Abstract
The nonparametric problem of estimating a variance based on a sample of sizen 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 eachn from 2 through 13, exceptn=4. For squared error loss weighted by the reciprocal of the variance, a constant multiple of the sample variance is minimax for eachn from 2 through 11. The least favorable distribution for these cases gives probability one to the Bernoulli distributions.
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Ferguson, T.S., Kuo, L. Minimax estimation of a variance. Ann Inst Stat Math 46, 295–308 (1994). https://doi.org/10.1007/BF01720586
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DOI: https://doi.org/10.1007/BF01720586