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Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Sankhya Ser A. Author manuscript; available in PMC 2010 December 24.
Published in final edited form as:
Sankhya Ser A. 2009; 71(1): 73–93.
PMCID: PMC3010272
NIHMSID: NIHMS175583

Unbiased Invariant Least Squares Estimation in A Generalized Growth Curve Model

SUMMARY

This paper is concerned with a generalized growth curve model. We derive the unbiased invariant least squares estimators of the linear functions of variance-covariance matrix of disturbances. Under the minimum variance criterion, we obtain the necessary and sufficient conditions of the proposed estimators to be optimal. Simulation studies show that the proposed estimators perform well.

Keywords: Generalized growth curve model, least squares estimation, maximum likelihood estimation, minimum norm estimation, optimality

1 INTRODUCTION

Consider the generalized growth curve model (GGCM)

Y=i=1mXiBiZi+U,
(1.1)

where Xi, Zi and U(≠ 0) are, respectively, known n × ki, p × li, n × s matrices and Bi is an unknown ki × li matrix of regression coefficients for i = 1, …, m; Y is an n × p matrix of observations; x2130 = (ε1, …, εs)′ is an s × p error matrix, ε1, …, εs are independent and identically distributed with E(ε1)=0,E(ε1ε1)=Σ.

There is a substantial literature on the inference for growth curve model, a simple version of (1.1) with m = 1, which was proposed by Potthoff and Roy (1964) and subsequently studied by numerous authors like Rao (1965, 1987), Lee (1988), and Lin and Lee (2003). See von Rosen (1991), Kshirsagar and Smith (1995), Pan and Fang (2002) and Kollo and von Rosen (2005) for a comprehensive survey on the growth curve model. The standard growth curve model actually requires that all treatment effects have the same profile which may not be true in practice. In an experiment analysis on grazing animals and perennial plants where the interest is the behavior of treatment effects over time, Evans and Roberts (1979) considered a GGCM with the form similar to (1.1). Later, von Rosen (1984) and Verbyla and Venables (1988) studied estimation problem of the parameters in Model (1.1), and demonstrated wide applicability of Model (1.1) through some examples such as the longitudinal data from designed experiments which is a common practice in biological and medical researches. See also Hamid and von Rosen (2006) for an interesting example on the model. In addition, Model (1.1) includes the MANOVA-GMANOVA model (Chinchilli and Elswick, 1985) as a special case.

In the literature of inference for the GGCM, a few efforts have been made for the estimations of the variance-covariance matrix Σ and its linear functions tr(CΣ) where C is a given symmetric matrix, although the corresponding investigation for the simpler and commonly used models has been conducted extensively. For example, for univariate linear models, a famous result is Hsu’s (1938) theorem which gives the necessary and sufficient conditions under which the usual unbiased estimator of error variance is uniformly minimum variance invariant quadratic unbiased estimator (UMVIQUE). For multivariate linear models, Kleffe (1979) derived an unbiased invariant minimum norm quadratic estimator (MINQE(U,I)) of the linear functions of the variance-covariance matrix, tr(CΣ), and found the conditions for the MINQE(U,I) to be UMVIQUE. His result shows that Hsu’s result continues to hold in multivariate linear models. Yang (1995) studied the topic similar to Kleffe’s for the growth curve models. He obtained the MINQE(U,I) of tr(CΣ) and considered the optimality of the proposed estimator. For the GGCM considered in this paper, the analysis is more difficult. Assuming the restriction condition R(Xm) [subset, dbl equals] (...) [subset, dbl equals] R(X1) (where R(Xi) denotes the range space of the matrix Xi, i = 1, …, m), von Rosen (1989) obtained the maximum likelihood estimator (MLE) of Σ under the assumptions of normal errors and U being an identity matrix. It is worthy pointing out that the implementation of von Rosen’s estimator is not very easy because there is no closed form and so the computation is somewhat cumbersome. Also, the normality of disturbances is required for MLE. Motivated by these facts, Wu et al. (2006) studied Model (1.1) by deriving an estimator of tr(CΣ) with closed form. They proved that the estimator is an MINQE(U,I) without the normal assumption. In addition, they also obtained the necessary and sufficient conditions for the MINQE(U,I) to be the UMVIQUE. However, although Wu’s et al. (2006) MINQE(U,I) is of closed form, its computational burden is still heavy, especially for the large sample size because it involves the calculation of generalized inverse of matrix.

