}^{+}) = Σ′_{<i]} = cov[X_{[i]}, X_{<i>}];
Σ_{<i>} : (p − c_{i}^{+}) × (p − c_{i}^{+}) = cov[X_{<i>}, X_{<i>}];
γ_{<i]} : (p − c_{i}^{+}) × c_{i} = Σ_{<i>}^{−1}Σ_{<i]};
μ′_{[i]•} : c_{i} × a_{i} = μ′_{[i]} − Σ_{[i>}Σ_{<i>}^{−1}μ′_{<i>}.  (13)

If we put Σ_{<0>} = Σ and

Σ̃_{<i−1>} : (p − c_{i−1}^{+}) × (p − c_{i−1}^{+}) = [ \begin{matrix}Σ_{[i]} & Σ_{<i]} \\ Σ_{<i]} & Σ_{<i>} \end{matrix} ] for i = 1, 2, …, m,

we can get the recursive formula for the covariance matrix

( \begin{matrix}(C̃_{i}C̃′_{i})^{−1}C̃_{i}Σ_{<i−1>}C̃′_{i}(C̃_{i}C̃′_{i})^{−1} & (C̃_{i}C̃′_{i})^{−1}C̃_{i}Σ_{<i−1>}D_{i} \\ D′_{i}Σ_{<i−1>}C̃′_{i}(C̃_{i}C̃′_{i})^{−1} & D′_{i}Σ_{<i−1>}D_{i} \end{matrix} ) = ( \begin{matrix}Σ_{[i]} & Σ_{[i>} \\ Σ_{<i]} & Σ_{<i>} \end{matrix} ) .

Then we have, for i = 1, 2, …, m,

X_{i} : a_{i} × (p − c_{i−1}^{+}) = (X_{[i]}, X_{<i>}) ∼ N([μ_{[i]}, μ_{<i>}], I_{a_{i}} ⊗ Σ̃_{<i−1>}),
Z_{i} : (n − a_{i}) × (p − c_{i−1}^{+}) = (Z_{[i]}, Z_{<i>}) ∼ N(0, I_{n−a_{i}} ⊗ Σ̃_{<i−1>}),  (14)

and

(X′_{[i]}, Z′_{[i]})′ : n × c_{i} ∼ N((μ′_{[i]}, 0′)′, I_{n} ⊗ Σ_{[i]•}),
(X′_{<m>}, Z′_{<m>})′ : n × (p − c_{m}^{+}) ∼ N(0, I_{n} ⊗ Σ_{<m>}).

For i = 1, 2, …, m, put

[ \begin{matrix}S_{[i]} & S_{[i>} \\ S_{<i]} & S_{<i>} \end{matrix} ] : (p − c_{i−1}^{+}) × (p − c_{i−1}^{+}) = [ \begin{matrix}Z′_{[i]}Z_{[i]} & Z′_{[i]}Z_{<i>} \\ Z′_{<i>}Z_{[i]} & Z′_{<i>}Z_{<i>} \end{matrix} ]  (15)

and

U_{<i>} : (p − c_{i}^{+}) × (p − c_{i}^{+}) = (Z_{<i>}^{∗})′Z_{<i>}^{∗}

where S_{[i]} are c_{i} × c_{i} matrices and we put U_{<0>} = 0 and S_{<0>} = S = Y′H′_{1}H_{1}Y. Then we have

S_{[i]} = (C̃_{i}C̃′_{i})^{−1}C̃_{i}(S_{<i−1>} + U_{<i−1>})C̃′_{i}(C̃_{i}C̃′_{i})^{−1}
S_{[i>} = (C̃_{i}C̃′_{i})^{−1}C̃_{i}(S_{<i−1>} + U_{<i−1>})D_{i}
S_{[i]•} : c_{i} × c_{i} = S_{[i]} − S_{[i>}S_{<i>}^{−1}S_{<i]}
S_{<i>} = D′_{i}(S_{<i−1>} + U_{<i−1>})D_{i}.

Further let

V_{<i]} : (p − c_{i}^{+}) × c_{i} = S_{<i>}^{−1}S_{<i]}.

Then we have, for i = 1, 2, …, m + 1, S_{<i−1>} + U_{<i−1>} ∼ W_{p−c_{i−1}^{+}}(n − a_{i}, Σ_{<i−1>}) and S_{<i−1>} ∼ W_{p−c_{i−1}^{+}}(n − a_{i−1}, Σ_{<i−1>}). Theorem 2.1. The loss function (4) is rewritten as

∑_{i=1}^{m} tr [{(A′_{i}A_{i})^{1/2}β̂_{i} + (A′_{i}A_{i})^{−1/2}A′_{i} ( ∑_{j=i+1}^{m} A_{j}β̂_{j}C_{j}) D_{0}D_{1}

× ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1} − μ_{[i]•}}Σ_{[i]•}^{−1}{ ⋯ }′].  (16)

Proof. Let

P_{i} : (p − c_{i−1}^{+}) × (p − c_{i−1}^{+}) = Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}C̃_{i}.

Decompose the term inside the curly bracket in (4) into

A_{1}(β̂_{1} − β_{1})C_{1} + (∑_{j=2}^{m} A_{j}(β̂_{j} − β_{j})C_{j}) P_{1} + (∑_{j=2}^{m} A_{j}(β̂_{j} − β_{j})C_{j}) (I_{p} − P_{1}). (17)

Noting that C_{1}Σ^{−1}(I_{p} − P′_{1}) = 0, (I_{p} − P_{1})Σ^{−1}P′_{1} = 0, and (I − A_{1}(A′_{1}A_{1})^{−1}A′_{1})A_{j} = 0 (j ≥ 2), we can see that the cross products between the term inside the curly bracket and the third term in (17) is zero and that

{ ∑_{i=1}^{m} A_{i}(β̂_{i} − β_{i})C_{i}}Σ^{−1}{ ⋯ }′

= {A_{1}(β̂_{1} − β_{1})C_{1} + A_{1}(A′_{1}A_{1})^{−1}A′_{1} (∑_{j=2}^{m} A_{j}(β̂_{j} − β_{j})C_{j}) P_{1}}Σ^{−1}{ ⋯ }′

+{ ∑_{j=2}^{m} A_{j}(β̂_{j} − β_{j})C_{j}}(I_{p} − P_{1})Σ^{−1}(I_{p} − P′_{1}){ ⋯ }′.  (18)

By virtue of the relation

Σ^{−1} − Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−1}C_{1}Σ^{−1} = D_{1}(D′_{1}ΣD_{1})^{−1}D′_{1}  (19)

for any p × p positive definite matrix Σ (see Siotani et al., 1985, p.311), we can see that

(A′_{1}A_{1})^{1/2}β_{1} + (A′_{1}A_{1})^{−1/2}A′_{1} (∑_{j=2}^{m} A_{j}β_{j}C_{j}) Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−1}

= (A′_{1}A_{1})^{1/2}β_{1} + (A′_{1}A_{1})^{−1/2}A′_{1} (∑_{j=2}^{m} A_{j}β_{j}C_{j}) [I − D_{1}(D′_{1}ΣD_{1})^{−1}D′_{1}Σ]C′_{1}(C_{1}C′_{1})^{−1}

= μ_{[1]•}.

Similarly we can see that

(C_{1}Σ^{−1}C′_{1})^{−1} = Σ_{[1]} − Σ_{[1>}Σ_{<1>}^{−1}Σ_{<1]} = Σ_{[1]•}
(I_{p} − P_{1})Σ^{−1}(I_{p} − P′_{1}) = (I_{p} − P_{1})Σ^{−1} = D_{1}(D′_{1}ΣD_{1})^{−1}D′_{1}.

From these, we can see that the left hand side of (18) equals

tr [{(A′_{1}A_{1})^{1/2}β̂_{1} + (A′_{1}A_{1})^{−1/2}A′_{1} (∑_{j=2}^{m} A_{j}β̂_{j}C_{j}) Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−1} − μ_{[1]•}}
Σ_{[1]•}^{−1}{ ⋯ }′] + tr [{ ∑_{j=2}^{m} A_{j}(β̂_{j} − β_{j})C_{j}D_{1}}Σ_{<1>}^{−1}{ ⋯ }′] .  (20)

Next similar decomposition of (17) with P_{2} shows that the second term in (20) equals

tr [{A_{2}(β̂_{2} − β_{2}) + A_{2}(A′_{2}A_{2})^{−1}A′_{2} (∑_{j=3}^{m} A_{j}(β̂_{j} − β_{j})C_{j}) D_{1}Σ_{<1>}^{−1}C̃′_{2}(C̃_{2}Σ_{<1>}^{−1}C̃′_{2})^{−1}}

×C̃_{2}Σ_{<1>}^{−1}C̃′_{2}{ ⋯ }′] + tr [{ (∑_{j=3}^{m} A_{j}(β̂_{j} − β_{j})C_{j}) D_{1}(I_{p−c_{1}^{+}} − P_{2})}Σ_{<1>}^{−1}{ ⋯ }′]

= tr [{(A′_{2}A_{2})^{1/2}β̂_{2} + (A′_{2}A_{2})^{−1/2}A′_{2} (∑_{j=3}^{m} A_{j}β̂_{j}C_{j}) D_{1}Σ_{<i>}^{−1}C̃′_{2}(C̃_{2}Σ_{<1>}^{−1}C̃′_{2})^{−1} − μ_{[2]•}}

×Σ_{[2]•}^{−1}{ ⋯ }′] + tr [{ (∑_{j=3}^{m} A_{j}(β̂_{j} − β_{j})C_{j}) D_{1}D_{2}}Σ_{<2>}^{−1}{ ⋯ }′] .

