Abstract
Suppose thatX n =(X 1,...X n) is a collection ofm-dimensional random vectorsX i forming a stochastic process with a parameter ϑ. Let\(\hat \theta \) be the MLE of ϑ. We assume that a transformationA(\(\hat \theta \)) of\(\hat \theta \) has thek-thorder Edgeworth expansion (k=2,3). IfA extinguishes the terms in the Edgeworth expansion up tok-th-order (k≥2), then we say thatA is thek-th-order normalizing transformation. In this paper, we elucidate thek-th-order asymptotics of the normalizing transformations. Some conditions forA to be thek-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 thek-th-order asymptotics of a modified signed log likelihood ratio in terms of the Edgeworth approximation.
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Research supported by the Office of Naval Research Contract N00014-91-J-1020.
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Taniguchi, M., Puri, M.L. Higher order asymptotic theory for normalizing transformations of maximum likelihood estimators. Ann Inst Stat Math 47, 581–600 (1995). https://doi.org/10.1007/BF00773402
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DOI: https://doi.org/10.1007/BF00773402