Another classical approach to deal with the estimation problem of Σ and tr(CΣ) is the least squares. The resulting estimators may be calculated easily. A discussion on the least squares approach in multivariate analysis can be found in Fang et al. (2006). For the classical growth curve model, the least squares estimators (LSEs) were derived in Xu and Yang (1983) and Yang and Xu (1985). However, for the GGCM, the derivation is by no means straightforward, even under the above nested condition. Especially, the analysis on the optimality would be quite complicated, although Seely and Zyskind (1971) established a useful result to assess the optimality of quadratic estimators. This can be seen from the following Theorem 3. The reason is that it is difficult to find the necessary and sufficient conditions for Seely and Zyskind’s (1971) theorem to hold. In this paper, we overcome the difficulty by using spectral decomposition. Our results can be regarded as the generalization of Hsu’s theorem in the GGCM.

The main objective of this paper is to study the LSE of tr(CΣ) under Model (1.1). We are interested in the unbiased invariance and optimality of the LSE under appropriate assumptions. This paper is organized as follows. In Section 2, we give the explicit expressions of the LSE and the unbiased invariant LSE (LSE(U,I)) of tr(CΣ). A necessary and sufficient condition for the LSE(U,I) of tr(CΣ) to be UMVIQUE is derived in Section 3. In Section 4, we present the results from simulation studies which make a comparison among our LSE(U, I) and the existing MINQE(U, I) and MLE of tr(CΣ) under both normal and non-normal assumptions. Some concluding remarks are given in Section 5. Lemmas and their proofs are provided in the Appendix.

2 LEAST SQUARES ESTIMATION

In this section, we derive the LSE of the linear function tr(CΣ) for any given symmetric matrix C, and then establish statistical properties for the proposed estimator. In what follows, we assume that R(Xm) [subset, dbl equals] (...) [subset, dbl equals] R(X1). This restriction was first imposed by von Rosen (1989) to derive the MLE of Σ and to establish associated statistical properties. The restriction can be checked readily. For example, as the design matrices often just consist of 0s and 1s, the nested condition can be checked by using a projection operator. On the other hand, we will sometimes know immediately the nested condition from the problems of interest. As an example, if we have two groups of individuals both having the linear growth in common and one group having additionally a quadratic term, then the nested condition is automatically met (see also Hamid and von Rosen, 2006; and Wu et al., 2006).

Let y=Vec(Y),X=(X1Z1,,XmZm),β=((Vec(B1)),,((Vec(Bm))),ε=Vec(), where Vec(·) creates a column vector by stacking the column vectors of below one another. Then model (1.1) becomes

y=Xβ+(UIp)ε,

where Ip is a p × p identity matrix.

Denote the unknown matrix E(ε1ε1ε1ε1) by Ψ, and assume that Ψ is finite. Write MA = IAA+ for the matrix A, where A+ is the Moore-Penrose inverse of A,M = MX, and G = UU′.

Then the induced linear model becomes

{MyyM,E(MyyM),(MyyM)},

where E(Myy′M) = M(G [multiply sign in circle] Σ)M, ћ(Myy′M) · MAM = 2M(G [multiply sign in circle] Σ)MAM(G [multiply sign in circle] Σ)M + M(U [multiply sign in circle] Ip)diag{SHk(Σ, Ψ)}(U[multiply sign in circle]Ip)M, A is an np × np symmetric matrix, and diag{SHk(Σ, Ψ)} is a block diagonal matrix of the form diag(SH1, …, SHs) with

SHk(Σ,Ψ)=i=1pj=1pe(i)Hke(j)(e(i)Ip)Ψ(e(j)Ip)2ΣHkΣtr(HkΣ)Σ,Hk=(ekIp)(UIp)MAM(UIp)(ekIp),

ei and e(j) are the ith and jth columns of the s × s and p × p identity matrices, respectively. ћ(Myy′M) is an operator in {MAM : A = A′} which satisfies that

tr[MAM{(MyyM)MBM}]=cov(yMAMy,yMBMy).

We first give the definitions of the LSE and estimability as follows.

Definition 1 Let Λ(Σ)=^MyyMM(GΣ)M2, where ΣΘ=^{A:Ais annp×npsymmetric matrix} and A2=^tr(AA) for the matrix A. If there exists [Sigma] [set membership] Θ such that Λ([Sigma]) = minΣ[set membership]Θ Λ(Σ), then [Sigma] is called the LSE of Σ. Also, for any symmetric matrix C(≠ 0), tr(C[Sigma]) is called the LSE of tr(CΣ).

