Estimation of the MSE Sample Clauses

Estimation of the MSE. The Mean Squared Error estimator (MSE) of θ˜B(ψ), depending on the variance components ψ= (σ2, σ2)T , γ u can be expressed as in Xxx (2003): where the first term MSE[θ˜B(ψ)] = g1(ψ) + g2(ψ) (6.5) g1(ψ) = Λ(ψ)R = R − RΣ−1(ψ)R (6.6) is due to the estimation of random effects and it is of order O(1), while the second term g2(ψ) = RΣ−1(ψ)X(XT Σ−1(ψ)X)−1XT Σ−1(ψ)R (6.7) is due to the estimation of β and it is of order O(m−1) for large m. The estimator θ˜B(ψ) depends on the unknown variance components σ2 and σ2. Replacing the pa- γ u rameters with estimators σˆ2, σˆ 2, a two stage estimator θ˜E (ψˆ ) is γ u θ˜E (ψˆ ) = Xβˆ(ψˆ ) + Λˆ (ψˆ )[θˆ − Xβˆ(ψˆ )] (6.8) where βˆ(ψˆ ) = (XT Σˆ−1(ψˆ )X)−1XT Σˆ−1(ψˆ )θˆ. Assuming normality of the random effects, σ2 and σ2 can be γ u estimated both by Maximum Likelihood (ML) and Restricted Maximum Likelihood (REML) procedures (see Xxxxxx and Xxx (1990)). × ∈ − The ML and REML estimators possess the following properties (see Datta et al. (2005)): (i) they are m1/2-consistent; (ii) they are even functions of θˆ, so that ψˆ ( θˆ) = ψˆ (θˆ); (iii) they are translation invariant functions, so that ψˆ (θˆ + Gc) = ψˆ (θˆ), for any m (g + 1) matrix, c Rg+1 and for all θˆ. For any ψˆ satisfying (ii) and (iii), the MSE of θ˜E (ψˆ ) can be decomposed as MSE[θ˜E (ψˆ )] = g1(ψ) + g2(ψ) + E([θ˜E (ψˆ ) − θ˜B(ψ)]2) = g1(ψ) + g2(ψ) + g3(ψ). (6.9) Under the model (6.1) with diagonal covariance matrix Σ(σ2), Xxxxxx and Xxx (1990) obtained an ap- proximation up to o(m−1) terms of g3(ψ) through Xxxxxx linearization. In case of the semiparametric Xxx & Xxxxxxx model the structure of the covariance matrix is not diagonal due to the introduction of the spline random component, then the results of Xxxxxx and Xxx (1990) can not be applied directly. The results of Opsomer et al. (2008) can be used for deriving a second order approximation to the g3(ψ) term. It can be given by g3(ψ) = LT (ψ)"I −1(ψ) ⊗ Σ(ψ)#L(ψ) + o(δm/m) (6.10) where L(ψ) = [Lσ2 (ψ), Lσ2 (ψ)]T , Li(ψ) = ∂Λ(ψ), i = 1, 2. γ u ∂ψi Here ⊗ represents Kronecker product, I −1(ψ) is the inverse of the information matrix with Ii−j 1(ψ) = −i j 1 0.5tr[P(ψ)B P(ψ)B ], i, j = 1, 2, P(ψ) = Σ−1(ψ) Σ−1(ψ)X(XT Σ−1(ψ)X)−1XT Σ−1(ψ), B = SST and B2 = ZZT and δm = o(√m). In practical applications, the EBLUP θ˜E (ψˆ ) should be accompanied with an estimate of the MSE. Again, under Xxx & Xxxxxxx models with diagonal covariance matrix, Xxxxxx and Xxx (1990) obtained an approximately unbiased estimator of the MS...