  We present a detailed derivation of some estimators of Shannon entropy for
discrete distributions. They hold for finite samples of N points distributed
into M "boxes", with N and M -> oo, but N/M < oo. In the high sampling regime
(<< 1 points in each box) they have exponentially small biases. In the low
sampling regime the errors increase but are still much smaller than for most
other estimators. One advantage is that our main estimators are given
analytically, with explicitly known analytical formulas for the biases.
