Bivariate location depth.(English)Zbl 0905.62002

Summary: The half-space depth of a point $$\theta$$ relative to a bivariate data set $$\{x_1, \dots, x_n\}$$ is given by the smallest number of data points contained in a closed half-plane of which the boundary line passes through $$\theta$$. A straightforward algorithm for the half-space depth needs $$O(n^2)$$ steps. The simplicial depth of $$\theta$$ relative to $$\{x_1, \dots, x_n\}$$ is given by the number of data triangles $$\Delta (x_i,x_j,x_k)$$ that contain $$\theta$$; this appears to require $$O(n^3)$$ steps. The algorithm proposed here computes both depths in $$O(n \log n)$$ time, by combining geometric properties with certain sorting and updating mechanisms. Both types of depth can be used for data description, bivariate confidence regions, $$p$$-values, quality indices and control charts. Moreover, the algorithm can be extended to the computation of depth contours and bivariate sign test statistics.

MSC:

 62-07 Data analysis (statistics) (MSC2010) 62-04 Software, source code, etc. for problems pertaining to statistics 65C99 Probabilistic methods, stochastic differential equations

AS 307
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