(Received March 10, 1994; revised January 26, 1995)
Abstract. Suppose that Xn = (X1, ..., Xn) is a collection of m-dimensional random vectors Xi forming a stochastic process with a parameter theta. Let ^theta be the MLE of theta. We assume that a transformation A(^theta) of ^theta has the k-th-order Edgeworth expansion (k = 2,3). If A extinguishes the terms in the Edgeworth expansion up to k-th-order (k > 2), then we say that A is the k-th-order normalizing transformation. In this paper, we elucidate the k-th-order asymptotics of the normalizing transformations. Some conditions for A to be the k-th-order normalizing transformation will be given. Our results are very general, and can be applied to the i.i.d. case, multivariate analysis and time series analysis. Finally, we also study the k-th-order asymptotics of a modified signed log likelihood ratio in terms of the Edgeworth approximation.
Key words and phrases: Normalizing transformation, higher-order asymptotic theory, variance stabilizing transformation, multivariate analysis, time series analysis, Edgeworth expansion, saddlepoint expansion, MLE, observed information, signed log likelihood ratio.
Source ( TeX , DVI , PS )