Definition 2 If there exists an np × np symmetric matrix A such that E(y′MAMy) = tr(CΣ), ∀Σ ≥ 0, then tr(CΣ) is said to be invariantly quadratically estimable and the quadratic estimator y′MAMy is called an unbiased invariant quadratic estimator of tr(CΣ).

For the convenience of our analysis, we introduce the following notation:

  • k0 = min {k : k = 1, …, m + 1, MXkG ≠ 0}, Mi, • = (MZi−1MZi) •′ MXiU, Wij,• = (MZi−1MZi) • (MZj−1MZj) (here • denotes any matrix), rij = tr(MXiGMXjG), Zi = (Z1, …, Zi) for i = 1, …, m, j = 1, …, m, MXm+1 = In, MZ0 = Ip and MZm+1 = 0p×p.

The following Theorem 1 provides the LSEs of Σ and tr(CΣ).

THEOREM 1 The LSEs of Σ and tr(CΣ) are given by

Σ^=i=k0m+1j=k0m+11rijMi,YMj,Y+ϒMZk01ϒMZk01,
(2.1)

and

tr(CΣ^)=yA*y+tr{C(ϒMZk01ϒMZk01)},

respectively, where [Upsilon] is an arbitrary p × p matrix which satisfies the condition [Upsilon]MZk0−1[Upsilon]MZk0−1 is a p × p symmetric matrix, and

A*=MA*M=i=k0m+1j=k0m+11rijMXiGMXjWij,C
(2.2)

Proof. According to Lemma A.1, we first show that [Sigma] given in (2.1) is the solution of the equation

tr{M(GQ)M(GΣ)}=yM(GΣ)My,Σ0.
(2.3)

It can be seen that

tr{M(GΣ^)M(GΣ)}=i=k0m+1j=k0m+1tr{MXiGMXjG(u=k0m+1v=k0m+11ruvWij,Mu,YMv,Y+Wij,ϒMZk01ϒMZk01)Σ}.

Since

Wij,ϒMZk01ϒMZk01=0, for i=k0,,m+1,j=k0,,m+1,

and

Wij,Mu,YMv,Y={Mi,YMj,Y, if i=u=k0,,m+1,j=v=k0,,m+1,0,otherwise,

it follows that ∀Σ ≥ 0,

tr{M(GΣ^)M(GΣ)}=i=k0m+1j=k0m+1tr(Mi,YMj,YΣ)=yM(GΣ)My.

which implies that [Sigma] is the solution of the equation (2.3).

Next we will show that [Sigma] is the general solution of the equation (2.3). Let [Sigma]0 be any solution of the equation (2.3). Then

i=k0m+1j=k0m+1rijtr(Wij,Σ^0Σ)=i=k0m+1j=k0m+1tr(Mi,YMj,YΣ), Σ0,

equivalently,

i=k0m+1j=k0m+1rijtr(Wij,Σ^0Σ)=i=k0m+1j=k0m+1tr(Mi,YMj,YΣ), ΣΣ,

which holds if and only if

i=k0m+1j=k0m+1rijWij,Σ^0=i=k0m+1j=k0m+1Mi,YMj,Y.
(2.4)

By using MZi−1MZi (i = k0, …, m + 1) to left multiply and MZj−1MZj (j = i, …, m + 1) to right multiply both sides of (2.4), we obtain

rijWij,Σ^0=Mi,YMj,Y.
(2.5)

Since MXiG ≠ 0, it follows that MXiGMXj ≠ 0 and then rij = tr{MXiGMXj (MXiGMXj)′} > 0.

Therefore,

Wij,Σ^0=1rijMi,YMj,Y,

which implies that

MZk01Σ^0MZk01=i=k0m+1j=k0m+1Wij,Σ^0=i=k0m+1j=k0m+11rijMi,YMj,Y,

and so

Σ^0=i=k0m+1j=k0m+11rijMi,YMj,Y+Σ^0MZk01Σ^0MZk01.

It can easily be seen that [Sigma] is equal to [Sigma]0 when [Upsilon] in [Sigma] is taken to be [Sigma]0. Hence, [Sigma] is a general solution of the equation (2.3) and this completes the proof of Theorem 1.

