Asymptotically Fully Efficient Fixed-Width Confidence Intervals for a Location Parameter

SYNOPTIC ABSTRACT

Let X₁, X₂, … be i.i.d. with distribution function F(x – θ), where F has a density f that is symmetric about the origin. f is assumed to be unknown. Assume further that the Fisher information is finite. For a given confidence coefficient, a two-stage procedure is proposed that gives an asymptotically consistent fixed-width confidence interval for θ for which the expected sample size is (asymptotically) as small as possible among all fixed-width confidence intervals based on translation equivariant estimators. The procedure does not depend on F and works under very mild regularity conditions. The results are based on use of Stone's (1975) adaptive maximum likelihood estimators, combined with the use of estimates of the Fisher information (truncated in a certain way) to determine an appropriate sample size.

Key Words and Phrases:

• Fixed-width confidence interval
• asymptotic consistency
• two-stage procedure
• asymptotic efficiency
• adaptive estimation