Estimation of the Cholesky decomposition of the covariance matrix for a conditional independent normal model

In this paper, we consider estimating the Cholesky decomposition (the lower triangular squared root) of the covariance matrix for a conditional independent normal model under four equivariant loss functions. Closed-form expressions of the maximum likelihood estimator and an unbiased estimator of the Cholesky decomposition are provided. By introducing a special group of lower-triangular block matrices, we obtain the best equivariant estimator of the Cholesky decomposition under each of the four losses. Because both the maximum likelihood estimator and the unbiased estimator belong to the class of equivariant estimators with respect to the special group, they are all inadmissible.