It should be noted that the LSE of Σ, [Sigma] given in (2.1), may not be positive semi-definite. In practice, we can choose a positive semi-definite estimator of Σ from (2.1) by selecting an appropriate [Upsilon]. From now on, we focus on the estimation problem of tr(CΣ). From Theorem 1, we see that there is a class of estimators which minimize Λ(Σ). We now show that there exists an estimator in the class which is of special interest by considering the unbiasedness of estimators.

THEOREM 2 The equality MZk0−1C = C is a necessary and sufficient condition for the LSE tr(C[Sigma]) of tr(CΣ) to be its unbiased invariant estimator.

Proof. According to Theorem 1, it can be seen that the LSE tr(C[Sigma]) is an unbiased estimator of tr(CΣ) if and only if

tr(CΣ)=E{tr(CΣ^)}=E(yMA*My)+tr{C(ϒMZk01ϒMZk01)},Σ0.
(2.6)

Since cov(y) = G [multiply sign in circle] Σ, it follows that

E(yMA*My)=tr{MA*ME(yy)}=tr{A*(GΣ)}=tr(MZk01CMZk01Σ).

Thus, (2.6) is equivalent to

tr(MZk01CMZk01Σ)=tr(CΣ),Σ0, and tr{C(ϒMZk01ϒMZk01)}=0,

which hold if and only if MZk0−1CMZk0−1 = C, or equivalently, MZk0−1C = C, and this implies that tr(C[Sigma]) = y′A*y where A* = MA*M, that is, tr(C[Sigma]) is invariant.

From Theorems 1 and 2, we can obtain some useful conclusions.

COROLLARY 1 The LSE(U,I) of tr(CΣ) is unique, and is given by tr(C[Sigma]) = y′A*y, where A* is given in (2.2).

Proof. The proof is straightforward from Theorems 1 and 2.

COROLLARY 2 Each of the following conditions forms the necessary and sufficient conditions for the LSE tr(C[Sigma]) to be the LSE(U,I) of tr(CΣ):

  1. CMZk0−1 = C.
  2. MZk0−1CMZk0−1 = C.
  3. MZiC = C, i = 1, …, k0 − 1.
  4. CMZi = C, i = 1, …, k0 − 1.
  5. MZiCMZi = C, i = 1, …, k0 − 1.

Proof. By using the result of Theorem 2 and the formulas MZiMZk0−1 = MZk0−1MZi = MZi for i = 1, …, k0 − 1, we can readily obtain the corollary.

COROLLARY 3 tr(CΣ) is estimable if and only if the LSE tr(C[Sigma]) is the LSE(U,I) of tr(CΣ).

Proof. From Wu et al. (2006), we see that MZk0−1C = C is a necessary and sufficient condition for tr(CΣ) to be estimable. So from Theorem 2, it is clear that the corollary holds.

3 THE OPTIMALITY OF THE LSE

In this section, we discuss the optimality of the LSE of tr(CΣ). We define N = max{k : k = k0 − 1, …, m, MZk ≠ 0} and Δ = MXN+1GMXN+1.

THEOREM 3 The necessary and sufficient conditions for the LSE tr(C[Sigma]) to be the UMVIQUE of tr(CΣ) are:

  1. MZk0−1C = C,
  2. There exists a constant α > 0 such that Δ2 = αΔ, or MZNCMZN = 0,
  3. There exist constants βi > 0 such that ΔMXiΔ = βiΔΔ+, or MZNC(MZi−1MZi) = 0, i = k0, ‥, N,
  4. There exist constants γij > 0 such that ΔMXiGMXjΔ = γijΔ, or (MZi−1MZi)C(MZj−1MZj) = 0, i = k0, …, N, j = i, …, N, and
  5. There exists a constant ρ > 0 such that MXN+1U diag(U′ Δ+U)UMXN+1 = ρΔ.

Proof. It can be seen from Lemma A.2 that M=i=k0N+1MXi(MZi1MZi). It follows that

(MyyM)A*=2M(GΣ)MA*M(GΣ)M+M(UIp)diag{SHk(Σ,Ψ)}(UIp)M,

where

diag{SHk(Σ,Ψ)}=i=k0N+1j=k0N+11rijdiag(UMXiGMXjU)SWij,C(Σ,Ψ).

