On the dependency for asymptotically independent estimates

Abstract

Suppose \({\widehat{\theta}_1}\) and \({\widehat{\theta}_2}\) are asymptotically independent non-lattice with a joint second order Edgeworth expansion in n−1/2. Then the α dependency coefficient is $$\alpha \left(\widehat{\theta}_1, \widehat{\theta}_2 \right) = n^{-1/2} C + O \left(n^{-1} \right),$$ where \({C = (4 \pi)^{-1}\exp (-1/2) (\tau^2_1 + \tau^2_2) ^{1/2}}\) for \({\tau_1, \tau_2}\) their joint skewness coefficients.