Rank Test Based On Depth Function

Rank test is a simple and practical', method ofnon-parametric statistics based on rank statistics, which is a kind of statistics based on ranks. There is a natural linear order relationship among the data of one-dimensional sample, so which can be ordered according to the size of the sample points, however, since there is no natural linear order relationship among data of multi-dimensional sample, it is impossible togain multi-dimensional order statistics in accordance with the order of the size of sample, neither is it impossible to extend many useful methods for one-dimensional nonparametric oiatistics to those for high dimensional ones. Thus occurred much trouble in the analysis of multi-dimensional data statistics. So it's a very important problem for statistician to develop the multi-dimensional nonparametric method. In document [1], a new method is introduced, that is, we are able to calculate therank based on depth function by introducing the depth functionfor high dimensional data. Rank test makes great difference theoretically as well as practically only because under certain conditions, rank vector have a uniform distribution over is a permution of free of underlying distribution. Rank vector of one-dimensional sample meets the property talked about previously if it's a simple random sample with continuous underlying distribution. However, rank vector of multi-dimensional sample based on depth function has something to do with depth function and underlying distribution, which limits its practice. So far, statistician haven't formed systematic theory to discuss rank statistics based on depth function and its application. Since linear rank statistics has the most popularity in application in nonparametric statistics, this article fabricate the linear rank statistics based on depth function, and discussed its asymptoticproperty and application. Particular result is as follows:1. The condition of R having a uniform distribution overR is given.2. When underlying distribution belongs to elliptical distribution family, rank statistics based on depth function having a uniform distribution over is proved.3.Under proper condition, the linear rank statistics basedon depth function having the sameasymptotic distribution as is proved.4. The independence of two d -dimensional random vectorsis tested by employing the linear rank statistics based on depth function.5. The position and scale problem of two d-dimensional samples is discussed by using the linear rank statistics based on depth function.