For (m − 1) step, we obtain the term corresponding to the second term of LHS in the above formula as

tr [{(A′_{m}A_{m})^{1/2}(β̂_{m} − β_{m})C_{m}D_{0} × ⋯ D_{m−1}}Σ_{<m−1>}^{−1}{ ⋯ }′]
= tr [{(A′_{m}A_{m})^{1/2}(β̂_{m} − β_{m})}C̃_{m}Σ_{<m−1>}^{−1}C̃′_{m}{ ⋯ }′]
= tr [{(A′_{m}A_{m})^{1/2}(β̂_{m} − β_{m})}Σ_{[m−1]•}^{−1}{ ⋯ }′] .

Finally recursive argument leads to the desired result.

Next we decompose the density function of Y into canonical form whose parameter space is expressed as a product of simpler subspaces. This parameterization is regarded as Mparameter space in [1]. Theorem 2.2. The density of Y can be factored as

∏_{i=1}^{m} \{ | Σ_{[i]•} |^{−a_{i}/2} exp (−\frac{1}{2}tr [\{X_{[i]} − X_{<i>}Σ_{<i>}^{−1}Σ_{<i]} − μ_{[i]•}\}Σ_{[i]•}^{−1}\{⋯\}′]) dX_{[i]}

× | Σ_{[i]•} |^{−(p−c_{i}^{+})/2}| S_{<i>} |^{a_{i}/2} exp (−\frac{1}{2}tr [S_{<i>}\{S_{<i>}^{−1}S_{<i]} − Σ_{<i>}^{−1}Σ_{<i]}\}Σ_{[i]•}^{−1}\{⋯\}′]) d(S_{<i>}^{−1}S_{<i]})

× | Σ_{[i]•} |^{−\{n−a_{i}−(p−c_{i}^{+})\}/2}| S_{[i]•} |^{\{n−a_{i}−(p−c_{1−1}^{+})−1\}/2} exp (−\frac{1}{2}tr [S_{[i]•}Σ_{[i]•}^{−1}]) d(S_{[i]•})\}

× | Σ_{<m>} |^{−a_{m}/2} exp (−\frac{1}{2}X_{<m>}Σ_{<m>}^{−1}X′_{<m>}) d(X_{<m>})

× | Σ_{<m>} |^{−(n−a_{i}−(p−c_{i}^{+}))/2}| S_{<m>}^{−(n−a_{i}−(p−c_{m}^{+})−1)/2} exp (−\frac{1}{2}S_{<m>}Σ_{<m>}^{−1}) d(S_{<m>}).  (21)

Proof. For i = 1, 2, …, m, let

Ỹ_{i} : a_{i−1} × (p − c_{i−1}^{+}) = Q_{i−1}(Y − ∑_{j=i}^{m} A_{j}β_{j}C_{j})D_{0} × ⋯ × D_{i−1};

Ỹ_{[i]} : a_{i} × c_{i} = Q_{i}(Y − ∑_{j=i}^{m} A_{j}β_{j}C_{j})D_{0} × ⋯ × D_{i−1}C̃′_{i}(C̃_{i}C̃′_{i})^{−1};
Ỹ_{<i>} : a_{i} × (p − c_{i}^{+}) = Q_{i}(Y − ∑_{j=i+1}^{m} A_{j}β_{j}C_{j})D_{0} × ⋯ × D_{i}

where a_{0} = n and Q_{0} = I_{n}. From the orthogonality of H_{i} to A_{i}, …, A_{m}, of H_{i}^{∗} to A_{i+1}, …, A_{m}, and of D_{i} to C̃_{i} and the definition of (8)-(10),

Z_{[i]} = H_{i}Ỹ_{1}D_{0}D_{1} × ⋯ × D_{i−1}C̃′_{i}(C̃_{i}C̃′_{i})^{−1};
Z_{<i>} = H_{i}Ỹ_{1}D_{0}D_{1} × ⋯ × D_{i};
Z_{<i>}^{∗} = H_{i}^{∗}Ỹ_{1}D_{0}D_{1} × ⋯ × D_{i}.

From

Q′_{1}Q_{1} + H′_{1}H_{1} = I_{n} and C′_{1}(C_{1}C′_{1})^{−1}C_{1} + D_{1}D′_{1} = I_{p},

we have

tr \{Ỹ_{1}Σ_{<0>}^{−1}Ỹ′_{1}\} = tr \{ [ Ỹ_{[1]} Ỹ_{<1>} ] [ \begin{matrix}C_{1}Σ^{−1}C′_{1} & C_{1}Σ^{−1}D_{1} \\ D′_{1}Σ^{−1}C′_{1} & D′_{1}Σ^{−1}D_{1} \end{matrix} ] [ Ỹ′_{[1]} ; Ỹ′_{<1>} ] \}
+tr \{ [ Z_{[1]} Z_{<1>} ] [ \begin{matrix}C_{1}Σ^{−1}C′_{1} & C_{1}Σ^{−1}D_{1} \\ D′_{1}Σ^{−1}C′_{1} & D′_{1}Σ^{−1}D_{1} \end{matrix} ] [ \begin{matrix}Z′_{[1]} \\ Z′_{<1>} \end{matrix} ] \}. (22)

Since

[ \begin{matrix}C_{1}Σ^{−1}C′_{1} & C_{1}Σ^{−1}D_{1} \\ D′_{1}Σ^{−1}C′_{1} & D′_{1}Σ^{−1}D_{1} \end{matrix} ] = [ \begin{matrix}Σ_{[1]} & Σ_{[1>} \\ Σ_{<1]} & Σ_{<1>} \end{matrix} ]^{−1}

= [ \begin{matrix}I & 0 \\ −Σ_{<1>}^{−1}Σ_{<1]} & I \end{matrix} ] [ \begin{matrix}Σ_{[1]•}^{−1} & 0 \\ 0 & Σ_{<1>}^{−1} \end{matrix} ] [ \begin{matrix}I & −Σ_{[1>}Σ_{<1>}^{−1} \\ 0 & I \end{matrix} ] ,

we can see that the first term of the right hand side in (22) equals

tr [{Ỹ_{[1]} − Ỹ_{<1>}Σ_{<1>}^{−1}Σ_{<1]}}Σ_{[1]•}^{−1}{⋯}′] + tr [Ỹ_{<1>}Σ_{<1>}^{−1}Ỹ′_{<1>}]

= tr [{X_{[1]} − X_{<1>}Σ_{<1>}^{−1}Σ_{<1]} − μ_{[1]•}}Σ_{[1]•}^{−1}{⋯}′] + tr [Ỹ_{<1>}Σ_{<1>}^{−1}(Ỹ_{<1>})′] = tr [{X_{[1]} − X_{<1>}Σ_{<1>}^{−1}Σ_{<1]} − μ_{[1]•}}Σ_{[1]•}^{−1}{⋯}′] + tr [Ỹ_{2}Σ_{<1>}^{−1}Ỹ′_{2}]

+tr [Z_{<1>}^{∗}Σ_{<1>}^{−1}(Z_{<1>}^{∗})′].  (23)

The last equality in (23) follows from Q′_{1}Q_{1} = (H_{1}^{∗})′H_{1}^{∗} +Q′_{2}Q_{2}. From (22)-(23), we conclude that

tr [Ỹ_{1}Σ_{<0>}^{−1}Ỹ′_{1}] = tr [{X_{[1]} − X_{<1>}Σ_{<1>}^{−1}Σ_{<1]} − μ_{[1]•}}Σ_{[1]•}^{−1}{⋯}′]
+tr [S_{[1]•}Σ_{[1]•}^{−1}] + tr [S_{<1>}{S_{<1>}^{−1}S_{<1]} − Σ_{<1>}^{−1}Σ_{<1]}}Σ_{[1]•}^{−1}{⋯}′]
+tr [(S_{<1>} + U_{<1>})Σ_{<1>}^{−1}] + tr [Ỹ_{2}Σ_{<1>}^{−1}Ỹ′_{2}].  (24)

Finally using the standard decomposition

| Σ |= ∏_{1}^{m+1} | Σ_{[i]•} |

and applying similar argument through (22) to (23), the formula (24) can be factored into (21).

From Theorem 2.2, we can see that a decomposition of the model (1) can be made as

X_{[i]} : a_{i} × c_{i} given X_{<i>} ∼ N(μ_{[i]•} + X_{<i>}γ_{<i]}, I_{a_{i}} ⊗ Σ_{[i]•})
V_{<i]} : (p − c_{i}^{+}) × c_{i} = S_{<i>}^{−1}S_{<i]} given S_{<i>} ∼ N(γ_{<i]}, S_{<i>}^{−1} ⊗ Σ_{[i]•})
S_{[i]•} = S_{[i]} − S_{[i>}S_{<i>}^{−1}S_{<i]} ∼ W_{c_{i}}(n − a_{i} − (p − c_{i}^{+}), Σ_{[i]•}),
X_{<m>} : a_{m} × (p − c_{m}^{+}) ∼ N(0, I_{a_{m}} ⊗ Σ_{<m>})
S_{<m>} ∼ W_{p−c_{m}^{+}}(n − a_{m} − (p − c_{m}^{+}), Σ_{<m>}) (25)

where (X_{[i]}, V_{<i]}, S_{[i]•}, i = 1, 2, …, m, are mutually independent. Furthermore, we have

X_{<i>} : a_{i} × (p − c_{i}^{+}) ∼ N(μ_{<i>}, I_{a_{i}} ⊗ Σ_{<i>}).

