Measures of Location and Asymmetry

Abstract

Three different approaches to the problem of defining a location parameter in the case of asymmetry are considered. First, the distribution F is approximated as closely as possible from above and below by two symmetric distribution functions, and the points 0 and 0 of symmetry of these two distributions are called lower and upper points of symmetry , respectively. The interval [O, V is now a set of location points for F. Next, the class of location parameters that are location-and scale-invariant and preserve order is considered. This class forms a location set which turns out to coincide with [0, 01. Finally, it is shown that by allowing parameters to be function valued, it is possible to find a parameter 0F(0) that has the defining property of the point of symmetry for general, continuous F. OF(-) is essentially unique. Its minimum value is 0 and its maximum value is 0. Estimates, confidence regions and tests for OF(-), and 0 are considered. Extensions to the case where one is interested only in the location of the central part of F are given. Applications to the measurement model with asymmetric errors, measures of asymmetry (skew-ness), and tests of symmetry, are treated.