Reduced-rank estimation of the difference between two covariance matrices

We consider m × m covariance matrices, Σ 1 and Σ 2 , which satisfy Σ 2 - Σ 1 = Δ , where Δ has a specified rank. Maximum likelihood estimators of Σ 1 and Σ 2 are obtained when sample covariance matrices having Wishart distributions are available and rank ( Δ ) is known. The likelihood ratio statistic for a test about the value of 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.