Consider the Poisson regression model with conditional mean ( mu exp left( mathbf x prime beta ight) ) 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 ight) ) and ( varepsilon sim operatorname iid left 0, sigma 2 ight ) (a) Derive the nonlinear least squares equations for ( left( boldsymbol beta , sigma 2 ight) ) 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

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Question: Consider the Poisson regression model with conditional mean (mu=exp left(mathbf{x}^{prime} beta ight)). Treat the estimation problem as an unweighted nonlinear squares problem in which (y=mathrm{E}[y

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.

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