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I plan to study the theory behind Targeted Maximum Likelihood Estimation, Doubly Robust Estimation, and Semiparametric Theory. I have a background in bioinformatics: I took courses in basic linear agebra and proof-based calculus (and a little bit of functional analysis).

Based on university classes on these topics, the necessary math background comprises: measure theory, measure-based probability, functional analysis. And perhaps others. I'm a PhD student with limited time to dedicate to each subject (and quite frankly, I am not super interested in spending time prooving theorems). I was wondering whether there is a textbook that might serve as a unique reference for all the necessary background. In physics and engineering, there are books like Advanced Engineering Mathematics by Erwin Kreyszig. Is there a similar reference for higher-level statistics?

WeakLearner
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1 Answers1

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Unfortunately there aren't a lot of good textbook references for semiparametrics relative to other areas of statistics. A good start are the following Lecture Notes by Bodhisattva Sen which are specific to semiparametric statistics. For more background, his page contains a set of notes on theoretical statistics that would be a good reference. Neither of these notes is geared to causal inference.

If you are interested more in applications (especially causal), then at a minimum you will need to understand the notion of an influence function and how to derive it for various models. The following two papers provide intuition and many examples:

  1. Demsityfing statistical learning based on efficient influence functions
  2. Visually communicating and teaching intuition for influence functions

I also found these slides by Kennedy to be quite helpful - he has other references so I would suggest looking them up.

There is also a new text on Targeted Learning which I haven't used much, but the authors are very active in the area so it is likely a good reference.

I want to point out that the above references are good but may not be enough for someone wanting more rigour, in which case I would recommend:

  1. Asymptotic Statistics by van der Vaart (Chapter 25)
  2. Introduction to empirical processes and semiparametric inference by Kosorok
WeakLearner
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