Abstract: For the class of nonnegative random variables with given mean, variance, and skewness and support bound, we present a sharp upper bound for the expectation of rounding due to the Jefferson rule. The result gives an estimate for average extra gains due to rounding down payments. Arguments of four-dimensional geometric moment theory implemented in the proof provide tools for refined evaluations of rates of convergence of probability distributions and positive linear operators.

1991 AMS Mathematics Subject Classification: Primary 60E15, 62P10; Secondary 90A28.

Key words and phrases: Gain of rounding, Jefferson rounding, mean-variance-skewness and support constraints, geometric moment theory, four-dimensional geometry.