Performance Analysis of Subspace-Based Algorithms in CES Data Models

Subspace-based algorithms that exploit the orthogonality between a sample subspace and a parameter-dependent subspace have proved very useful in many applications in signal processing. The statistical performance of these subspace-based algorithmsSubspace-based algorithm depends on the deterministic and stochastic statistical model of the noisy linear mixtureNoisy linear mixture of the data, the estimate of the projector associated with different estimates of the scatter/covariance of the data, and the algorithm that estimates the parameters from the projector. This chapter presents various different complex circularCircular (C-CES) and non-circularNoncircular (NC-CES) elliptically symmetric models of the data, as well as different associated non-robust and robust covariance estimators. These estimators include the sample covariance matrixSample covariance matrix (SCM), the maximum likelihood (ML) estimate, robust M-estimates, Tyler’s M-estimate and the sample sign covariance matrixSample sign covariance matrix (SSCM) (SSCM). The asymptotic distributionsAsymptotic distribution of these estimators are also derived. This enables us to unify the asymptotic distributionAsymptotic distribution of subspace projectorsSubspace projector adapted to the different models of the data and demonstrate various invariance properties that have impacts on the parameters to be estimated. Particular attention is paid to the comparison between the projectors derived from Tyler’s M-estimate and SSCMSample sign covariance matrix (SSCM). Finally, this study investigates the asymptotic distributionsAsymptotic distribution of paratmeter estimates of parameters characterized by the principal subspace derived from the distributions of subspace-based parameter estimates. In particular, the efficiencyEfficiency with respect to the stochastic and semiparametric Cramér–Rao boundSemiparametric Cramer-Rao bound (SCRB) is considered.