An IPW estimator may be implemented as: (1) Estimate a logistic model for R given X (2) Compute the IPW for missingness (3) Take the Horvitz-Thompson or Hajek weighted mean - ThreadSky

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pausalz.bsky.social • 15 days ago

An IPW estimator may be implemented as:
(1) Estimate a logistic model for R given X
(2) Compute the IPW for missingness
(3) Take the Horvitz-Thompson or Hajek weighted mean

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pausalz.bsky.social•15 days ago

To build our M-estimator, we first need a logistic regression model. Luckily for us, the logistic model has a simple score function (which gives us the estimating function directly), which is shown below

pausalz.bsky.social•15 days ago

For the Horvitz-Thompson, we can get the corresponding estimating equation after doing a little math (left as an exercise for the reader)

pausalz.bsky.social•15 days ago

For the Hajek, we have a similar process of deriving the estimating equation

pausalz.bsky.social•15 days ago

One item you might recall is that the Hajek is bounded in the parameter space, while the Horvitz-Thompson is not. Here, you can see this feature in the estimating equations. This results from where the difference between Y and \mu is computed

pausalz.bsky.social•15 days ago

To complete our estimator, we can simply stack together the logistic regression estimating equations with our chosen weighted mean estimator. Again the sandwich gives us the variance directly

pausalz.bsky.social•15 days ago

Unlike the usual 'robust' or 'GEE' trick used with IPW estimator, this variance estimator is not conservative. We will return to this in a future week

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