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THE 3/2TH AND 2ND ORDER ASYMPTOTIC EFFICIENCY OF

MAXIMUM PROBABILITY ESTIMATORS IN

NON-REGULAR CASES

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MASAFUMI AKAHIRA

*Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan*
(Received September 28, 1989; revised February 19, 1990)

**Abstract.**
In this paper we consider the estimation problem on
independent and identically distributed observations from a location parameter
family generated by a density which is positive and symmetric on a finite
interval, with a jump and a nonnegative right differential coefficient at
the left
endpoint. It is shown that the maximum probability estimator (MPE) is
3/2th order two-sided asymptotically efficient at a point in the sense
that it has the most concentration probability around the true parameter
at the point in the class of 3/2th order asymptotically median unbiased
(AMU) estimators only when the right differential coefficient vanishes
at the left endpoint. The second order upper bound for the concentration
probability of second order AMU estimators is also given. Further, it is
shown that the MPE is second order two-sided asymptotically efficient at
a point in the above case only.

*Key words and phrases*:
Higher order two-sided asymptotic efficiency,
maximum probability estimator, non-regular distributions, asymptotically
median unbiased estimator, asymptotic concentration probability.

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