Consider the Poisson regression model with conditional mean (mu=exp left(mathbf{x}^{prime} beta ight)). Treat the estimation problem as

Consider the Poisson regression model with conditional mean \(\mu=\exp \left(\mathbf{x}^{\prime} \beta\right)\). Treat the estimation problem as an unweighted nonlinear squares problem in which \(y=\mathrm{E}[y \mid \mathbf{x}]+\varepsilon\), where \(\mathrm{E}[y \mid \mathbf{x}]=\exp \left(\mathbf{x}^{\prime} \beta\right)\) and \(\varepsilon \sim \operatorname{iid}\left[0, \sigma^{2}\right]\).

(a) Derive the nonlinear least-squares equations for \(\left(\boldsymbol{\beta}, \sigma^{2}\right)\). Compare the leastsquares and the maximum likelihood equations for \(\beta\) and explain the difference between them.

(b) Derive the weighted nonlinear least-squares equations for \(\beta\). Explain your choice of weights. [Weights are used to handle heteroskedasticity].

(c) Compare the weighted nonlinear least-squares and the maximum likelihood equations and explain the similarities, if any.