Good resources (online or book) on the mathematical foundations of statistics

Before I ask my question, let me give you a bit of background about what I know about statistics so that you have a better sense of the types of resources that I’m looking for.

I’m a graduate student in psychology, and as such, I use statistics almost every day. By now I’m familiar with a pretty broad array of techniques, mostly as they are implemented in the general structural equation modeling framework. However, my training has been in the use of these techniques and the interpretation of results — I don’t have much knowledge of the formal mathematical foundations of these techniques.

However, increasingly, I’ve had to read papers from statistics proper. I’ve found that these papers often assume a working knowledge of mathematical concepts that I don’t know much about, such as linear algebra. I have therefore become convinced that if I wish to do more than blindly use the tools that I have been taught, it would be useful for me to learn some of the mathematical basis of statistics.

So, I have two related questions:

  1. What mathematical techniques would be useful for me to know if I want to brush up on the mathematical foundation of statistics? I’ve encountered linear algebra pretty often, and I’m sure that learning about probability theory would be useful, but are there any other areas of math that would be useful for me to learn about?
  2. What resources (online or in book form) can you recommend to me as someone who wants to know more about the mathematical foundations of statistics?

Answer

Maths:

Grinstead & Snell, Introduction to Probability (it’s free)

Strang, Introduction to Linear Algebra

Strang, Calculus

Also check out Strang on MIT OpenCourseWare.

Statistical theory (it’s more than just maths):

Cox, Principles of Statistical Inference

Cox & Hinkley, Theoretical Statistics

Geisser, Modes of Parametric Statistical Inference

And I second @Andre’s Casella & Berger.

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Scortchi – Reinstate Monica

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