In [1], the family of parameters (μ_{[i]•}, Σ_{<i>}^{−1}Σ_{<i]}, Σ_{[i]•}) is termed as M-parameters. As noted in [1], there is a one-to-one correspondence between the original parameters and Mparameters and reconstruction algorithm from M-parameter to the original parameter is given in [1].

3. ESTIMATION OF β WHEN THE COVARIANCE IS KNOWN In this section, we consider the problem of estimating β under the loss function (4) when the covariance matrix is known. From Theorem 2.2 we can see that the maximum likelihood estimators for μ_{[i]•} is X_{[i]} − X_{<i>}Σ_{<i>}^{−1}Σ_{<i]}. Now using (6), (7), (11)-(13), (19), we can reconstruct the estimators (β_{1}…. , β_{m}) from the estimators for M-parameters in the following: Step 1:

(A′_{m}A_{m})^{1/2}B̂_{m}^{Σ} = μ̂_{[m]•} = X_{[m]} − X_{<m>}Σ_{<m>}^{−1}Σ_{<m]}
= Q_{m}Y D_{0}D_{1} × … × D_{m−1}Σ_{<m−1>}^{−1}C̃′_{m}(C̃_{m}Σ_{<m−1>}^{−1}C̃′_{m})^{−1}.

Step i (2 ≤ i ≤ m):

(A′_{m−i+1}A_{m−i+1})^{1/2}B̂_{m−i+1}^{Σ} = μ̂_{[m−i+1]•} − Q_{m−i+1} ( ∑_{j=m−i+2}^{m} A_{j}B̂_{j}^{Σ}C_{j}) D_{0}D_{1} × ⋯ × D_{m−i}

×Σ_{<m−i>}^{−1}C̃′_{m−i+1}(C̃_{m−i+1}Σ_{<m−i>}^{−1}C̃′_{m−i+1})^{−1}

= X_{[m−i+1]} − X_{<m−i+1>}Σ_{<m−i+1>}^{−1}Σ_{<m−i+1]} − Q_{m−i+1} ( ∑_{j=m−i+2}^{m} A_{j}B̂_{j}^{Σ}C_{j})

×D_{0}D_{1} × ⋯ × D_{m−i}Σ_{<m−i>}^{−1}C̃′_{m−i+1}(C̃_{m−i+1}Σ_{<m−i>}^{−1}C̃′_{m−i+1})^{−1}.

From this argument we can see that the MLE’s are given by

B̂_{i}^{Σ} = (A′_{i}A_{i})^{−1}A′_{i}(Y − ∑_{j=i+1}^{m} A_{j}B̂_{j}^{Σ}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}.

In the light of the factorization theorem of the density function, we consider a family of estimators of the form

β̂_{i}^{Σ} = (A′_{i}A_{i})^{−1}A′_{i}(Y − ∑_{j=i+1}^{m} A_{j}β̂_{j}^{Σ}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}
+(A′_{i}A_{i})^{−1/2}L_{i}(X_{[i]}, X_{<i>}γ_{<i]}, Σ_{[i]•}),  (26)

where L_{i} is an a_{i} × c_{i} matrix. Use the relation (19) to obtain

[C̃′_{i}(C̃_{i}C̃′_{i})^{−1} − D_{i}γ_{<i]}] = [I − D_{i}(D′_{i}Σ_{<i−1>}D_{i})^{−1}D′_{i}Σ_{<i−1>}]C̃′_{i}(C̃_{i}C̃′_{i})^{−1}
= Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}.  (27)

Then the estimators (26) can be rewritten

(A′_{i}A_{i})^{1/2}β̂_{i}^{Σ} = X_{[i]} − X_{<i>}Σ_{<i>}^{−1}Σ_{<i]} + L_{i} (X_{[i]}, X_{<i>}γ_{<i]}, Σ_{[i]•})
−Q_{i} ( ∑_{j=i+1}^{m} A_{j}β̂_{j}^{Σ}C_{j}) D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}.  (28)

By Theorem 2.1 and (25), we can see that the risk function for β^{Σ} = (β_{1}^{Σ}, …, β_{m}^{Σ}) is

R(β̂^{Σ}, β) = E [∑_{i=1}^{m} tr ({X_{[i]} + L_{i} − X_{<i>}Σ_{<i>}^{−1}Σ_{<i]} − μ_{[i]•}}Σ_{[i]•}^{−1}{ … }′)]
= ∑_{i=1}^{m} E^{X_{<i>}}E^{X_{[i]}|X_{<i>}} [tr ({X_{[i]} + L_{i} − (μ_{[i]•} + X_{<i>}Σ_{<i>}^{−1}Σ_{<i]})}Σ_{[i]•}^{−1}{ … }′)] (29)

where

X_{[i]} : a_{i} × c_{i} given X_{<i>} ∼ N(μ_{[i]•} + X_{<i>}γ_{<i]}, I_{a_{i}} ⊗ Σ_{[i]•}).  (30)

Hence the estimation problem is decomposed into m independent conditional problems of estimating μ_{[i]•} + X_{<i>}Σ_{<i>}^{−1}Σ_{<i]} based on X_{[i]} given X_{<i>} under usual loss function. Since each problem is regarded as conditional problem of estimating the mean matrix with known covariance matrix which was considered by [5, 18, 23, 7], the result in this problem is applicable.

To find L_{i}’s for which R(β̂^{Σ}, β) ≤ ∑_{i=1}^{m} a_{i}c_{i}, we apply Stein’s method, i.e., integration by parts and unbiased risk estimate, (conditioning on X_{<i>}) to the each term in (29). To this end, we need the following notation. Denote by V_{kl} the (k, l) element of a matrix V and let ∇_{V} be a differential operator whose (k, l) element is given by (∂/∂V_{kl}). For a matrix L, we define

(∇′_{V} L)_{kl} = ∑_{j} \frac{∂}{∂V_{jk}} L_{jl}

as a formal product followed by differentiation at the component level. Lemma 3.1. Assume that E{(L_{i})_{kl}}^{2} < ∞ and E|(∂/∂(X_{[i]})_{k_{2}l_{2}})(L_{i})_{k_{1}l_{1}}| < ∞ and that (L_{i})_{k_{1}l_{1}} is absolutely continuous as a function of (X_{[i]})_{k_{2}l_{2}} for almost all (X_{[i]}, X_{<i>}) except (X_{[i]})_{k_{2}l_{2}} . Then unbiased estimate of the risk R(β̂^{Σ}, β) is given by

R̂(β̂^{Σ}, β) = ∑_{i=1}^{m} {a_{i}c_{i} + 2tr (∇′_{X_{[i]}}L_{i}) + tr {(L_{i})′L_{i}Σ_{[i]•}^{−1}}} .

Proof. The result is immediately obtained by conditioning on X_{<i>} (hence the distribution of X_{[i]} is given by (30)) and applying [2, Lemma 2.1] to the term

tr [{X_{[i]} − (μ_{[i]•} + X_{<i>}γ_{<i]})}Σ_{[i]•}^{−1}L′_{i}].

Now define

X_{[i]}^{∗} = X_{[i]}T_{i}, L_{i}^{∗} = L_{i}T_{i}, and γ_{<i]}^{∗} = γ_{<i]}T_{i}  (31)

where Σ_{[i]•}^{−1} = T_{i}T′_{i}. Noting that

∇′_{X_{[i]}} = T_{i}∇′_{X_{[i]}^{∗}},  (32)

we have that

R̂(β̂^{Σ}, β) = ∑_{i=1}^{m} [a_{i}c_{i} + 2tr {∇′_{X_{[i]}^{∗}}L_{i}^{∗}} + tr {(L_{i}^{∗})′(L_{i}^{∗})}]  (33)

and that

R(β̂^{Σ}, β) = ∑_{i=1}^{m} E^{X_{<i>}}E^{X_{[i]}^{∗}|X_{<i>}} [a_{i}c_{i} + 2tr {∇′_{X_{[i]}^{∗}}L_{i}^{∗}} + tr {(L_{i}^{∗})′(L_{i}^{∗})}]

where

X_{[i]}^{∗} ∼ N((μ_{[i]•} + X_{<i>}γ_{<i]})T_{i}, I_{a_{i}} ⊗ I_{c_{i}}).

Then our estimators become

(A′_{i}A_{i})^{1/2}β̂_{i}^{Σ}T_{i} = X_{[i]}^{∗} − X_{<i>}γ_{<i]}^{∗} + L_{i}^{∗}

−Q_{i} ( ∑_{j=i+1}^{m} A_{j}β̂_{j}^{Σ}C_{j}) D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}T_{i}.

From these argument, we can see that each estimation problem can be reduced to the estimation problem of matrix mean considered in in [18, 23].

