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THIRD ORDER EFFICIENCY IMPLIES FOURTH ORDER

EFFICIENCY: A RESOLUTION OF

THE CONJECTURE OF J. K. GHOSH

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MASAFUMI AKAHIRA

*Institute of Mathematics, University of Tsukuba, Ibaraki 305,
Japan*
(Received September 8, 1994; revised June 15, 1995)

**Abstract.**
Under suitable regularity conditions, it is
shown that a third order asymptotically efficient estimator is fourth
order asymptotically efficient in some class of estimators in the
sense that the estimator has the most concentration probability in
any symmetric interval around the true parameter up to the fourth
order in the class. This is a resolution of the conjecture by Ghosh
(1994, *Higher Order Asymptotics*, Institute of Mathematical
Statistics, Hayward, California). It is also shown that the
bias-adjusted maximum likelihood estimator is fourth order
asymptotically efficient in the class.

*Key words and phrases*:
Asymptotically median
unbiasedness, fourth order asymptotically symmetric efficiency,
concentration probability, asymptotic cumulants, Edgeworth
expansion, maximum likelihood estimator.

**Source**
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