  This paper considers the regularized Tyler's scatter estimator for elliptical
distributions, which has received considerable attention recently. Various
types of shrinkage Tyler's estimators have been proposed in the literature and
proved work effectively in the "small n large p" scenario. Nevertheless, the
existence and uniqueness properties of the estimators are not thoroughly
studied, and in certain cases the algorithms may fail to converge. In this
work, we provide a general result that analyzes the sufficient condition for
the existence of a family of shrinkage Tyler's estimators, which quantitatively
shows that regularization indeed reduces the number of required samples for
estimation and the convergence of the algorithms for the estimators. For two
specific shrinkage Tyler's estimators, we also proved that the condition is
necessary and the estimator is unique. Finally, we show that the two estimators
are actually equivalent. Numerical algorithms are also derived based on the
majorization-minimization framework, under which the convergence is analyzed
systematically.