Therefore, from Lemma A.4, it can be seen that the necessary and sufficient conditions for tr(C[Sigma]) to be the UMVIQUE of tr(CΣ) are: tr(C[Sigma]) is the unbiased estimator of tr(CΣ) and

2M(GΣ)MA*M(GΣ)M+M(UIp)diag{SHk(Σ,Ψ)}(UIp)M{M(GΣ)M:Σ0}=^{M(GV)M:VΩ},

where Ω = {V : V is a p × p symmetric matrix}, i.e.,

2M(GΣ)MA*M(GΣ)M+M(UIp)diag{SHk(Σ,ψ)}(UIp)M=M(GV)M,VΩ.
(3.1)

Note that (2.6) holds for all error distributions with mean zero and the same Σ and Ψ. In particular, it is true for the normal case in which SHk(Σ, Ψ) = 0 for Σ ≥ 0 and Ψ ≥ 0. Thus, formula (2.6) implies that

M(GΣ)MA*M(GΣ)M=M(GV1)M,V1Ω,
(3.2)

and so

M(UIp)diag{SHk(Σ,Ψ)}(UIp)M=M(GV2)M,V2Ω.
(3.3)

Clearly, if (3.2) and (3.3) hold, then (3.1) also holds.

Necessity

Since tr(C[Sigma]) is the unbiased estimator of tr(CΣ), it is evident to see from Theorem 2 that MZk0−1C = C. Now we assume that formulas (3.2) and (3.3) hold. Using ηkMZN to left multiply both sides of (3.2) and using ηk [multiply sign in circle] MZN to right multiply both sides of (3.2), we obtain

i=k0N+1j=k0N+1μkηkMXiGMXjηkrijMZNΣWij,CΣMZN=MZNV1MZN,

equivalently,

i=k0N+1j=k0N+1μkηkMXiGMXjηkrijMZNΣWij,CΣMZN     =i=k0N+1j=k0N+1μ1η1MXiGMXjη1rijMZNΣWij,CΣMZN.
(3.4)

Since MZN ≠ 0, it follows from Lemma A.5 that (3.4) is equivalent to

i=k0N+1j=k0N+1μkηkMXiGMXjηkrijWij,C=i=k0N+1j=k0N+1μ1η1MXiGMXjη1rijWij,C.
(3.5)

Note that MZN+1 = 0, it can be readily shown that (3.5) holds if and only if there exist α > 0, βi > 0, and γij > 0 such that

{μk2MZNCMZN=α2MZNCMZN,μk2ηkMXiηkMZNC(MZi1MZi)=βiMZNC(MZi1MZi),μkηkMXiGMXjηkWij,C=γijWij,C.
(3.6)

In the same way, we left multiply both sides of (3.2) by ηkMZN and right multiply both sides of (3.2) by ηl [multiply sign in circle] MZN to obtain

i=k0N+1j=k0N+1ηkMXiGMXjηlrijMZNΣWij,CΣMZN=0,

which holds if and only if

i=k0N+1j=k0N+1ηkMXiGMXjηlrijWij,C=0,

or equivalently,

{ηkMXiηlMZNC(MZi1MZi)=0,ηkMXiGMXjηlWij,C=0.
(3.7)

Combining (3.6) and (3.7) and from Lemma A.3, we see that

{Δ2=αΔorMZNCMZN=0,ΔMXiΔ=βiΔΔ+orMZNC(MZi1MZi)=0,ΔMXiGMXjΔ=γijΔor(MZi1MZi)C(MZj1MZj)=0,
(3.8)

which are the same as (ii)–(iv) in the theorem.

Given (3.8), we proceed to show that (v) holds. Note first that at least one of

MZNCMZN,MZNC(MZi1MZi),(MZi1MZi)C(MZj1MZj),

for i = k0, …, N + 1, j = i, …, N + 1, is nonzero. Otherwise,

MZk01CMZk01=i=k0N+1j=k0N+1Wij,C=0,

which contradicts the unbiased condition of tr(C[Sigma]). Therefore, it can be derived from (3.8) that Δ2 = αΔ or there exists an i0 ([set membership] {k0, ‥, N}) such that ΔMXi0Δ = βi0ΔΔ+, or there exist i0 ([set membership] {k0, ‥, N}) and j0 ([set membership] {i0, …, N}) such that ΔMXi0GMXj0Δ = γi0j0Δ, which implies that U′ΔU = αU′Δ+U or UMXi0GMXN+1U = βi0U′Δ+U or UMXi0GMXj0 U = γi0j0U′Δ+U.

