Reduced-rank estimation of the difference between two covariance matrices

We consider m×mm×m covariance matrices, Σ1Σ1 and Σ2Σ2, which satisfy Σ2-Σ1=ΔΣ2-Σ1=Δ, where ΔΔ has a specified rank. Maximum likelihood estimators of Σ1Σ1 and Σ2Σ2 are obtained when sample covariance matrices having Wishart distributions are available and rank(Δ)rank(Δ) is known. The likelihood ratio statistic for a test about the value of rank(Δ)rank(Δ) is also given and some properties of its null distribution are obtained. The methods developed in this paper are illustrated through an example.