A generalized maximum likelihood characterization of the normal distribution

LetP be a probability measure on ℝ andI _x be the set of alln-dimensional rectangles containingx. If for allx ∈ ℝⁿ and θ ∈ ℝ the inequality holds,P is a normal distributioin with mean 0 or the unit mass at 0. The result generalizes Teicher’s (1961) maximum likelihood characterization of the normal density to a characterization ofN(0, σ²) amongall distributions (including those without density). The m.l. principle used is that of Scholz (1980).