Using MXN+1 [multiply sign in circle] MZN to left multiply both sides of (3.3) and to right multiply both sides of (3.3), we obtain

i=k0N+1j=k0N+11rijMXN+1Udiag(UMXiGMXjU)UMXN+1MZNSWij,C(Σ,Ψ)MZN=ΔMZNV2MZN,

which leads to

MXN+1Udiag(UΔ+U)UMXN+1MZNSΛ(Σ,Ψ)MZN=ΔMZNV2MZN,

where

Λ=α2rN+1,N+1WN N,C+i=k0N{βiri,N+1(WiN,C+WiN,C)+γiiriiWii,C}+Σk0i<jNγijrij(Wij,C+Wij,C).

It is readily seen that Λ ≠ 0, which derives from Lemma A.5 that MZNSΛ(Σ, Ψ)MZN ≠ 0. Thus, by using Lemma A.6, we get

MXN+1Udiag(UΔ+U)UMXN+1=ρΔ,ρ>0.
(3.9)

The necessity is proved.

Sufficiency

Assume that (i)–(v) hold. It is evident to see that tr(C[Sigma]) is the unbiased estimator of tr(CΣ). We will next show that (3.2) and (3.3) hold. Note that

M(MXN+1I)=(MXN+1I)M=M.

By using a way similar to the discussion of the necessity, we can obtain

M(GΣ)MA*M(GΣ)M=M(MXN+1GΣ)A*(GMXN+1Σ)M    =M{i=k0N+1j=k0N+11rijΔMXiGMXjΔΣWij,CΣ}M    =M(ΔΣΛΣ)M.

Letting V1 be ΣΛΣ, we see that (3.2) holds.

Furthermore, from Lemma A.5, it can be seen that SWij, C(Σ, Ψ) = 0 if Wij, C = 0. Therefore, it follows from (i)–(v) and Lemma A.3 that

M(UIp)diag{SHk(Σ,Ψ)}(UIp)M    =M{i=k0N+1j=k0N+11rijMXN+1Udiag(UMXiGMXjU)UMXN+1SWij,C(Σ,Ψ)}M    =M{MXN+1Udiag(UΔ+U)UMXN+1SΛ(Σ,Ψ)}M    =M{ΔρSΛ(Σ,Ψ)}M.

Taking V2 = ρSΛ(Σ, Ψ) implies that (3.3) holds. Thus, the sufficiency of the theorem is proved.

Clearly, from Corollary 2, Theorem 3 essentially discusses the optimality of the LSE(U,I) of tr(CΣ). In this case, tr(C[Sigma]) can be expressed as tr(C[Sigma])=y′A*y.

Finally, we consider a special case of quasi-normal disturbances. Model (1.1) is called quasi-normal if ε1, …, εs are independent and identically distributed and have the same first four moments as a random vector normally distributed with mean zero and covariance matrix Σ.

COROLLARY 4 Assume that Model (1.1) is quasi-normal. Then the necessary and sufficient conditions for the LSE tr(C[Sigma]) to be UMVIQUE of tr(CΣ) are:

  1. MZk0−1C = C,
  2. There exists a constant α > 0 such that Δ2 = αΔ, or MZNCMZN = 0,
  3. There exist constants βi > 0 such that ΔMXiΔ = βiΔΔ+, or MZNC(MZi−1MZi) = 0, i = k0, ‥, N, and
  4. There exist constants γij > 0 such that ΔMXiGMXjΔ = γijΔ, or (MZi−1MZi)C(MZj−1MZj) = 0, i = k0, …, N, j = i, …, N.

Proof. If Model (1.1) is quasi-normal, then it follows that

(MyyM)A*=2M(GΣ)MA*M(GΣ)M.

From this and using the proof similar to that of Theorem 3, we can see that the corollary is true.

