Higher order asymptotic theory for normalizing transformations of maximum likelihood estimators

Masanobu Taniguchi*, Madan L. Puri

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


Suppose that Xn=(X1,... Xn) is a collection of m-dimensional random vectors Xi forming a stochastic process with a parameter θ{symbol}. Let {Mathematical expression} be the MLE of θ{symbol}. We assume that a transformation A( {Mathematical expression}) of {Mathematical expression} has the k-thorder 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.

Original languageEnglish
Pages (from-to)581-600
Number of pages20
JournalAnnals of the Institute of Statistical Mathematics
Issue number3
Publication statusPublished - 1995 Sept
Externally publishedYes


  • Edgeworth expansion
  • MLE
  • Normalizing transformation
  • higher-order asymptotic theory
  • multivariate analysis
  • observed information
  • saddlepoint expansion
  • signed log likelihood ratio
  • time series analysis
  • variance stabilizing transformation

ASJC Scopus subject areas

  • Statistics and Probability


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