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## Rank, Sign, and Permutation Statistics

Statistics that depend on the observations only through their ranks can be used to test hypotheses on departures from the null hypothesis that the observations are identically distributed. Such rank statistics are attractive, because they are distribution-free under the null hypothesis and need not be less efficient, asymptotically. In the case of a sample from a symmetric distribution, statistics based on the ranks of the absolute values and the signs of the observations have a similar property. Rank statistics are a special example of permutation statistics.

### 13.1 Rank Statistics

The order statistics XN(1)XN(2) ≤ … ≤ XN(N) of a set of real-valued observations X1,..., XN ith order statistic ...

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