4 A SIMULATION STUDY

To evaluate the performance of the proposed LSE(U,I), tr(C[Sigma]) = y′A*y, and compare it with two existing estimators: MINQE(U,I) in Wu et al. (2006) and MLE in Rosen (1989), we conduct a moderate scale simulation experiment. We generate n = 20, 50, 100, 150, 300 observations from the following growth curve model:

Yj=i=1mXi,jBiZi+εj,  j=1,,n,
(4.1)

where Yj = (Yj1, …, Yjp)′, Xi, j = (xj1, …, xjki), i = 1, …, m, ki = m+2−i which guarantees the nested condition, and each element xjk is generated from a standard normal distribution. We take the matrix C to be the p × p identity matrix and consider the following four scenarios:

  1. p = 3, m = 2, B1 = (3, 6, 9)′, B2 = (6, 9)′,
    (Z1,Z2)=(102531),Σ=(1.50.50.50.52.00.50.50.52.5),
    and the error vector εj follows a normal distribution N(0, Σ).
  2. p = 4, m = 3, B1 = (3, 6, 9, 12)′, B2 = (6, 9, 12)′, B3 = (9, 12)′,
    (Z1,Z2,Z3)=(110215311438),Σ=(2.00.50.50.50.52.50.50.50.50.53.00.50.50.50.53.5),
    and the error vector εj follows a normal distribution N(0, Σ).
  3. The parameters are the same as Case (a), but the error vector εj follows the same distribution as the random vector 0.5Σ1/22 − 2), where χ2 is a 3 × 1 vector, each element of which follows a chi-squared distribution with the degree of freedom of 2 and is independent of others.
  4. The parameters are the same as Case (b), but the error vector ε follows the same distribution as the random vector 0.5Σ1/22 − 2), where χ2 is a 4 × 1 vector, each element of which follows a chi-squared distribution with the degree of freedom of 2 and is independent of others.

The last two cases are used to see the effect of non-normal error on the three estimators.

We first generate 1000 data sets in each of the configurations, then compute the mean and variance of 1000 estimated values based on least squares (ls), minimum norm (mn), and maximum likelihood (ml) methods, and present the results in Table 1. From Table 1, we see that overall, the three estimators are all valid for the cases considered. In particular, under normal disturbances (Cases (a) and (b)), the proposed LSE(U,I) in this paper has a substantial bias advantage over the other two estimators. It is also observed that among the three estimators, LSE(U,I) has largest variance, but its mean squared error (MSE) is smallest. Under the non-normal case, LSE(U,I) still has least bias. But in this case, its MSE is often largest for not large sample sizes (n ≤ 100). However, the difference in magnitude is ignorable. It is interesting to note that MLE derived under the normal assumption performs very well indicating that it may be robust for various error structures. For different values of m, the similar conclusions can be observed (data not shown).

Table 1
Comparison of LSE(U,I), MINQE(U,I) and MLE

5 CONCLUDING REMARKS

In this article, we obtained the LSE(U,I) of the linear functions of the variance-covariance matrix, tr(CΣ) for the GGCM. The proposed LSE(U,I) is of closed form and its calculation is straightforward. The necessary and sufficient conditions for the LSE(U,I) to be the UMVIQUE were derived. Our results extended Hsu’s theorem from the univariate linear model to the GGCM. Simulation studies showed that our proposed LSE(U,I) can outperform von Rosen’s (1989) MLE even under the normality assumption of the disturbances and Wu’s et al. (2006) MINQE(U,I).

In general, the existing estimators of tr(CΣ) such as von Rosen’s (1989) MLE, Wu’s et al. (2006) MINQE(U,I) and our proposed LSE(U,I) here can take negative values as the estimates, even C is the identity matrix. Clearly, a nonnegative estimator is more interesting. It should also be noted that following von Rosen (1989), we have assumed the restriction condition R(Xm) [subset, dbl equals] (...) [subset, dbl equals] R(X1). When the nested condition does not hold, how to derive the LSE(U,I) of tr(CΣ) warrants our future work.

Acknowledgements

The authors are grateful to the Co-Editor and the Reviewer for their valuable comments and suggestions which substantially improved the original manuscript. We also thank Professor D. von Rosen for helpful discussion. Liang’s research was partially supported by the two grants AI62247 and AI59773 from the National Institute of Allergy and Infectious Diseases, and DMS0806097 from NSF. Zou’s research was partially supported by the three grants 70625004, 10721101 and 70221001 from the National Natural Science Foundation of China.

APPENDIX

Here we state several preliminary lemmas which are used for the proofs of the main results. Throughout the appendix, our assertions hold, unless specified, for k = 1, …, τ, l = 1, …, τ, kl, i = k0, …, N, and j = i, …, N, where τ = rank(Δ) > 0 and Δ was defined in Section 3.

LEMMA A.1 [Sigma] is the LSE of Σ if and only if

tr{M(GΣ^)M(GΣ)}=yM(GΣ)My,Σ0.

