Different powers of coefficient - solvable within GLM?

Question

I have model where coefficient is of different powers:

$$\mbox{log} ( \mu_{i} ) = \alpha + \beta x_1 + \beta^2x_2 + \beta^3x_3 + ... + \beta^nx_n \\ \\ N_{i} \sim \mbox{Poiss} ( \mu_{i} ) $$

$N_i$ and $x_i$ is the data, $\alpha$ and $\beta$ are model coefficients.

1. Can this problem be somehow transformed so that it can be solved using GLM model? I.e. so that I can use the glm() function in R.
2. If not, how can I solve this problem using some frequentist package in R? I can write model in WinBUGS but prefer to have "frequentist" solution in R, because it is much faster and inference is easier (one has p-values, t-tests, F-tests...). But which function or package use to fit it?

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If you are interested how this problem was created: actually taking model from this question and adding coefficient $\beta$ for $\mbox{log} ( \mu_{i,j} )$ on the predictor side of the equation. While the transformation of the original model for use in GLM was elegant and without problem, adding $\beta$ coefficient complicated things and leads to the model presented here.

Answer 1

This appears to be a generalized nonlinear model.

One could always use nonlinear least squares, and then iteratively reweight the observations according to their variance under the model (akin to using IRLS to estimate GLMs).

However, there are packages that can fit generalized nonlinear models in R.

For example, the gnm function in the gnm package seems to be a likely choice.

Another alternative would be to maximize the likelihood directly; there are only two parameters, and even more conveniently, conditional on the value of $\beta$, the likelihood of $\alpha$ can be readily maximized. As a result, this reduces it in effect to a univariate problem (for a given $\beta$, one can compute the ML estimate for $\alpha$ and evaluate the (log-)likelihood). The derivative is also computable, if need be, and there are several optimization functions in R.