The maximum-likelihood parameter estimates \(\hat{\beta}\) satisfy a self-consistency relationship: they are the coefficients of a weighted least squares fit, where responses are (4.29): \[ z_i = x_i^T\hat{\beta} + \cfrac{(y_i - \hat{p}_i)}{\hat{p}_i(1-\hat{p}_i)} \] and the weights are \(w_i=\hat{p}_i(1-\hat{p}_i)\). This connection has more to offer:
The weight RSS is the familiar Pearson chi-square statistic (4.30) \[ \sum_{i=1}^N\cfrac{(y_i-\hat{p}_i)^2}{\hat{p}_i(1-\hat{p}_i)} \]
a quadratic approximation to the deviance.
Asymptotic likelihood theory says that if the model is correct, then \(\hat{\beta}\) converges to the true \(\beta\).
A central limit theorem then shows that the distribution of \(\hat{\beta}\) converges to \(N(\beta, (\mathbf{X}^T\mathbf{XW})^{-1})\). (Can be derived from weighted least squares fit).