Proof. If [Sigma] is the LSE of Σ, then

Λ(Σ^+λΣ)Λ(Σ^),λ,Σ0,

i.e.,

λ2M(GΣ)M2+2λ[tr{M(GΣ^)M(GΣ)}yM(GΣ)My]0.

So we must have

tr{M(GΣ^)M(GΣ)}=yM(GΣ)My,Σ0.

Conversely, if the above equation holds, then as it is equivalent to

tr{M(GΣ^)M(GΣ)}=yM(GΣ)My,Σ=Σ,

we have

Λ(Σ)=Λ(Σ^)+M{G(Σ^Σ)}M2,Σ=Σ,

which implies that ∀Σ = Σ′, Λ(Σ) ≥ Λ([Sigma]) and therefore [Sigma] is the LSE of Σ.

LEMMA A.2 If R(Xm) [subset, dbl equals] (...) [subset, dbl equals] R(X1), then

M=i=1m+1MXi(MZi1MZi).

Proof. The proof can be found in Wu et al. (2006).

LEMMA A.3 Let η1, …, ητ be an orthogonal basis of the matrix Δ, and μ1, …, μτ be the corresponding non-zero eigenvalues. Then we have

  1. μk = ρ holds for some constant ρ > 0 if and only if Δ2 = ρΔ, which is equivalent to U′ΔU = ρ2U′Δ+U;
  2. μk2ηkMXiηk=ρ for some constant ρ > 0 and ηkMXiηl=0 hold if and only if ΔMXiΔ = ρΔΔ+, which is equivalent to UMXiGMXN+1U = ρU′Δ+U;
  3. μkηkMXiGMXjηk=ρ for some constant ρ > 0 and ηkMXiGMXjηl=0 hold if and only if ΔMXjΔ = ρΔ, which is equivalent to UMXiGMXjU = ρU′Δ+U.

Proof. (i) It is evident that μk = ρ holds if and only if

k=1τμk2ηkηk=ρk=1τμkηkηk,

i.e., Δ2 = ρΔ, or equivalently, Δ3 = ρ2ΔΔ+Δ. This is in turn equivalent to

UMXN+1ΔMXN+1U=ρ2UMXN+1Δ+MXN+1U,

or equivalently,

UΔU=ρ2UΔ+U.

(ii) It can be seen that μk2ηkMXiηk=ρ and ηkMXiηl=0 hold if and only if

k=1τl=1τμkμlηkηkMXiηlηl=ρk=1τηkηk,

i.e., ΔMXiΔ = ρΔΔ+, or equivalently,

ΔMXiGMXN+1Δ=ρΔΔ+Δ.

This is in turn equivalent to

UMXiGMXN+1U=ρUΔ+U.

(iii) It can be readily seen that μkηkMXiGMXjηk=ρ and ηkMXiGMXjηl=0 hold if and only if

k=1τl=1τμkμlηkηkMXiGMXjηlηl=ρk=1τμkηkηk,

i.e.,

ΔMXiGMXjΔ=ρΔ=ρΔΔ+Δ.

This is equivalent to

UMXiGMXjU=ρUΔ+U.

This completes the proof of Lemma A.3.

LEMMA A.4 If tr(A*Myy′M) is an unbiased estimator of tr(CΣ), then tr(A*Myy′M) is the uniformly minimum variance linear (in Myy′M) unbiased estimator of tr(CΣ) if and only if

(MyyM)MA*MSpan{M(GΣ)M:Σ0}.

Proof. The proof is referred to Seely and Zyskind (1971).

LEMMA A.5 Let B be a nonzero symmetric and idempotent matrix, and F be a symmetric matrix. Then (i) BΣFΣB = 0, ∀Σ ≥ 0 if and only if F = 0. (ii) BSF (Σ, Ψ)B = 0, ∀Σ ≥ 0 if and only if F = 0.

Proof. See Yang (1995).

LEMMA A.6 Let F1 and F2 be the p1 × p2 matrices, and B(x) and D(x) be the functions defined in the set S. If F2 ≠ 0, then F1 [multiply sign in circle] B(x) = F2 [multiply sign in circle] D(x), ∀x [set membership] S if and only if (i) there exists a λ such that F1 = λF2 and B(x) = λD(x), or (ii) B(x) [equivalent] 0 and D(x) [equivalent] 0.

Proof. See Yang (1995).

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