Pseudo confidence regions for the solution of a multivariate linear functional relationship

Abstract

Abstract The problem of finding confidence regions (CR) for a q-variate vector γ given as the solution of a linear functional relationship (LFR) Λγ = μ is investigated. Here an m-variate vector μ and an m × q matrix Λ = (Λ1, Λ2,…, Λq) are unknown population means of an m(q+1)-variate normal distribution N m(q+1) (ζΩ⊗Σ) , where ζ′ = (μ′, Λ1′, Λ2′,…, Λq′ Σ is an unknown, symmetric and positive definite m × m matrix and Ω is a known, symmetric and positive definite (q+1) × (q+1) matrix and ⊗ denotes the Kronecker product. This problem is a generalization of the univariate special case for the ratio of normal means. A CR for γ with level of confidence 1 − α, is given by a quadratic inequality, which yields the so-called ‘pseudo’ confidence regions (PCR) valid conditionally in subsets of the parameter space. Our discussion is focused on the ‘bounded pseudo’ confidence region (BPCR) given by the interior of a hyperellipsoid. The two conditions necessary for a BPCR to exist are shown to be the consistency conditions concerning the multivariate LFR. The probability that these conditions hold approaches one under ‘reasonable circumstances’ in many practical situations. Hence, we may have a BPCR with confidence approximately 1 − α. Some simulation results are presented.