  An explicit representation of an arbitrary zero-mean distribution as the
mixture of (at-most-)two-point zero-mean distributions is given. Based in this
representation, tests for (i) asymmetry patterns and (ii) for location without
symmetry conditions can be constructed. Exact inequalities implying
conservative properties of such tests are presented. These developments extend
results established earlier by Efron, Eaton, and Pinelis under a symmetry
condition.
