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Published online by Cambridge University Press: 05 June 2012
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.
Rank Statistics
The order statistics XN(1) XN of a set of real-valued observations order statistic are the values of the observations positioned in increasing order. The of among is its position number in the order statistics. More precisely, if are all different, then is defined by the equation
If Xi is tied with some other observations, this definition is invalid. Then the rank is defined as the average of all indices such that (sometimes called the or alternatively as (which is something like an uprank).
In this section it is assumed that the random variables have continuous distribution functions, so that ties in the observations occur with probability zero. We shall neglect the latter null set. The ranks and order statistics are written with double subscripts, because N varies and we shall consider order statistics of samples of different sizes. The vectors of order statistics and ranks are abbreviated to X N () and R N, respectively.
A rank statistic is any function of the ranks. A linear rank statistic is a rank statistic of the special form for a given matrix.
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