Sankhya: The Indian Journal of Statistics

2004, Volume 66, Pt. 4,  707-732

The Asymptotic Minimax Risk for the Estimation of  Constrained Binomial and Multinomial Probabilities

By

 Dietrich Braess and Holger Dette, Ruhr-Universit\"at Bochum, Germany

SUMMARY.  In this paper we present a direct and simple approach to obtain bounds on the asymptotic minimax risk for the estimation of  constrained binomial and multinomial proportions. Quadratic,  normalized quadratic and entropy loss are considered and it is demonstrated that in all  cases linear estimators are asymptotically minimax optimal. For the quadratic loss function the asymptotic minimax risk does not change unless a neighborhood of the point $1/2$ is excluded by the restrictions on the parameter space. For the two other loss functions the asymptotic behavior of the minimax risk is not changed by such additional knowledge about the location of the unknown probability. The results are also extended to the problem of minimax estimation of a vector of constrained multinomial probabilities.

AMS (1991) subject classification. 62C20 ;

Key words and phrases. Binomial  distribution, multinomial distribution, entropy loss, quadratic loss, constrained parameter space, least favourable distribution.

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