THIRD ORDER EFFICIENCY IMPLIES FOURTH ORDER EFFICIENCY: A RESOLUTION OF THE CONJECTURE OF J. K. GHOSH

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.