Structured dispersion and validity in linear inference

Abstract

Structured matrices ∑(γ) = [I n + eγ′ + γe′ − \\ ̄ ggee′] arise in nonstandard linear models, where e′ = [1, …, 1], γ′ = [ γ 1, …, γ n ], and \\ ̄ gg = (γ 1 + … + γ n) n . Their properties are studied, including expressions for eigenvalues, conditions for positive definiteness, and conditioning of bE(γ) as γ varies. It is shown that if γ majorizes γ 0, then the condition numbers are ordered as c φ (∑( γ)) ⩾ c φ (∑( γ 0)) for every condition number { c φ (·); φ ∈ ϱ} generated by the unitarily invariant matrix norms. Applications are noted in linear inference and in outlier detection.