or at least most register values are in the range $[1,\symRegRange]$, which is the case if $2^\symPrecision \ll \symPoissonRate \ll 2^{\symPrecision
+\symRegRange}$, the approximation of by is valid. This explains why the raw estimator works best for intermediate cardinalities. However, for small and large cardinalities many register values are equal to 0 or $\symRegRange+1$, respectively, which contradicts and ends up in the observed bias.

Improved Estimator
==================

If we knew the values $\symRegValVariate'_1,\ldots,\symRegValVariate'_\symNumReg$ for which transformation led to the observed register values $\symRegValVariate_1,\ldots,\symRegValVariate_\symNumReg$, we would be able to use the raw estimator $$\label{equ:raw_estimate_with_multiplicities}
\symPoissonRateEstimate 
= 
\frac{\symAlpha_\infty\,\symNumReg^2}{\sum_{\symRegVal=-\infty}^\infty \symCountVariate'_\symRegVal 2^{-\symRegVal}}$$ where $\symCountVariate'_\symRegVal := \vert\lbrace \symIndexI\vert \symRegVal = \symRegValVariate'_\symIndexI\rbrace\vert$ are the multiplicities of value $\symRegVal$ in $\lbrace\symRegValVariate'_1,\ldots,\symRegValVariate'_\symNumReg\rbrace$. Due to , the multiplicities $\symCountVariate'_\symRegVal$ and the multiplicities $\symCountVariate_\symRegVal$ for the observed register values have following relationships $$\label{equ:multiplicity_transformation}
\symCountVariate_0 = \textstyle\sum_{\symRegVal = -\infty}^{0} \symCountVariate'_\symRegVal,\
\symCountVariate_\symRegVal =  \symCountVariate'_\symRegVal\ (1\leq\symRegVal\leq\symRegRange),\
\symCountVariate_{\symRegRange+1} = \textstyle\sum_{\symRegVal = \symRegRange + 1}^{\infty} \symCountVariate'_\symRegVal.$$ The idea is now to find estimates $\hat{\symCount}'_\symRegVal$ for all $\symRegVal\in\mathbb{Z}$ and use them as replacements for $\symCountVariate'_\symRegVal$ in . For $\symRegVal\in[1,\symRegRange]$ we can use the trivial estimators $\hat{\symCount}'_\symRegVal := \symCountVariate_\symRegVal$. Estimators for $\symRegVal\leq 0$ and $\symRegVal\geq\symRegRange+1$ can be found by considering the expectation of $\symCountVariate'_\symRegVal$ $$\symExpectation(\symCountVariate'_\symRegVal)
=
\symNumReg\,