For i = 1, …, m, let

F_{i∗}^{Σ} : a_{i} × a_{i} = (X_{[i]} − X_{<i>}γ_{<i]})Σ_{[i]•}^{−1}(X_{[i]} − X_{<i>}γ_{<i]})′ = (f_{kl}^{Σ(i∗)})

and let F̃_{i∗}^{Σ} = diag (f_{11}^{Σ(i∗)}, …, f_{a_{i}a_{i}}^{Σ(i∗)})). Further, let

M_{i∗} = diag (g_{1}^{(i∗)}(f_{11}^{Σ(i∗)}), …, g_{a_{i}}^{(i∗)}(f_{a_{i}a_{i}}^{Σ(i∗)})

and put

L_{i∗} = −M_{i∗}(F̃_{i∗}^{Σ})^{−1}(X_{[i]} − X_{<i>}γ_{<i]}).

Consider a class of estimators of the form

(A′_{i}A_{i})^{1/2}β̂_{i}^{ΣBA} = X_{[i]} − X_{<i>}γ_{<i]} + L_{i∗}

−Q_{i} ( ∑_{j=i+1}^{m} A_{j}β̂_{j}^{ΣBA}C_{j}) D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}.

Theorem 3.2. Assume that max(c_{1}, …, c_{m}) ≥ 3. If

(i)  0 ≤ g_{j}^{(i∗)} ≤ \frac{2}{a_{i}} min(c_{i} − 2, 0)

(ii)  (g_{j}^{(i∗)})′ ≥ 0, j = 1, …, a_{i}, i = 1, …, m,

then β̂^{ΣBA} is minimax where β̂^{ΣBA} = (β̂_{1}^{ΣBA}, …, β̂_{m}^{ΣBA}). Proof. Note that

tr {∇′_{X_{[i]}}M_{i∗}(F̃_{i∗}^{Σ})^{−1}(X_{[i]} − X_{<i>}γ_{<i]})}

= tr [{∇′_{X_{[i]}}M_{i∗}(F̃_{i∗}^{Σ})^{−1}} (X_{[i]} − X_{<i>}γ_{<i]})] + tr [{M_{i∗}(F̃_{i∗}^{Σ})^{−1}}′ ∇_{X_{[i]}}(X_{[i]} − X_{<i>}γ_{<i]})′]

= 2 tr (∂M_{i∗}) + (c_{i} − 2) tr {M_{i∗}(F̃_{i∗}^{Σ})^{−1}}

where ∂M_{i∗} = diag ((g_{1}^{(i∗)})′, …, (g_{a_{i}}^{(i∗)})′). From this, we have

R̂(β̂^{ΣBA}, β) = ∑_{i=1}^{m} [a_{i}c_{i} − 4 tr (∂M_{i∗}) − 2(c_{i} − 2) tr {M_{i∗}(F̃_{i∗}^{Σ})^{−1}}
+ tr {M_{i∗}^{2}(F̃_{i∗}^{Σ})^{−1}(F_{i∗}^{Σ})(F̃_{i∗}^{Σ})^{−1}}] .  (34)

Finally note that a_{i}(F̃_{i∗}^{Σ})^{−1} − (F̃_{i∗}^{Σ})^{−1}(F_{i∗}^{Σ})(F̃_{i∗}^{Σ})^{−1} is positive definite to see that the RHS of (34) is bounded above by ∑ a_{i}c_{i} if the conditions of this theorem holds.

For i = 1, …, m, let

L_{i∗∗} = −\frac{g^{(i∗∗)}(w_{i})}{w_{i}}(X_{[i]} − X_{<i>}γ_{<i]})

where w_{i} = tr F_{i∗}^{Σ}. Theorem 3.3. Assume that max(a_{1}c_{1}, …, a_{m}c_{m}) ≥ 3. Put

(A′_{i}A_{i})^{1/2}β̂_{i}^{ΣBA} = X_{[i]} − X_{<i>}γ_{<i]} + L_{i∗∗}

−Q_{i} ( ∑_{j=i+1}^{m} A_{j}β̂_{j}^{ΣBA}C_{j}) D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}.

Then β̂^{ΣSR} = (β̂_{1}^{ΣSR}, …, β̂_{m}^{ΣSR}) is minimax if

(i)  0 ≤ g^{(i∗∗)}(w_{i}) ≤ 2 max(a_{i}c_{i} − 2, 0)

(ii)  (g^{(i∗∗)})′ ≥ 0, i = 1, …, m.

Proof. Since

tr \{∇′_{X_{[i]}} \frac{g^{(i∗∗)}(w_{i})}{w_{i}}(X_{[i]} − X_{<i>}γ_{<i]})\} = 2(g^{(i∗∗)})′ + (a_{i}c_{i} − 2)\frac{g^{(i∗∗)}(w_{i})}{w_{i}},

we have

R̂(β̂^{ΣSR}, β) = ∑_{i=1}^{m} [a_{i}c_{i} − 4(g^{(i∗∗)})′ − 2(a_{i}c_{i} − 2)\frac{g^{(i∗∗)}}{w_{i}} + \frac{(g^{(i∗∗)})^{2}}{w_{i}} ] .  (35)

It is easily seen that the conditions in this theorem guarantee that RHS of (35) is bounded above by ∑ a_{i}c_{i}.

Using (31) and (33) and following [18, 23], we consider a class of estimators of the form

(A′_{i}A_{i})^{1/2}β̂_{i}^{Σ} = X_{[i]} − X_{<i>}γ_{<i]} + (∇_{X_{[i]}}h_{i}(l _{i})) Σ_{[i]•}
−Q_{i} ( ∑_{j=i+1}^{m} A_{j}β̂_{j}^{Σ}C_{j}) D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1} (36)

where l _{i} = (l_{i1}, …, l_{i min(a_{i},c_{i})}), the characteristic roots of (X_{[i]}−X_{<i>}γ_{<i]})′(X_{[i]}−X_{<i>}γ_{<i]})Σ_{[i]•}^{−1} and h_{i}(·) is a function from R^{min(a_{i},c_{i})} to R^{+}. Furthermore, let

h_{ik} = ∂h_{i}/∂l_{ik}, h_{ikk} = ∂^{2}h_{i}/∂l_{ik}^{2}, k = 1, 2, …, min(a_{i}, c_{i}), i = 1, 2, …, m.

Lemma 3.4. Assume that E[{tr {(∇_{X_{[i]}}h_{i})′(∇_{X_{[i]}}h_{i})}] < ∞ and E|(∂^{2}h_{i}/∂(X_{[i]})_{k_{1}l_{1}})^{2}| < ∞ and that (∇_{X_{[i]}}h_{i}(l ))_{k_{1}l_{1}} is absolutely continuous as a function of (X_{[i]})_{k_{2}l_{2}} for almost all (X_{[i]}, X_{<i>}) except (X_{[i]})_{k_{2}l_{2}}. Then, if h = (h_{1}, …, h_{m}) satisfies

∑_{i=1}^{m} ∑_{k=1}^{min(a_{i},c_{i})} \{2(max(a_{i}, c_{i}) − min(a_{i}, c_{i}) + 1)h_{ik} + 4l_{ik}h_{ikk} + 4 ∑_{j>k} \frac{l_{ik}h_{ik} − l_{ij}h_{ij}}{l_{ik} − l_{ij}}
+l_{ik}h_{ik}^{2}} ≤ 0,  (37)

the estimator β̂^{Σ} = (β̂_{1}^{Σ}, …, β̂_{m}^{Σ}) where β̂_{i}^{Σ} is given by (36) beats B̂^{Σ}. Proof. This is immediately obtained from [18, pp. 368, (35)].

Using Lemma 3.4, we may obtain a generalization of estimators given in [7, 23]. Theorem 3.5. Assume that | a_{i} − c_{i} |> 1 and put

h_{i}^{EM}(l_{i}) = −{(max(a_{i}, c_{i}) − min(a_{i}, c_{i}) − 1)/2} ∏_{j=1}^{min(a−i,c_{i})} l_{ij}

in (36). Then the estimators β̂^{ΣEM} = (β̂_{1}^{ΣEM}, …, β̂_{m}^{ΣEM}) is minimax. Proof. Proceeding as in [18], the result immediately follows from Lemma 3.4.

We demonstrate a generalization of a crude Efron-Morris type estimator and state the case when m = 2. Corollary 3.5. Assume that m = 2 and that | a_{i} − c_{i} |> 1, i = 1, 2. Then the estimators in Theorem 3.5 are expressed as

β̂_{1}^{ΣEM} = (A′_{1}A_{1})^{−1}A′_{1} [{Y Σ^{−1} − A_{1}β̂_{2}^{ΣEM}C_{1}} Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−1}
−\frac{a_{1} − c_{1} − 1}{2}Y Σ^{−1}C′_{1} \{C_{1}Σ^{−1}Y′A_{1}(A′_{1}A_{1})^{−1}A′_{1}Y Σ^{−1}C′\}^{−1} (C_{1}Σ^{−1}C′_{1})]  (38)
if a_{i} > c_{i} + 1,
β̂_{1}^{ΣEM} = (A′_{1}A_{1})^{−1}A′_{1} {Y Σ^{−1} − A_{1}β̂_{2}^{ΣEM}C_{1}} Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−1}

−\frac{c_{1} − a_{1} − 1}{2} \{(A′_{1}A_{1})^{−1}A′_{1}Y Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−2}C′_{1}Σ^{−1}Y′A_{1}(A′_{1}A_{1})^{−1}\}^{−1}

×(A′_{1}A_{1})^{−1}A′_{1}Y Σ^{−1}C′_{1} if c_{1} > a_{1} + 1,  (39)
β̂_{2}^{ΣEM} = (A′_{2}A_{2})^{−1}A′_{2} [Y Σ_{<2>}^{−1}C′_{2}(C_{2}Φ^{−1}C′_{2})^{−1} − \frac{a_{2} − c_{2} − 1}{2}Y Φ^{−1}C′_{2}
× {C_{2}Σ_{<2>}^{−1}Y′A_{2}(A′_{2}A_{2})^{−1}A′_{2}Y Φ^{−1}C′_{2}}^{−1} (C_{2}Φ^{−1}C′_{2})]  (40)

if a_{i} > c_{i} + 1,

β̂_{2}^{ΣEM} = (A′_{2}A_{2})^{−1}A′_{2}Y Φ^{−1}C′_{2}(C_{2}Φ^{−1}C′_{2})^{−1}

−\frac{c_{2} − a_{2} − 1}{2} \{(A_{2})′(A_{2})^{−1}A′_{2}Y Φ^{−1}C′_{2}(C_{2}Φ^{−1}C′_{2})^{−2}C_{2}Φ^{−1}Y A_{2}(A_{2})′(A_{2})^{−1}\}^{−1}

×(A_{2})′(A_{2})^{−1}A′_{2}Y Φ^{−1}C′_{2} if c_{1} > a_{1} + 1,  (41)

where Φ^{−1} = D_{1}(D′_{1}Σ_{<1>}^{−1}D_{1})^{−1}D′_{1} = Σ^{−1} − Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−1}C_{1}Σ^{−1}. Proof. Use (31) and (32) to see that

(∇_{X_{[i]}}h_{i}^{EM}(l_{i})) Σ_{[i]•} = (∇_{X_{[i]}}^{∗}h_{i}^{EM}(l_{i})) T′_{i}.  (42)

For a_{i} > c_{i} + 1, use [18, pp. 369, (37) and (38)] to obtain

∇_{X_{[i]}}^{∗}h_{i}^{EM}(l_{i}) = −\frac{a_{i} − c_{i} − 1}{2}(X_{[i]}^{∗} − X_{<i>}γ_{<i]}^{∗}) \{(X_{[i]}^{∗} − X_{<i>}γ_{<i]}^{∗})′(X_{[i]}^{∗} − X_{<i>}γ_{<i]}^{∗})\}^{−1} (43)

Put (43) into (42) to obtain

(∇_{X_{[i]}}h_{i}^{EM}( l_{i})) Σ_{[i]•} = −\frac{a_{i} − c_{i} − 1}{2}(X_{[i]}−X_{<i>}γ_{<i]}) \{(X_{[i]} − X_{<i>}γ_{<i]})′(X_{[i]} − X_{<i>}γ_{<i]})\}^{−1} .

Then use

(A′_{i}A_{i})^{1/2}(X_{[i]} − X_{<i>}γ_{<i]}) = (A′_{i}A_{i})^{−1}A′_{i}Y D_{0} × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}.  (44)

to obtain (38) and (40).

For c_{i} > a_{i} + 1, we may similarly obtain that

∇_{X_{[i]}}^{∗}h_{i}^{EM}(l_{i}) = −\frac{c_{i} − a_{i} − 1}{2} \{(X_{[i]}^{∗} − X_{<i>}γ_{<i]}^{∗})(X_{[i]}^{∗} − X_{<i>}γ_{<i]}^{∗})′\}^{−1} (X_{[i]}^{∗} − X_{<i>}γ_{<i]}^{∗})

from which it follows that

(∇_{X_{[i]}}h_{i}^{EM}( l_{i})) Σ_{[i]•} = −\frac{c_{i} − a_{i} − 1}{2} \{(X_{[i]} − X_{<i>}γ_{<i]})(X_{[i]} − X_{<i>}γ_{<i]})′\}^{−1}
×(X_{[i]} − X_{<i>}γ_{<i]})Σ_{[i]•}^{−1}.

Finally use (44) and Σ_{[i]•}^{−1} = C̃_{i}Σ_{[i]•}^{−1}C̃′_{i} to obtain (39) and (41).

4. ESTIMATION OF β WHEN THE COVARIANCE IS UNKNOWN In this Section, we make clear the intrinsic structure of the estimation problem for β in the model (1) and decompose the estimation problems into sum of some conditional problems, which double shrinkage estimators are derived for the first component β_{1}.

Using Theorem 2.2 we can easily see that the MLE’s for M-parameters (μ_{[i]•}, γ_{<1]}, Σ_{[i]•}), i = 1, 2, …, m, are given by

μ̂_{[i]•} = X_{[i]} − X_{<i>}S_{<i>}^{−1}S_{<i]}, γ̂_{<i]} = S_{<i>}^{−1}S_{<i]}, Σ̂_{[i]•} = S_{[i]•}.

Now using (6), (7), (11)-(13), (15) and (19), we can reconstruct the estimators (β_{1}…. , β_{m}) from the estimators for M-parameters in the following: Step 1:

(A′_{m}A_{m})^{1/2}B̂_{m} = μ̂_{[m]•} = X_{[m]} − X_{<m>}S_{<m>}^{−1}S_{<m]}
= Q_{m}Y D_{0}D_{1} × … × D_{m−1}(S_{<m−1>} + U_{<m−1>})^{−1}C̃′_{m}
×{C̃_{m}(S_{<m−1>} + U_{<m−1>})^{−1}C̃′_{m}}^{−1}

Step i (2 ≤ i ≤ m):

(A′_{m−i+1}A_{m−i+1})^{1/2}B̂_{m−i+1} = μ̂_{[m−i+1]•} − Q_{m−i+1} ( ∑_{j=m−i+2}^{m} A_{j}B̂_{j}C_{j}) D_{0}D_{1} × ⋯ × D_{m−i}

×(S_{<m−i>} + U_{<m−i>})^{−1}C̃′_{m−i+1}{C̃_{m−i+1}(S_{<m−i>} + U_{<m−i>})^{−1}C̃′_{m−i+1}}^{−1}

= X_{[m−i+1]} − X_{<m−i+1>}S_{<m−i+1>}^{−1}S_{<m−i+1]} − Q_{m−i+1} ( ∑_{j=m−i}^{m} A_{j}B̂_{j}C_{j}) D_{0}D_{1}

× ⋯ × D_{m−i}(S_{<m−i>} + U_{<m−i>})^{−1}C̃′_{m−i+1}{C̃_{m−i+1}(S_{<m−i>} + U_{<m−i>})^{−1}C̃′_{m−i+1}}^{−1}.

From this argument we can see that the MLE’s are given by

B̂_{i} = (A′_{i}A_{i})^{−1}A′_{i}(Y − ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}(S_{<i−1>} + U_{<i−1>})^{−1}C̃′_{i}
×{C̃_{i}(S_{<i−1>} + U_{<i−1>})^{−1}C̃′_{i}}^{−1}.

Now we demonstrate a decomposition of the MLE. To this end, let

X̃_{<i>} : a_{i} × (p − c_{i}^{+}) = Q_{i}(Y − ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i}
= X_{<i>} − Q_{i}( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i}  (45)

and

U^{(i)} = (U_{<i>}, U_{<i+1>}, …, U_{<m>}).  (46)

From (27) we have that

Q_{i}( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}

= Q_{i}( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}C̃′_{i}(C̃_{i}C̃′_{i})^{−1}

−Q_{i}( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i}γ_{<i]}.  (47)

From (6), (7), (45) and (47), a decomposition of the MLE B̂_{i} can be made with

(A′_{i}A_{i})^{1/2}B̂_{i} = Q_{i}(Y − ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}

×[I − D_{i}{D′_{i}(S_{<i−1>} + U_{<1−1>})D_{i}}^{−1}D′_{i}(S_{<i−1>} + U_{<1−1>})]C̃′_{i}(C̃_{i}C̃′_{i})^{−1}

= Q_{i}(Y − ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}

× [C̃′_{i}(C̃_{i}C̃′_{i})^{−1} − D_{i}γ_{<i]} − D_{i}V_{<i]} + D_{i}γ_{<i]}]
= X_{[i]} − X_{<i>}γ_{<i]} − X̃_{<i>}(V_{<i]} − γ_{<i]})
−Q_{i}( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}. (48)

Here note that, given (X_{<i>}, S_{<i>}, U^{(i)}),

X_{[i]} ∼ N(μ_{[i]•} + X_{<i>}γ_{<i]}, I_{a_{i}} ⊗ Σ_{[i]•})
X̃_{<i>}(V_{<i]} − γ_{<i]}) ∼ N(0, T̃_{i} ⊗ Σ_{[i]•}),

and X_{[i]} and X̃_{<i>}(V_{<i]} − γ_{<i]}) are independent, where T̃_{i} : a_{i} × a_{i} = X̃_{<i>}S_{<i>}^{−1}X̃′_{<i>}. Next we evaluate the conditional risk of MLE. To this end, define

(A′_{i}A_{i})^{1/2}β_{i}^{∗} = μ_{[i]•} − Q_{i}( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}.

Recall (11) - (13) and use (27) to obtain

E[(A′_{i}A_{i})^{1/2}β_{i}^{∗}] = μ_{[i]•} − Q_{i}( ∑_{j=i+1}^{m} A_{j}E[B̂_{j}]C_{j})D_{0}D_{1}
× ⋯ × D_{i−1}Σ_{<i−1>}^{−1}C̃′_{i}(C̃_{i}Σ_{<i−1>}^{−1}C̃′_{i})^{−1}
= (A′_{i}A_{i})^{1/2}β_{i}.

The last equality follows from the fact E[B̂_{i}] = β_{i}. Hence, given (X_{<i>}, S_{<i>}, U^{(i)}), the MLE B̂_{i} is conditionally distributed as

B̂_{i} ∼ N((A′_{i}A_{i})^{1/2}β_{i}^{∗}, (I + T̃_{i}) ⊗ Σ_{[i]•}),

and the risk function of the MLE is evaluated as

R(B̂, β) = ∑_{i=1}^{m} tr {E[(X_{[i]} − X_{<i>}γ_{<i]} − μ_{[i]•})Σ_{[i]•}^{−1}(X_{[i]} − X_{<i>}γ_{<i]} − μ_{[i]•})′]
+E[X̃_{<i>}(V_{<i]} − γ_{<i]})Σ_{[i]•}^{−1}(V_{<i]} − γ_{<i]})′X̃′_{<i>}]}
= ∑_{i=1}^{m} {a_{i}c_{i} + a_{i}E[tr T̃_{i}]} .

In order to decompose the estimation problem, we put W_{i} = [ X′_{[i]}, (V_{<i]}^{0})′ ]′ where

V_{<i]}^{0} = S_{<i>}^{1/2}V_{<i]}, J_{i}^{∗∗} = [I, −I], J_{i}^{∗} = [ \begin{matrix}I & 0 \\ 0 & X̃_{<i>}^{∗} \end{matrix} ] , X̃_{<i>}^{∗} = X̃_{<i>}S_{<i>}^{−1/2},

and we view the MLE’s B̂ as

(A′_{i}A_{i})^{1/2}B̂_{i} = J_{i}^{∗∗}J_{i}^{∗}W_{i} − Q_{i} ( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j}) D_{0}D_{1} × ⋯ × D_{i−1}C̃′_{i}(C̃_{i}C̃′_{i})^{−1}.  (49)

Note that conditionally on (X̃_{<i>}, S_{<i>}, U^{(i)}), we have

W_{i} ∼ N(ξ_{i}, I_{a_{i}+(p−c_{i}^{+})} ⊗ Σ_{[i]•}),  (50)

where

ξ_{i} = [ \begin{matrix}η_{[i]} \\ δ_{<i]} \end{matrix} ] = [ \begin{matrix}μ_{[i]•} + X_{<i>}γ_{<i]} \\ S_{<i>}^{1/2}γ_{<i]} \end{matrix} ] .  (51)

Because of complex nature of the MLE, we consider the estimators which modify B̂_{1} only. Problem I. We look at the estimation problem of ξ_{1} based on (W_{1}, S_{[1]•}). Consider a class

of estimators of the form

ξ̂_{1} = W_{1} + G_{1} with [ \begin{matrix}G_{[1]}(X_{[1]}, S_{[1]•}) \\ G_{<1]}(V_{<1]}^{0}, S_{[1]•} | X̃_{<1>}, S_{<1>}) \end{matrix} ]  (52)

as an estimator of ξ_{1} and assume that

G_{[1]} does not depend on X̃_{<1>} and G_{<1]}(· | X̃_{<1>}, S_{<1>}) = G_{<1]}(· | −X̃_{<1>}, S_{<1>}). (53)

Let

R_{1}^{c}[ξ̂_{1}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})] = E[(ξ̂_{1} − ξ_{1})Σ_{[1]•}^{−1}(ξ̂_{1} − ξ_{1})′ | (X̃_{<1>}, S_{<1>}, U^{(1)})] (54)

be the conditional risk matrix of ξ̂_{1}, where U^{(1)} is given in (46). Here

R_{1}^{c}[ξ̂_{1}^{∗}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})] ≤ R_{1}^{c}[ξ̂_{1}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})]

means

R_{1}^{c}[ξ̂_{1}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})] − R_{1}^{c}[ξ̂_{1}^{∗}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})]

is positive semidefinite matrix. Theorem 4.1. Assume that (53). If, given (X̃_{<1>}, S_{<1>}, U^{(1)}),

tr [J_{1}^{∗∗}J_{1}^{∗}R_{1}^{c}[ξ̂_{1}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})](J_{1}^{∗})′(J_{1}^{∗∗})′]
≤ tr [J_{1}^{∗∗}J_{1}^{∗}R_{1}^{c}[W_{1}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})](J_{1}^{∗})′(J_{1}^{∗∗})′] , a.e

holds uniformly in (ξ_{1}, Σ_{[1]•}), then the risk function R(β̂, β) is uniformly smaller than the risk function R(B̂, β) where β̂ = (β̂_{1}, B̂_{2}, …, B̂_{m}) with

(A′_{1}A_{1})^{1/2}β̂_{1} = (A′_{1}A_{1})^{1/2}B̂_{1} + G_{[1]}(X_{[1]}, S_{[1]•}) − X̃_{<1>}^{∗}G_{<1]}(V_{<1]}^{0}, S_{[1]•} | X̃_{<1>}, S_{<1>})

−Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) C′_{1}(C_{1}C′_{1})^{−1}

= J_{1}^{∗∗}J_{1}^{∗}ξ̂_{1} − Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) C′_{1}(C_{1}C′_{1})^{−1}  (55)

and B̂ = (B̂_{1}, …, B̂_{m}). Proof. In the light of (16), we have that the risk difference R(β̂, β) − R(B̂, β) is rewritten as

E (tr [{(A′_{1}A_{1})^{1/2}β̂_{1} + Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−1} − μ_{[1]•}}Σ_{[1]•}^{−1}{ ⋯ }′]

−tr [{(A′_{1}A_{1})^{1/2}B̂_{1} + Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−1} − μ_{[1]•}}Σ_{[1]•}^{−1}{ ⋯ }′]) .

From (48), we have

(A′_{1}A_{1})^{1/2}B̂_{1} + Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) Σ^{−1}C′_{1}(C_{1}Σ^{−1}C′_{1})^{−1} − μ_{[1]•}

= X_{[1]} − X_{<1>}γ_{<1>} − X̃_{<1>}^{∗}(V_{<1]}^{0} − δ_{<1]}) − μ_{[1]•}

= X_{[1]} − {μ_{[1]•} + X_{<1>}γ_{<1]}} − X̃_{<1>}^{∗}(V_{<1]}^{0} − δ_{<1]}).  (56)

From this and (55), we can get the assertion of Theorem.

Following is the immediate consequence of Theorem 4.1. Corollary 4.2. Assume that (53). If, given (X̃_{<1>}, S_{<1>}, U^{(1)}),

R_{1}^{c}[ξ̂_{1}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})] ≤ R_{1}^{c}[W_{1}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})], a.e

holds uniformly in (ξ_{1}, Σ_{[1]•}), then the risk function R(β̂, β) is uniformly smaller than the risk function R(B̂, β).

Lemma 4.3. If the assumption (53) holds, the the cross term of the risk matrix is zero:

E [(X_{[1]} + G_{[1]} − η_{[1]})Σ_{[1]•}^{−1}(V_{<1]}^{0} + G_{<1]} − δ_{<1]})′(X̃_{<1>}^{∗})′]] = 0.

Proof. Note that X̃_{<1>}^{∗} depends only on (X̃_{<1>}, S_{<1>}, U^{(1)}) and that

E[X̃_{[1]} − η_{[1]} | (X̃_{<1>}, S_{<1>})] = 0, E[V_{<1]}^{0} − δ_{<1]} | (X̃_{<1>}, S_{<1>})] = 0

and X̃_{[1]} − η_{[1]} and V_{<1]}^{0} − δ_{<1]} are independent. Also note that

E[X̃_{<i>}] = Q_{i}(E[Y] − ∑_{j=i+1}^{m} A_{j}E[B̂_{j}]C_{j})D_{0}D_{1} × ⋯ × D_{i}
= Q_{1}( ∑_{j=1}^{i} A_{j}β_{j}C_{j})D_{0}D_{1} × ⋯ × D_{i} = 0, i = 1, 2, …, m

since E[B̂_{j}] = β_{j}. Hence the term E[G_{[1]}Σ_{[1]•}^{−1}(G_{<1]})′(X̃_{<1>}^{∗})′] is zero because −X̃_{<1>} have the same unconditional distribution as X̃_{<1>} with E[X̃_{<1>}] = 0 and it is an odd function of X̃_{<1>}. Proceeding the proof of [9, Lemma 2.1] by replacing X_{1}, X_{2}, C_{0}, η, δ, Σ_{11.2} by X_{[1]}, X̃_{<1>}, V_{<1]}^{0}, η_{[1]}, δ_{<1]}, Σ_{[1]•}, then we can get the desired result.

By Lemma 4.3, the conditional risk matrix (54) is expressed as

R_{1}^{c}[ξ̂_{1}, (ξ_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})]
= R_{11}^{c}[η̂_{[1]}, (η_{[1]}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})]
+X̃_{<1>}^{∗}R_{12}^{c}[δ̂_{<1]}, (δ_{<1]}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})](X̃_{<1>}^{∗})′.

Here we define

R_{11}^{c}[η̂_{[1]}, (η_{[1]}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})] = E[(η̂_{[1]} − η_{[1]})Σ_{[1]•}^{−1}(⋯)′ | (X̃_{<1>}, S_{<1>}, U^{(1)})(57)

R_{12}^{c}[δ̂_{<1]}, (δ_{<1]}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})]

= E[(δ̂_{<1]} − δ_{<1]})Σ_{[1]•}^{−1}(⋯)′ | (X̃_{<1>}, S_{<1>}, U^{(1)})]  (58)

where η̂_{[1]} = X_{[1]} + G_{[1]} and δ̂_{<1]} = V_{<1]}^{0} + G_{<1]}. Theorem 4.4. The conditional problem (50)-(52) of estimating ξ_{1} is decomposed into the two independent conditional problems of estimating η_{[1]} by η̂_{[1]} under (57) and of estimating δ_{<1]} by δ̂_{<1]} under (58).

Hence we need to find G_{[1]} for which

tr R_{11}^{c}[η̂_{[1]}, (η_{[1]}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})] ≤ tr R_{11}^{c}[X_{[1]}, (η_{1}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})] (59) and G_{<1]} for which

tr {((X̃_{<1>}^{∗})′X̃_{<1>}^{∗})R_{12}^{c}[δ̂_{<1]}, (δ_{<1]}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})]}
≤ tr {((X̃_{<1>}^{∗})′X̃_{<1>}^{∗})R_{12}^{c}[V_{<1]}^{0}, (δ_{<1]}, Σ_{[1]•}) | (X̃_{<1>}, S_{<1>}, U^{(1)})]},  (60)

respectively. As in GMANOVA estimation problem, we can enjoy the double shrinkage effects: one is the mean part of X_{[1]} and the other part is the correlation of X_{[1]} and X_{<1>}.

Problem II. The second shrinkage problem suggested by the structure (49)-(51) is based on the conditional model

J_{i}^{∗}W_{i} = [ \begin{matrix}X_{[i]} \\ X̃_{<i>}V_{<i]} \end{matrix} ] ∼ N ([ \begin{matrix}μ_{[i]•} + X_{<i>}γ_{<i]} \\ X̃_{<i>}γ_{<i]} \end{matrix} ] , ( \begin{matrix}I & 0 \\ 0 & T̃_{i} \end{matrix} ) ⊗ Σ_{[i]•}) , i = 1, 2, …, m.

(61)

Since a_{i} > p − c_{i}^{+} the model (61) reduces to the model which is equivalent to the model (50) (see [9, pp. 249]), we may assume that a_{i} ≤ p − c_{i}^{+} and we put

W̃_{i} = ( X_{[i]} ; Ṽ_{[i]} ) and Ṽ_{[i]} = T̃_{i}^{−1/2}X̃_{<i>}V_{<i]}

and we view the MLE B_{i} as

(A′_{i}A_{i})^{1/2}B_{i} = [I, −T̃_{[i]}^{1/2}] ( X_{[i]} ; Ṽ_{<i]}^{∗} ) − Q_{i} ( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j}) D_{0}D_{1} × ⋯ × D_{i−1}C′_{i}(C_{i}C′_{i})^{−1}
= J_{i}^{∗∗∗}Z_{i}^{∗} − Q_{i} ( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j}) D_{0}D_{1} × ⋯ × D_{i−1}C′_{i}(C_{i}C′_{i})^{−1},  (62)

where J_{i}^{∗∗∗} = [I, −T̃_{[i]}^{1/2}]. Then conditional on (X̃_{<i>}, S_{<i>}, U^{(i)}),

W̃_{i} ∼ N(ν_{i}, I_{2a_{i}} ⊗ Σ_{[i]•}) with ν_{i} = ( \begin{matrix}μ_{[i]•} + X̃_{<i>}γ_{<i]} \\ T̃_{i}^{−1/2}X̃_{<i>}γ_{<i]} \end{matrix} ) .  (63)

To modify B̂_{1}, let

ν̂_{1} = W̃_{1} + K_{1} with K_{1} = ( \begin{matrix}G_{[1]}(X_{[1]}, S_{[1]•}) \\ K_{[1]}(Ṽ_{[1]}, S_{[1]•} | X̃_{<1>}, S_{<1>}) \end{matrix} )  (64)

and

R_{2}^{c}[(ν̂_{1}, (ν_{1}, Σ_{[i]•}) | (X̃_{<i>}, S_{<i>}, U^{(i)})] = E[(ν̂_{1} − ν_{1})Σ_{[1]•}^{−1}(ν̂_{1} − ν_{1})′ | (X̃_{<i>}, S_{<i>}, U^{(i)})].

From (62) and J_{i}^{∗∗∗}ν_{i} = β_{i}, it follows that

R(β̂, β) − R(B̂, β) = tr E[J_{1}^{∗∗∗}R^{c}[(ν̂_{1}, (ν_{1}, Σ_{[i]•}) | (X̃_{<i>}, S_{<i>}, U^{(i)})(J_{1}^{∗∗∗})′]
−tr E[J_{1}^{∗∗∗}R^{c}(W̃_{1}, (ν_{1}, Σ_{[i]•}) | (X̃_{<i>}, S_{<i>}, U^{(i)})(J_{1}^{∗∗∗})′],

where β̂ = (β̂_{1}, B̂_{2}, …, B̂_{m}) with

(A′_{1}A_{1})^{1/2}β̂_{1} = (A′_{1}A_{1})^{1/2}B_{1} + K_{[1]}(X_{[1]}, S_{[1]•}) − T̃_{1}^{1/2}K_{[1]}(Ṽ_{[1]}, S_{[1]•} | X̃_{<1>}, S_{<1>})
= J_{1}^{∗∗∗}ν̂_{1} + K_{[1]}(X_{[1]}, S_{[1]•}) − T̃_{1}^{1/2}K_{[1]}(Ṽ_{[1]}, S_{[1]•} | X̃_{<1>}, S_{<1>})
−Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) C′_{1}(C_{1}C′_{1})^{−1}  (65)

and B̂ = (B̂_{1}, …, B̂_{m}). If K_{[1]} satisfies

K_{[1]}(−Ṽ_{[1]}, S_{[1]•} | −X̃_{<1>}, S_{<2>}) = −K_{[1]}(Ṽ_{[1]}, S_{[1]•} | X̃_{<1>}, S_{<2>}),

this problem boils down to Problem I because T̃_{i}^{1/2}K_{[1]} is regarded as a special case of X̃_{<1>}^{1/2}G_{<1]}.

Problem III. The third shrinkage problem is based on

(A′_{i}A_{i})^{1/2}B̂_{i} = J_{i}^{∗∗}J_{i}^{∗}W_{i} − Q_{i} ( ∑_{j=i+1}^{m} A_{j}B̂_{j}C_{j}) D_{0}D_{1} × ⋯ × D_{i−1}C′_{i}(C_{i}C′_{i})^{−1}.

In the view of the fact that, given (X̃_{<i>}, S_{<i>}, U^{(i)}),

J_{i}^{∗∗}J_{i}^{∗}W_{i} ∼ N(μ_{[i]•} + (X_{<1>} − X̃_{<1>})γ_{<1]}, (I + T̃_{i}) ⊗ Σ_{[i]•}),  (66)

we may consider a class of estimators of the form

(A′_{1}A_{1})^{1/2}β̂_{1} = (A′_{1}A_{1})^{1/2}B̂_{1} + L_{1}(X_{[1]} − X_{<1>}V_{<1]}, I + T̃_{1}, S_{[1]•}).
= J_{1}^{∗∗}J_{1}^{∗}W_{1} + L_{1}(X_{[1]} − X̃_{<1>}V_{<1]}, I + T̃_{1}, S_{[1]•})
−Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) C′_{1}(C_{1}C′_{1})^{−1}  (67)

Then the risk difference between β̂ = (β̂_{1}, B̂_{2}, …, B̂_{m}) and B̂ = (B̂_{1}, B̂_{2}, …, B̂_{m}) is expressed as

R(β̂, β) − R(B̂, β) = E [ tr {(X_{[1]} − X̃_{<1>}V_{<1]} + L_{1} − {μ_{[1]•} + (X_{<1>} − X̃_{<1>})γ_{<1]}})Σ_{[1]•}^{−1}( ⋯)′}]

−E [ tr {(X_{[1]} − X̃_{<1>}V_{<1]} − {μ_{[1]•} + (X_{<1>} − X̃_{<1>})γ_{<1]}})Σ_{[1]•}^{−1}( ⋯)′}] .

Hence we need to find L_{1}’s for which, given (X̃_{<1>}, S_{<1>}, U^{(1)}),

E^{c} [ tr {(X_{[1]} − X̃_{<1>}V_{<1]} + L_{1} − {μ_{[1]•} + (X_{<1>} − X̃_{<1>})γ_{<1]}})
×Σ_{[1]•}^{−1}( ⋯)′} |(X̃_{<1>}, S_{<1>}, U^{(1)})]
≤ E^{c} [ tr {(X_{[1]} − X̃_{<1>}V_{<1]} − {μ_{[1]•} + (X_{<1>} − X̃_{<1>})γ_{<1]}})
×Σ_{[1]•}^{−1}( ⋯)′} |(X̃_{<1>}, S_{<1>}, U^{(1)})] .

Now applying the result in [9] we demonstrate how to derive minimax estimators of β from the conditional problems since given (X̃_{<1>}, S_{<1>}, U^{(1)}), each of conditional problems can be reduced to the GMANOVA problem by replacing (X_{1}, X_{2}, S_{11.2}, C, S_{22}) and (β+X_{2}γ, γ, Σ_{11.2}) in [9] with (X_{[1]}, X̃_{<1>}, S_{[1]•}, V_{<1]}, S_{<1>}) and (μ_{[1]•} + X_{<1>}γ_{<1]}, γ_{<1]}, Σ_{[1]•}).

