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MINIMAX ESTIMATION OF A VARIANCE

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THOMAS S. FERGUSON^{1 } AND L YNN KUO^{2}

^{1} *Mathematics Department, UCLA, Los Angeles, CA 90024, U.S.A.*

^{2} *Statistics 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.

**Source**
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