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A Parametric Approach to Nonparametric Statistics pp 163–186Cite as

Optimal Rank Tests

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Abstract

Lehmann and Stein (1949) and Hoeffding (1951b) pioneered the development of an optimal theory for nonparametric tests, parallel to that of Neyman and Pearson (1933) and Wald (1949) for parametric testing. They considered nonparametric hypotheses that are invariant under permutations of the variables in multi-sample problems so that rank statistics are the maximal invariants, and extended the Neyman-Pearson and Wald theories for independent observations to the joint density function of the maximal invariants. Terry (1952) and others subsequently implemented and refined Hoeffding’s approach to show that a number of rank tests are locally most powerful at certain alternatives near the null hypothesis. We shall first consider Hoeffding’s change of measure formula and derive some consequences with respect to the two-sample problem. This formula assumes knowledge of the underlying distribution of the random variables and leads to an optimal choice of score functions and subsequently to locally most powerful tests. Hence, for any given underlying distributions, we may obtain the optimal test statistic.

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Notes

  1. 1.

    Lehmann & Stein considered the case for two samples and Hoeffding for multi-samples.

  2. 2.

    See also Royston (1982) who provides the approximation \(E\left [X_{\left (i\right )}\right ] =\mu +\sigma \varPhi ^{-1}\left ( \frac{i} {m+1}\right )\left [1 + \frac{\left ( \frac{i}{m+1} \right )\left (1- \frac{i}{m+1} \right )} {2\left (m+2\right )\left [\phi \left [\varPhi ^{-1} \frac{i} {m+1} \right ]\right ]^{2}} \right ]\).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, Canada

    Mayer Alvo

  2. Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong, China

    Philip L. H. Yu

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  1. Mayer Alvo
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  2. Philip L. H. Yu
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Alvo, M., Yu, P.L.H. (2018). Optimal Rank Tests. In: A Parametric Approach to Nonparametric Statistics. Springer Series in the Data Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-94153-0_8

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