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Measures of location and asymmetry in the plane

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K. Doksum (1975), using three different approaches, considered the problem of defining a location parameter for asymmetric univariate probability distributions. These approaches are extended to the bivariate case. For a given bivariate distribution function F, each approach yields a closed, convex set in the plane, any point of which serves as a measure of location for F. The first method is to construct a symmetric bivariate distribution function and use it to approximate F from above and below. To do this, three stochastic orderings of bivariate random vectors are introduced: the standard ordering, based on the joint distribution functions, an ordering based on the marginal distribution functions, and an ordering based on both the marginal and conditional distribution functions. Using each ordering, F, and every rotation of F through an angle (alpha) is approximated from above and below to obtain a location rectangle for F in every direction. These rectangles are rotated back through an angle -(alpha) and their intersection defines the location region for F. It is shown that the standard and marginal orderings yield exactly the same sets. The second approach is axiomatic: four axioms of location are defined and all functionals satisfying these axioms make up the location region for F. Finally, Doksum's function of symmetry is generalized to both marginal and conditional functions of symmetry. The intersection of the respective ranges of these functions in each direction constitutes a third location region for F. As in the univariate case, all three approaches define essentially the same set. The estimation of the location region as well as the consistency of the estimate are discussed for both the marginal and conditional orderings. Using a fourth stochastic ordering which is a stronger form of the conditional ordering, it is possible to give examples of location parameters by solving a minimization problem similar in spirit to Bayesian estimation;Two applications of the location region are given. The first uses the size of the region to characterize the degree of asymmetry in F, and the second uses points in the location region to define stochastic orderings of distribution functions. The multivariate generalization of these results is also discussed;Reference;Doksum, K. A. 1975. Measures of Location and Asymmetry. Scand. J. Stat. 2:11-22.