Assume an i.i.d sample of a scalar dependent variable $y_i$ and a $k$-dimensional regressor $x_i$. Assume that $\mathbb{E}[y_i \mid x_i] = \exp\left(x_i'\beta_0\right)$ and $Var(y_i \mid x_i) = \exp\left(x_i'\gamma_0\right)$. Also assume that the non-linear least squares estimator $\widehat{\beta}$ of $\beta_0$ is $\sqrt{N}$-consistent and asymptotically normal, i.e., $$\sqrt{N}\left(\widehat{\beta} - \beta_0\right) \overset{d}{\rightarrow} N(0, H_0^{-1}V_0 H_0^{-1}) $$ Find an expression for $H_0$ and $V_0$.

I know the general formulas for $H_0$ and $V_0$ for non-linear least squares, that is, for the non-linear least squares estimator $$\widehat{\theta}_N = \arg\min_{\theta} \frac{1}{N} \sum_{i=1}^{N} (y_i - g(x_i, \theta))^2 $$ where $g(x_i, \theta) = \mathbb{E}\left[y_i \mid x_i = x \right]$, we have: $$V_0 = 4\mathbb{E}\left[(y_i - g(x_i, \theta_0))^2 \frac{\partial g(x_i, \theta_0)}{\partial \theta} \frac{\partial g(x_i, \theta_0)}{\partial \theta'} \right] $$ and $$H_0 = 2\mathbb{E}\left[ \frac{\partial g(x_i, \theta_0)}{\partial \theta} \frac{\partial g(x_i, \theta_0)}{\partial \theta'}\right] $$

where $\theta_0$ denotes the true parameter.

I'm unsure of how to compute the expressions using these results.


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