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How to Attain Minimax Risk with Applications to Distribution-Free Nonparametric Estimation and Testing

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  • Karl H. Schlag

Abstract

We show how to a derive exact distribution-free nonparametric results for minimax risk when underlying random variables have known finite bounds and means are the only parameters of interest. Transform the data with a randomized mean preserving transformation into binary data and then apply the solution to minimax risk for the case where random variables are binary valued. This shows that minimax risk is attained by a linear strategy and the the set of binary valued distributions contains a least favorable prior. We apply these results to statistics. All unbiased symmetric non-randomized estimates for a function of the mean of a single sample are presented. We find a most powerful unbiased test for the mean of a single sample. We present tight lower bounds on size, type II error and minimal accuracy in terms of expected length of confidence intervals for a single mean and for the difference between two means. We show how to transform the randomized tests that attain the lower bounds into non-randomized tests that have at most twice the type I and II errors. Relative parameter efficiency can be measured in finite samples, in an example on anti-selfdealing indices relative (parameter) efficiency is 60% as compared to the tight lower bound. Our method can be used to generate distribution-free nonparametric estimates and tests when variance is the only parameter of interest. In particular we present a uniformly consistent estimator of standard deviation together with an upper bound on expected quadratic loss. We use our estimate to measure income inequality.

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Bibliographic Info

Paper provided by European University Institute in its series Economics Working Papers with number ECO2007/04.

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Date of creation: 2007
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Handle: RePEc:eui:euiwps:eco2007/04

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Keywords: exact; distribution-free; nonparametric inference; finite sample theory;

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  1. Djankov, Simeon & La Porta, Rafael & Lopez-de-Silanes, Florencio & Shleifer, Andrei, 2008. "The law and economics of self-dealing," Journal of Financial Economics, Elsevier, vol. 88(3), pages 430-465, June.
  2. Karl H. Schlag, 2006. "Designing Non-Parametric Estimates and Tests for Means," Economics Working Papers ECO2006/26, European University Institute.
  3. Karl Schlag, 2006. "ELEVEN - Tests needed for a Recommendation," Economics Working Papers ECO2006/2, European University Institute.
  4. Stoye, Jörg, 2009. "Minimax regret treatment choice with finite samples," Journal of Econometrics, Elsevier, vol. 151(1), pages 70-81, July.
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