The geometry of inadmissibility of independent observations for estimating a single parameter in two-parameter ordered symmetric problems

We consider settings where: (i) from one "primary" study we obtain an observation from a continuous symmetric distribution, and where the goal is to estimate the center of symmetry with absolute loss; (ii) from a second study, we are aware of an independent observation from a symmetric distribution with the same shape as in the first study, but with a possibly different center of symmetry; and (iii) we know the order of the two centers, but not their values. Practically, the two observations can be estimators of the locations, obtained separately from the two studies; the symmetry can be arising from the central limit behaviour of the estimators, and the known order often arises (e.g., in public health or medicine) from knowledge of the types of populations studied. In the literature, this problem has been dealt algebraically, and most often with combination of losses across parameters. In this paper we provide a geometric proof that, in order to estimate even only the location for the first "primary" population with absolute loss, without combining losses with the second population, the first observation alone is inadmissible in the presence of the second independent observation from the different unknown location. The geometric result provides a fresh understanding of the problem and its practical implications.