Asymptotic variance

Asymptotic variance is one criterion for choosing the function G to be used in the contrast function. Comparison of, say, the traces of the asymptotic covariance matrices of two estimators enables direct comparison of the mean-square error of the estimators. In [18], evaluation of asymptotic variances was addressed using a related family of contrast functions. In fact, it can be seen that the results in [18] are valid even in this case, and thus we have the following theorem:

Theorem 2   The trace of the asymptotic (co)variance of $\hat{{\bf w}}$ is minimized when G is of the form

$$G_{opt}(u)=k_1\log f_i(u)+k_2 u^2+k_3$$ (10)

where f_i(.) is the density function of s_i, and k₁,k₂,k₃ are arbitrary constants.

For simplicity, one can choose $G_{opt}(u)=\log f_i(u)$. Thus the optimal contrast function is the same as the one obtained by the maximum likelihood approach [34], or the infomax approach [3]. Almost identical results have also been obtained in [5] for another algorithm. The theorem above treats, however, the one-unit case instead of the multi-unit case treated by the other authors.