Mighell-Poisson weighted average

When $O_j=C_j+\min{\ensuremath{\left({1,C_j}\right)}}$ and $\sigma_j^2=C_j+1$

$$\langle x \rangle_{\!\mathrm{w(2)}}={\ensuremath{\displaystyle{\f... ...uremath{\displaystyle{1 }}}}{{\ensuremath{\displaystyle{ C_j+1 }}}}}}} }}}}}}}$$

$$\sigma_{\langle x \rangle_{\!\mathrm{w(2)}}} = {\ensuremath{\disp... ..._{\!\mathrm{w(2)}}}}}}{{\ensuremath{\displaystyle{ N_{\mathrm{obs}}^* }}}}}}}}$$

$$\mathsf{GoF}_{(2)}= \sqrt{ {\ensuremath{\displaystyle{\frac{{\ens... ...{\left({ \langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(2)}}+1 }\right)}} }$$

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

$${\sigma}_{\langle x \rangle_{\!\mathrm{w(2)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(2)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(2)}}}$$