Straight Poisson (zero-skipping) weighted average

When $O_j=C_j$ and $\sigma_j^2=C_j$

$$\langle x \rangle_{\!\mathrm{w(1)}}={\ensuremath{\displaystyle{\f... ...nsuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ C_j }}}}}}} }}}}}}}$$

Here we need to eliminate the singularity when $C_j=0$ . In order to do so, we skip data points which are zero. Then if $N_{\mathrm{obs}}^*$ is the number of non-zero data points,

$$\langle x \rangle_{\!\mathrm{w(1)}}={\ensuremath{\displaystyle{\f... ...nsuremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ C_j }}}}}}} }}}}}}}$$

$$\sigma_{\langle x \rangle_{\!\mathrm{w(1)}}} = {\ensuremath{\disp... ..._{\!\mathrm{w(1)}}}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^* }}}}}}}}$$

$$\mathsf{GoF}_{(1)}= \sqrt{ {\ensuremath{\displaystyle{\frac{{\ens... ...th{\left({ \langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(1)}} }\right)}} }$$

where $\langle x\rangle^*$ is the simple average of the non-zero data points; and of course

$${\sigma}_{\langle x \rangle_{\!\mathrm{w(1)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(1)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(1)}}}$$