I'm posting a difficult general linear model question which I would like to solve.

Question: Consider a probit regression model for $y \in ${$0,1$}:$E(y|x)=\Phi(x'b)$, where $\Phi$ is the standard normal cdf and x is a vector of covariates.The parameter vector of interest is b. Let $\hat{b}$ be the maximum likelihood estimator based on iid data $(y_i,x_i)$, $=1,...,n$ generated from the probit model.

We wish to construct a weighted least square algorithm which iterates as $b^{t+1}=b^t+\delta^t$, $t=0,1,2,3,...$, where

$\delta^t = $ arg $min_{\delta}\Sigma_{i=1}^n w_i(y_i-x_i\delta)^2$,

so that any limiting point of this iteration is equal to the maximum likelihood estimator $b^{\infty} = \hat{b}$, $(*)$ under some regularity conditions.

(i) Find the formula for the weight $w_i$'s

(ii) Write down the regulatory conditions

(iii)Prove $(*)$

I'm just trying to get some insight as to how to get started.


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