Show that inference about (beta) based on (mathrm{EE}) is valid even when the NB model does not

Question:

Show that inference about $\beta$ based on $\mathrm{EE}$ is valid even when the NB model does not describe the distribution of the count variable $y_{i}$, provided that the systematic component $\log \left(\mu_{i}\right)=\mathbf{x}_{i}^{\top} \boldsymbol{\beta}$ specified in (6.22) is correct.

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