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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.

Suggested Citation

  • Karl H. Schlag, 2007. "How to Attain Minimax Risk with Applications to Distribution-Free Nonparametric Estimation and Testing," Economics Working Papers ECO2007/04, European University Institute.
  • Handle: RePEc:eui:euiwps:eco2007/04
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    References listed on IDEAS

    as
    1. Stoye, Jörg, 2009. "Minimax regret treatment choice with finite samples," Journal of Econometrics, Elsevier, vol. 151(1), pages 70-81, July.
    2. Karl Schlag, 2006. "ELEVEN - Tests needed for a Recommendation," Economics Working Papers ECO2006/2, European University Institute.
    3. Karl H. Schlag, 2006. "Designing Non-Parametric Estimates and Tests for Means," Economics Working Papers ECO2006/26, European University Institute.
    4. 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.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

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

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

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