MINIMAX ESTIMATION OF A VARIANCE

THOMAS S. FERGUSON^{1} AND LYNN KUO^{2}

1Mathematics Department, UCLA, Los Angeles, CA 90024, U.S.A. 2Statistics Department, University of Connecticut, Storrs, CT 06269, U.S.A.

(Received October 5, 1992; revised August 6, 1993)

Abstract. The nonparametric problem of estimating a variance based on a sample of size n from a univariate distribution which has a known bounded range but is otherwise arbitrary is treated. For squared error loss, a certain linear function of the sample variance is seen to be minimax for each n from 2 through 13, except n = 4. For squared error loss weighted by the reciprocal of the variance, a constant multiple of the sample variance is minimax for each n from 2 through 11. The least favorable distribution for these cases gives probability one to the Bernoulli distributions.

Key words and phrases: Admissible, minimax, nonparametric, linear estimator, moment conditions.