# Bias of the Hajek estimator (potential errors in Technical Point 12.1 of Causal inference: what if)

# Summary

This article claims that the Hajek (modified Horvitz-Thompson) estimator is not an unbiased estimator of the expectation of potential outcome.

Specifically, I point out that Technical Point 12.1 of `Causal inference: what if`

would contain potential error about the bias.

PDF version of the book is published on this page

The link of Goodreads is here:

I also propose to refer the name of the `Hajek estimator`

when introducing the modified Horvitz-Thompson estimator.

I emphasize my proposition by code-block as follows:

<Proposition 1> I propose ...

I would appreciate it if the experts of causal inference could discuss this issue with me and reflect the discussion to the book, `Causal inference: what if`

.

# Notation

- : Random variable of outcome
- : Random variable of treatment. This article focus on discrete treatment.
- : Random variable of covariate
- : Potential outcome of treatment level
- : Indicator function
- : Conditional probability of treatment given .

# Introducing Technical Point 12.1

This chapter introduces related statements in Technical Point 12.1

In Technical Point 12.1, IP weighted (IPW) mean for treatment level is defined as follows:

In Technical Point 2.3 and 3.1, this expectation is shown to be equal to the counterfactual mean (expectation of potential outcome), ].

The estimator (empirical approximation) of this weighted mean is known as the Horvitz-Thompson estimator,

Next, the modified Horvitz-Thompson estimator is defined as follows:

<Proposition 1> I found that the estimator of equation (3) is named the Hajek estimator. Therefore, I propose to refer the name of the Hajek estimator when introducing the modified Horvitz-Thompson estimator.

In Technical Point 12.1, the Hajek estimator is introduced as an unbiased estimator of the following ratio of expectations **(this statement might be incorrect)**:

Under positivity, this ratio of expectations is proved to be equal to the IP weighted mean because

# Bias of Hajek estimator

In this chapter, I point out that the Hajek estimator is not an unbiased estimator of the expectation of potential outcome.

## Ratio estimator

Ratio estimator is defined to be the ratio of means of two random variables (https://en.wikipedia.org/wiki/Ratio_estimator).

In general, ratio estimates are known to be biased.

The bias is shown as equation (8.5) of the link: https://jkim.public.iastate.edu/teaching/book8.pdf.

## Hajek estimator and Ratio estimator

As the equation (8.14) of the teaching book above, the Hajek estimator is a special case of the ratio estimator, where in the ratio estimator.

<Potential Error 1> Therefore, the Hajek estimator should be a biased estimator, and the bias would be as follows: