value of the loss distribution, but also on its degree of non-uniformity in a way that is reminiscent of a mean square deviation. We observe that Eq. (\[eq:p-lower-bound\]) provides a useful lower bound only for $\{\eta\}$ distributions violating uniformity to only a small degree. When the expression inside the curly braces becomes negative, the assumption made on the magnitude of the second term of Eq. (\[eq:inhomo-prob-def\]) becomes invalid, and therefore the formula does not give a correct lower bound.

The estimated lower bound (\[eq:p-lower-bound\]) decreases with increasing degree of non-uniformity, in accordance with a naive expectation. However, as we shall show later, numerical simulations taking into account the full complexity of the problem provide evidence to the contrary: departure from uniformity can result in improved efficiency.

![The coefficients for the $\left<\eta\right>$ dependent second order term in the Taylor series expansion of $p_n^{\mathrm{\max}}(\{\eta\})$. The coefficients have been obtained by second order fitting to numerically generated values for a hypercube sized $n=8$. The confidence of each fit is represented on the graph by an error bar. It is clearly visible that the coefficients are always greater than $2^{-n}$. Another feature, that is more suggestive on a linear scale, is that the points 