Special nasty cases

Here we explore some special cases to see the robustness of the method.

1) If no experimental observation contributes to bin $B_\ell$ according to one of the criteria above, then we shall find $X_\ell=0$ and especially $Y_\ell=0$ . The latter condition is valid as an exclusion condition (meaning that we discard that point and we do not perform further operations on it, neither do we output it).

2) if only one experimental observation - call it interval $b$ , dropping indices - contributes to bin $B_\ell$ , then we have

$$X_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyl... ...nsuremath{\displaystyle{\vert b\vert m}}}}{{\ensuremath{\displaystyle{e}}}}}}}$$

$$Y_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyl... ...\displaystyle{\vert b\vert^2m^2}}}}{{\ensuremath{\displaystyle{e^2(C+1)}}}}}}}$$

and so

$$R_\ell={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyl... ...emath{\displaystyle{e(C+1)}}}}{{\ensuremath{\displaystyle{m\vert b\vert}}}}}}}$$

that is the experimental rate as in pixel $b$ ;

$$\sigma_{R_\ell}={\ensuremath{\displaystyle{\frac{{\ensuremath{\di... ...isplaystyle{e\sqrt{(C+1)}}}}}{{\ensuremath{\displaystyle{\vert b\vert m}}}}}}}$$

that is the same s.d. that can be calculated directly for $b$ , augmented by factor

$$\sqrt{{\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle... ...ath{\displaystyle{{\ensuremath{\left\vert{ b\cap B_\ell }\right\vert}}}}}}}}}}$$

that takes into account the extrapolation error.