A Characterization of Population Weighted-Symmetry and Related Results

The concept of population weighted-symmetry (a generalization of population symmetry) is discussed and a characterization of weighted-symmetry provided. For the special case of population symmetry, this characterization yields a strong converse to the well-known result: if the distribution of a continuous random variable X is symmetric about θ, then Z= |X – θ| and the indicator function for the event {X ≥ θ} are stochastically independent. Implications and possible applications of the general weighted-symmetry results are considered.