Sankhya: The Indian Journal of Statistics

1995, Volume 57, Series A, Pt. 3, pp. 472--485

 AN ANALOGY BETWEEN NONPARAMETRIC PROBLEMS OF ESTIMATING A DISTRIBUTION FUNCTION AND THEIR PARAMETRIC VERSIONS

By

  QIQING YU, State University of New York at Birmingham

and

LYNN KUO, University of Connecticut

SUMMARY.   The connection between the nonparametric problem of estimating an unknown continuous (or discrete) distribution function and the parametric problem of estimating the probability of a success of a binomial distribution is considered. It is found that a class of step function estimators in the nonparametric problem and the linear estimator in the binomial problem have the same decision theoretical properties (minimaxity, admissibility and Bayes). Consequently, it is proved that a class of step function estimators are all admissible under a wide class of integrated weighted square error loss functions for either continuous or discrete distributions. In particular, the empirical distribution function, is both minimax and admissible under a weighted loss function.

AMS (1991) subject classification.   Primary 62C15, secondary 62C10.

Key words and phrases. Admissibility, multiple beta prior, stepwise Bayes procedure, nonparametric estimation, parametric estimation, binomial distribution.

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