# Variance of maximum likelihood estimators for Poisson distribution

I wish to find the variance of the ML estimators for a linear model for data which is distributed according to a Poisson distribution.

$$p_i(O_i;E) = \frac{E^{O_i} \exp(-E)}{O_i!}$$

I've found how to construct the MLE and find its variance[1] when I only wish to find its expected value from $$N$$ observations $$O_1, \dots, O_N$$:

$$\hat{E} = \frac 1 N \sum_{i=1}^N O_i$$ $$\mathrm{var}(\hat{E}) = \left. \left[ -\frac{\partial^2 \ln L}{\partial E^2} \right]^{-1} \right |_{E=\hat{E}} = \frac{\hat{E}^2}{\sum_{i=1}^N O_i}$$

I would like to generalise this to where the expected value is now a linear function.

$$P_i(O_i;a_1, a_2;x_i) = \frac{(a_1+a_2x_i)^{O_i} \exp(-(a_1+a_2x_i))}{O_i!}$$ where $$x$$ is some independent variable.

I can find $$\frac{\partial L}{\partial a_1}=0$$, and $$\frac{\partial L}{\partial a_2}=0$$, and see that I must numerically solve

$$N=\sum_{i=1}^N\frac{O_i}{a_1+a_2x_i}$$ and $$\sum_{i=1}^Nx_i = \sum_{i=1}^N\frac{O_ix_i}{a_1+a_2x_i}$$ to find $$a_1$$ and $$a_2$$.

How can I now find the variance of $$a_1$$ and $$a_2$$?

In $$\S$$7.3 of [1], Martin states that the variance matrix, $$\mathbf{V}$$, for a multivariate distribution is the inverse of the $$\mathbf{M}$$ matrix where $$M_{ij} = -N \mathrm{E}\left[ \frac{\partial^2 \ln P(O; \theta_1, \theta_2,\dots,\theta_p)}{\partial \theta_i \, \partial \theta_j} \right]$$ where $$\mathrm{E}[\dots]$$ is the expectation value and $$N$$ is the number of observations.

If I use this definition, I get a whole lot of $$x$$'s which I don't know what to do with.

I've also just found [2; eqn 47], in which the author also says that the variance matrix, $$\mathbf{V}$$, for a multivariate distribution is the inverse of the $$\mathbf{M}$$ matrix, except this time, where

$$M_{ij} = -\frac{\partial^2 \ln L}{\partial \theta_i\,\partial\theta_j}$$

Which is right? Is there another method?

[1] Martin (2012), Statistics for Physical Scientists: An Introduction, $$\S$$7.2

$$M_{ij} = \left. -\frac{\partial^2 \ln L}{\partial a_i \partial a_j}\right|_{a=\hat{a}}$$
It all seems self-consistent, and is valid in the limit of large $N$.
[1] R.J. Barlow (1989) "Statistics: A Guide to the Use if Statistical Methods in the Physical Sciences", $\S$5.3.4