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A general easily checkable Cochran theorem is obtained for a normal random operator Y . This result does not require that the covariance, Σ Y , of Y is nonsingular or is of the usual form A ⊗ Σ ; nor does it assume that the mean, μ, of Y is equal to zero. Indeed, { Y ′ W i Y } (with nonnegative definite W i ′s) is a family of independent Wishart random operators Y ′ W i Y of parameter ( m i , Σ , λ i ) if and only if for some nonnegative definite A and for all i ≠ j : (a)( W i ⊗ I )( Σ Y − A ⊗ Σ )( W i ⊗ I ) = 0; (b) AW i AW i = AW i , r ( AW i ) = m i , (c) λ i = μ′ W i μ = μ′ W i AW i μ; and (d) ( W i ⊗ I ) Σ Y ( W j ⊗ I ) = 0. The usual multivariate versions of Cochran′s theorem are contained in a special case of our result where Σ Y = A ⊗ Σ . The A in our version of Cochran′s theorem can actually be constructed from Σ , Σ Y , and the sum of the W i ′s.

Let θθ̂ = (θθ̂1, θθ̂2, …, θθ̂n)′ be the least-squares estimator of θ = (θ1, θ2, …, θn)′ by the realization of the process y(t) = Σk = 1n θkfk(t) + ξ(t) on the interval T = [a, b] with f = (f1, f2, …, fn)′ belonging to a certain set X. The process satisfies E(ξ(t))≡0 and has known continuous covariance r(s, t) = E(ξ(s)ξ(t)) on T × T. In this paper, A-, D-, and Ds-optimality are used as criteria for choosing f in X. A-, D-, and Ds-optimal models can be constructed explicitly by means of r.

For a normal random matrix Y with mean zero, necessary and sufficient conditions are obtained for Y′WkY to be Wishart–Laplace distributed and {Y′WkY} to be independent, where each Wk is assumed to be symmetric rather than nonnegative definite.

For a normally distributed random matrix Y with mean zero and general covariance matrix ΣY and for a symmetric matrix W, necessary and sufficient conditions are derived for the Wishartness of Y′WY.

Let E, V be n-, p-dimensional inner product spaces over the real field, let L(V, E) be the set of all linear maps of V into E, and let Y be a normal random vector in L(V, E) with mean μ = 0 and covariance ΣY such that S1 ⊏ S2(≠ }0{) is the image set, Im ΣY of ΣY, where S1, S2 are linear subspaces of E, V, respectively, and ⊏ is the outer product. Let }Wi{ be a family of self-adjoint operators in L(E, E). Then (∗): }Y′WiY{ is an independent family of Wishart random operators Y′WiY with parameter (mi, Σ, λi), each mi > 0 and λi = 0,” if and only if Im Σ = S2 and for any distinct i,j∈I, ΣY(Wi⊗Σ+)ΣY(Wi⊗Σ+)ΣY = ΣY(Wi⊗Σ+)ΣY, tr(ΣY(Wi⊗Σ+)) ≠ 0, and ΣY(Wi⊗Σ+)ΣY(Wj⊗Σ+)ΣY = 0. A necessary and sufficient condition for (∗) is also obtained for the general case where no condition whatever is imposed on (μ, ΣY). This generalizes a recent result of Pavur who considered the case where Σ is nonsingular and each Wi is nonnegative definite.

Moments of generalized Wishart distributions are obtained through tensor differential forms and reindexing.