  In this paper we study the theoretical properties of the deflation-based
FastICA method, the original symmetric FastICA method, and a modified symmetric
FastICA method, here called the squared symmetric FastICA. This modification is
obtained by replacing the absolute values in the FastICA objective function by
their squares. In the deflation-based case this replacement has no effect on
the estimate since the maximization problem stays the same. However, in the
symmetric case a novel estimate with unknown properties is obtained. In the
paper we review the classic deflation-based and symmetric FastICA approaches
and contrast these with the new squared symmetric version of FastICA. We find
the estimating equations and derive the asymptotical properties of the squared
symmetric FastICA estimator with an arbitrary choice of nonlinearity.
Asymptotic variances of the unmixing matrix estimates are then used to compare
their efficiencies for large sample sizes showing that the squared symmetric
FastICA estimator outperforms the other two estimators in a wide variety of
situations.
