HIGHER ORDER ASYMPTOTIC THEORY FOR NORMALIZING TRANSFORMATIONS OF MAXIMUM LIKELIHOOD ESTIMATORS

MASANOBU TANIGUCHI^{1} AND MADAN L. PURI^{2}

1Department of Mathematical Science, Faculty of Engineering Science, Osaka University, Toyonaka 560, Japan

2Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A.

(Received March 10, 1994; revised January 26, 1995)

Abstract. Suppose that X_{n} = (X_{1}, …, X_{n}) is a collection of m-dimensional random vectors X_{i} forming a stochastic process with a parameter θ. Let θ̂ be the MLE of θ. We assume that a transformation A(θ̂) of θ̂ 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.