On Rank Estimation In Symmetric Matrices: The Case Of Indefinite Matrix Estimators

In this paper we consider estimating the rank of an unknown symmetric matrix based on a symmetric, asymptotically normal estimator of the matrix. The related positive definite limit covariance matrix is assumed to be estimated consistently and to have either a Kronecker product or an arbitrary structure. These assumptions are standard although they exclude the case when the matrix estimator is positive or negative semidefinite. We adapt and reexamine here some available rank tests and introduce a new rank test based on the sum of eigenvalues of the matrix estimator. We discuss two applications where rank estimation in symmetric matrices is of interest, and we also provide a small simulation study.The first author acknowledges the support of an Alfred P. Sloan Foundation Research Fellowship and NSF Grant SES-0196372. We thank the co-editor and the two referees for useful comments and suggestions. CEMPRE—Centro de Estudos Macroeconómicos e Previsão—is supported by the Fundação para a Ciência e a Tecnologia, Portugal, through the Programa Operacional Ciência, Tecnologia e Inovação (POCTI) of the Quadro Comunitário de Apoio III, which is financed by FEDER and Portuguese funds.