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My test (All of Statistics, second edition, Wasserman) contains the following:

Reweighted Least Squares Algorithm (for Logistic Regression)

  1. Choose starting values $\hat{\beta}^0 = (\hat{\beta}_0^0, \ldots, \hat{\beta}_k^0)$, and compute $p_i^0$ using the logistic regression model.

  2. Set

$$Z_i = \operatorname{logit}(p_i^s) + \frac{Y_i - p_i^s}{p_i^s(1-p_i^s)}$$

  1. Let $W$ be a diagonal matrix with $(i,i)$ element equal to $p_i^s(1-p_i^s)$.

  2. Set:

$$\hat{\beta}^s = (X^TWX)^{-1}X^TWY$$

This corresponds to doing a weighted linear regression of $Z$ on $Y$.

  1. Set $s = s + 1$ and go back to step 1.

What is curious about this is $Z_i$ is defined but then never used. Am I correct in thinking the formula for the coefficients is supposed to instead be:

$$\hat{\beta}^s = (X^TWX)^{-1}X^TWZ$$

and that it should read, "This corresponds to doing a weighted linear regression of $X$ on $Z$." ? The fact that $Z$ is defined with only one index makes me think it's supposed to be a transformed version of $Y$ rather than $X$.

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