Stabilizing the asymptotic covariance of an estimate

Abstract

Suppose $n^{1/2}(\widehat{\theta}_{n}-\theta)\rightarrow \mathcal{N}_{p}(0,V(\theta))$ as n→∞ for some estimate $\widehat{\theta}_{n}$ of θ in R^p. If p=1 and g(θ)=∫₀^θV(x)^−1/2dx, it is well known that $n^{1/2}(g(\widehat{\theta}_{n})-g(\theta))\rightarrow \mathcal{N}(0,1)$ as n→∞, the distribution often being less skew so that inference based on the approximation $n^{1/2}(g(\widehat{\theta}_{n})-g(\theta))\sim \mathcal{N}(0,1)$should be more accurate than inference based on the approximation $V(\widehat{\theta}_{n})^{-1/2}n^{1/2}(\widehat{\theta}_{n}-\theta)\sim \mathcal{N}(0,1)$. If p>1 there is generally no such one to one transformation g(⋅). We consider three different types of stabilization of V(θ). We also consider the problem of finding g(⋅) so that the components of $g(\widehat{\theta}_{n})$ are asymptotically independent.