Problem I. The estimation problem summarized by (52) and (54) corresponds to the problem described by [9, (2.11a) and (2.11b)]. Hence we derive (η̂_{[1]}, δ̂_{<1]}) for which (59) and (60) hold uniformly. Let

F_{[1]} = X_{[1]}S_{[1]•}^{−1}X′_{[1]} = (f_{ij}^{[1]})
F_{<1>} = V_{<1]}^{0}S_{[1]•}^{−1}(V_{<1]}^{0})′ = (f_{ij}^{<1>})

and

F̃_{k} = diag(f_{11}^{k}, …, f_{r_{k}r_{k}}^{k}) (k = [1], < 1 >)

with r_{[1]} = c_{1} and r_{<1>} = p − c_{1}. Further, let

M_{k} = diag(h_{1}^{k}(f_{11}^{k}), …, h_{r_{k}}^{k}(f_{r_{k}r_{k}}^{k})) (k = [1], < 1 >)

and h_{j}^{(k)} satisfy

0 ≤ h_{j}^{[1]} ≤ \frac{2(c_{1} − 2)}{a_{1}(n − a_{1} − p + 3) + 2a_{1}^{2}}
0 ≤ h_{j}^{<1>} ≤ \frac{2(c_{1} − 2)}{(p − c_{1})(n − a_{1} − p + 3) + 2(p − c_{1})^{2}}
0 ≤ \frac{∂h_{j}^{k}}{∂f_{jj}^{k}} ≤ \frac{2}{∂f_{jj}^{k}} , j = 1, 2, …, r_{k}, and k = [1], < 1 > .

Theorem 4.5. Let c_{1} ≥ 3 and let

(A′_{1}A_{1})^{1/2}β̂_{1} = (A′_{1}A_{1})^{1/2}B̂_{1} + G_{[1]} − X̃_{<1>}^{∗}G_{<1]} − Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) C′_{1}(C_{1}C′_{1})^{−1} (68)

where

G_{[1]} = −M_{[1]}F̃_{[1]}^{−1}X_{[1]} and G_{<1]} = −M_{<1>}F̃_{<1>}^{−1}V_{<1]}^{0}.

Then the estimator β̂ = (β̂_{1}, B̂_{2}, …, B̂_{m}) is minimax.

Let w^{k} = tr (F_{k}) and let g^{k}(w^{k}, S_{[1]•}) satisfy

0 ≤ g^{k} ≤ \frac{2(a_{1}c_{1} − 2)}{n − a_{1} − p + 3}

∂g^{k}(w^{k}, S_{[1]•})/∂w^{k} ≥ 0

((1/(2 − δ_{ij}))∂g^{k}(w^{k}, S_{[1]•})/∂(S_{[1]•})_{ij}) is nonnegative definite

for k = [1], < 1 > and a Kronecker’s delta δ_{ij}. Theorem 4.6. Assume that a_{1}c_{1} ≥ 3 and c_{1}(p − c_{1}) ≥ 3. Let β̂_{1} be of the form (68) with

G_{[1]} = −(1/w^{[1]})g^{[1]}(w^{[1]}, S_{[1]•})_{[1]}X_{[1]} and G_{<1]} = −(1/w^{<1>})g^{<1>}(w^{<1>}, S_{[1]•})V_{<1]}^{0}. Then the estimator β̂ = (β̂_{1}, B̂_{2}, …, B̂_{m}) is minimax.

Problem II. To estimate ν_{i} in (63), let a_{1} ≤ p − c_{1} and put

F_{1} = T̃_{1}^{−1/2}X̃_{<1>}V_{<1]}S_{[1]•}^{−1}V′_{<1]}X̃′_{<1>}T̃_{1}^{−1/2} = (f_{ij}^{(1)}).

Further define

F̃_{1} = diag(f_{11}^{(1)}, …, f_{a_{1}a_{1}}^{(1)}) and M_{1} = diag(h_{1}^{(1)}(f_{11}^{(1)}), …, h_{a_{1}}^{(1)}(f_{a_{1}a_{1}}^{(1)}))

with

0 ≤ h_{j}^{(1)} ≤ \frac{2(a_{1} − 2)}{c_{1}(n − a_{1} − p + 3) + 2(p − c_{1})^{2}}
0 ≤ \frac{∂h_{j}^{(1)}}{∂f_{jj}^{(1)}} ≤ \frac{2}{∂f_{jj}^{(1)}} , j = 1, 2, …, a_{1}.

Theorem 4.7. Let c_{1} ≥ 3 and a_{1} ≤ p − c_{1}. Let

(A′_{1}A_{1})^{1/2}β̂_{1} = (A′_{1}A_{1})^{1/2}B̂_{1} + G_{[1]} − T̃_{1}^{1/2}K_{[1]} − Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) C′_{1}(C_{1}C′_{1})^{−1}  (69)

where

G_{[1]} = −M_{[1]}F̃_{[1]}^{−1}X_{[1]} and K_{[1]} = −M_{1}(F̃_{1}^{(1)})^{−1}T̃_{1}^{−1/2}X̃_{<1>}V_{<1]}.

Then the estimator β̂ = (β̂_{1}, B̂_{2}, …, B̂_{m}) is minimax.

Let w^{(1)} = tr (F_{1}) and let g^{(1)}(w^{(1)}, S_{[1]•}) satisfy

0 ≤ g^{(1)} ≤ \frac{2(a_{1}c_{1} − 2)}{n − a_{1} − p + 3}

∂g^{(1)}(w^{(1)}, S_{[1]•})/∂w^{(1)} ≥ 0

((1/(2 − δ_{ij}))∂g^{(1)}(w^{(1)}, S_{[1]•})/∂(S_{[1]•})_{ij}) is nonnegative definite.

Theorem 4.8. Let a_{1}c_{1} ≥ 3 and a_{1} ≤ p − c_{1}. Let β̂_{1} be of the form (69) with G_{[1]} = −(1/w^{[1]})g^{[1]}(w^{[1]}, S_{[1]•})_{[1]}X_{[1]} and K_{[1]} = −(1/w^{(1)})g^{(1)}(w^{(1)}, S_{[1]•})T̃_{1}^{−1/2}X̃_{<1>}V_{<1]}. Then the estimator β̂ = (β̂_{1}, B̂_{2}, …, B̂_{m}) is minimax.

Problem III. To estimate μ_{[1]•} + (X_{<1>} − X̃_{<1>})γ_{<1]} in (66), let

F_{1∗} = (I + T̃_{1})^{−1/2}(X_{[1]} − X̃_{<1>}V_{<1]})S_{[1]•}^{−1}(X_{[1]} − X̃_{<1>}V_{<1]})′(I + T̃_{1})^{−1/2} = (f_{ij}^{(1∗)}). Further define

F̃_{1∗} = diag(f_{11}^{(1∗)}, …, f_{a_{1}a_{1}}^{(1∗)}) and M_{1∗} = diag(h_{11}^{(1∗)}(f_{11}^{(1∗)}), …, h_{a_{1}a_{1}}^{(1∗)}(f_{a_{1}a_{1}}^{(1∗)}) with

0 ≤ h_{j}^{(1∗)} ≤ \frac{2(c_{1} − 2)}{a_{1}(n − a_{1} − p + 3) + 2a_{1}^{2}}
0 ≤ \frac{∂h_{j}^{(1∗)}}{∂f_{jj}^{(1∗)}} ≤ \frac{2}{∂f_{jj}^{(1∗)}} , j = 1, 2, …, c_{1}.

Theorem 4.9. Let c_{1} ≥ 3 and let

(A′_{1}A_{1})^{1/2}β̂_{1} = (A′_{1}A_{1})^{1/2}B̂_{1} + (I + T̃_{1})^{1/2}L_{1} − Q_{1} (∑_{j=2}^{m} A_{j}B̂_{j}C_{j}) C′_{1}(C_{1}C′_{1})^{−1}  (70)

where

L_{1} = −M_{1∗}F̃_{1∗}^{−1}(I + T̃_{1})^{−1/2}(X_{[1]} − X̃_{<1>}V_{<1]}).

Then the estimator β̂ = (β̂_{1}, B̂_{2}, …, B̂_{m}) is minimax.

Let w^{(1∗)} = tr (F_{1∗}) and let g^{(1∗)}(w^{(1∗)}, S_{[1]•}) satisfy

0 ≤ g^{(1∗)} ≤ \frac{2(a_{1}c_{1} − 2)}{n − a_{1} − p + 3}

∂g^{(1∗)}(w^{(1∗)}, S_{[1]•})/∂w^{(1∗)} ≥ 0

((1/(2 − δ_{ij}))∂g^{(1∗)}(w^{(1∗)}, S_{[1]•})/∂(S_{[1]•})_{ij}) is nonnegative definite.

Theorem 4.10. Let a_{1}c_{1} ≥ 3 and let β̂_{1} be of the form (70) with

L_{1} = −(1/w^{(1∗)})g^{(1∗)}(w^{(1∗)}, S_{[1]•})(X_{[1]} − X̃_{<1>}V_{<1]}).

Then the estimator β̂ = (β̂_{1}, B̂_{2}, …, B̂_{m}) is minimax.

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