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HIGHER ORDER ASYMPTOTIC THEORY FOR NORMALIZING

TRANSFORMATIONS OF MAXIMUM LIKELIHOOD ESTIMATORS

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

^{1} *Department of Mathematical Science, Faculty of
Engineering
Science,*

Osaka University, Toyonaka 560, Japan

^{2} *Department 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 *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.

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