  This paper investigates regularized estimation of Kronecker-structured
covariance matrices (CM) for polarization radar in sea clutter scenarios where
the data are assumed to follow the complex, elliptically symmetric (CES)
distributions with a Kronecker-structured CM. To obtain a well-conditioned
estimate of the CM, we add penalty terms of Kullback-Leibler divergence to the
negative log-likelihood function of the associated complex angular Gaussian
(CAG) distribution. This is shown to be equivalent to regularizing Tyler's
fixed-point equations by shrinkage. A sufficient condition that the solution
exists is discussed. An iterative algorithm is applied to solve the resulting
fixed-point iterations and its convergence is proved. In order to solve the
critical problem of tuning the shrinkage factors, we then introduce two methods
by exploiting oracle approximating shrinkage (OAS) and cross-validation (CV).
The proposed estimator, referred to as the robust shrinkage Kronecker estimator
(RSKE), is shown to achieve better performance compared with several existing
methods when the training samples are limited. Simulations are conducted for
validating the RSKE and demonstrating its high performance by using the IPIX
1998 real sea data.
