Inference about dependencies in a multiway data array can be made using the array normal model which corresponds to the class of multivariate normal distributions with separable covariance matrices. uniformly minimum risk equivariant estimator (UMREE) can be obtained via a generalized Bayes procedure. Although this UMREE is minimax and dominates the MLE it can be improved upon via an orthogonally equivariant modification. Numerical comparisons of the risks of these estimators show that the equivariant estimators can have substantially lower risks than the MLE. {= {indexing units indexing variables and indexing time.|= indexing units indexing indexing and variables time. Another example is multivariate relational data where is the type-relationship between person and person = Θ + represents additive residual variation about Θ. Standard models for Θ include regression models additive effects models (such as those estimated by ANOVA decompositions) and unconstrained mean models if replicate observations are available. Another popular approach is to model Θ as being a low-rank array. For such models ordinary least-squares estimates of Θ can be obtained via various types of tensor decompositions depending on the definition of rank being used (De Lathauwer et al. 2000 a; de Silva and Lim 2008 Less attention has been given to the analysis of the residual variation taking values in ?has an array normal distribution if is a random array in ?having i.i.d. {standard normal entries and is a × nonsingular matrix for each ∈ {1 … and “?|standard normal entries and is a × nonsingular matrix for each ∈ 1 “ and …?” denote the Kronecker product we write in a small simulation study. A discussion follows in Section 5. Proofs are contained Dapoxetine hydrochloride in an appendix. 2 An invariant measure for the array normal model 2.1 The array normal model The array normal model on ?consists of the distributions of random ∈ ?for which ∈ ?= 1 … and a random array with i.i.d. standard normal entries. Here “×” denotes the = vec(operation which reshapes an array into a matrix along an index set or of is the × (∏along the = × {is a scalar. This shows that can be interpreted as the covariance among the slices of the array along its and covariance Cov[vec(? ? ? Σ1) where σ2 > 0 and for each × positive definite matrices. To make the parameterization identifiable we restrict the determinant of each Σto be one. Denote by this parameter space that is the values of (σ2 Σ1 … Σ= 1 … ~ (Θ σ2(Σ? ? ? Σ1)) if and only if is a matrix such that ~ i.i.d. ? ? ? Σ1)) the (+1)-array obtained by “stacking” along a (+1)st mode also has an array normal distribution is the × 1 vector of ones and “?” denotes the outer product. If > Dapoxetine hydrochloride 1 then covariance estimation for the array normal model can be reduced to the case that Θ = 0. To see this let be a (? 1) × matrix such that = = 0. This implies that = × {+ 1) matricization Rabbit Polyclonal to Merlin (phospho-Ser10). of = 0 and so is mean-zero. Dapoxetine hydrochloride Using identity (3) the covariance of vec(? Σ? ? Dapoxetine hydrochloride ? Σ1) = σ2(? ? ? Σ1) and so ~ × (? ? ? Σ1)). For the remainder of this paper we consider covariance estimation in the case that Θ = 0. 2.2 Model invariance and a right invariant measure Consider the model for an i.i.d. sample of size from a ~ ? Σ) ~ ? by elements of mapping the sample space ?to is said to be equivariant under this group if for all and ∈ ?/is equivariant and different from the UMREE the MLE is dominated by the UMREE. We pursue analogous results for the array normal model by first reparameterizing in terms of the parameter Σ1/2 = (σ Ψ1 … Ψis in the set of lower triangular matrices with positive diagonals and determinant 1. In this parameterization Ψis the lower triangular Cholesky square root of the mode-covariance matrix Σdescribed in Section 2.1. Define the group as consists of the same set as the parameter space for the model as parameterized in (5). If the group acts on the sample space by to the parameter space is equivariant if is the estimator of Ψwhen Dapoxetine hydrochloride observing is the estimator when observing × {in have determinant 1 and so one of the nonzero elements of can be expressed as a function of the others. For the rest of this section and the next we parameterize in terms of the elements {≤ ≤ therefore take values in the space &.