Minimax Estimation of a Variance

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. AMS 1980 subject classifications. Primary 62C20; Secondary 62G05.