Optimal estimation for doubly multivariate data in blocked compound symmetric covariance structure

Abstract

a b s t r a c tThe paper deals with the best unbiased estimators of the blocked compound symmetric covariance structure for m-variate observations over u sites under the assumption of multivariate normality. The free-coordinate approach is used to prove that the quadratic estimation of covariance parameters is equivalent to linear estimation with a properly defined inner product in the space of symmetric matrices. Complete statistics are then derived to prove that the estimators are best unbiased. Finally, strong consistency is proven. The proposed method is implemented with a real data set.

Testing Hypotheses About Structure of Parameters in Models With Block Compound Symmetric Covariance Structure

“…It was presented how the methods of estimation and testing for a univariate model can be extended to the multivariate case. Estimators of the parameters of the presented BCS covariance structure model and the data presenting measures of mineral content of bones can be found in Roy et al (2016). The power of the proposed tests for expectation and covariance parameters, in the multivariate case, is compared with well-known tests such as LRT and Roy's test in Fonseca et al (2018) and Zmyślony et al (2018).…”

“…For the proof that for the model (9) cov( ) is a quadratic subspace and assumption that commutativity of E( ) holds see Roy et al (2016).…”

“…Using the formula (4.13) and Theorem 1 from Roy et al (2016) we get that BLUE for iŝ = 0.84380 1.79268 0.70440 0.81832 1.73484 0.69384 , where, in accordance with the order of variables, the first three values are the means for measurements of mineral content in dominant side of radius, humerus and ulna, respectively, while the last three values are the means for measurements of mineral content in non-dominant side for these bones. From the same paper, using formulas (3.4) and (3.5) and Theorem 1 we get that BQUE for Γ Γ Γ 0 and Γ Γ Γ 1 arê Table 1 follow from the fact that in case = 2 both tests are equivalent.…”

“…One can prove that quadratic estimator of ∑ =2 ( ) ( ) ′ is a function of complete sufficient statistics (see Roy et al (2016)) and has the following form:…”

“…whereΓ Γ Γ 0 andΓ Γ Γ 1 are best unbiased estimators (BUE) for Γ Γ Γ 0 and Γ Γ Γ 1 , respectively. For details see Roy et al (2016). Note that…”