Material for the LTTC module `Fundamental Theory of Statistical Inference’.

The course descriptor, with suggested reading, is here. A full set of notes, subject to correction, is available here. The notes contain a set of problems.

The material is arranged in six chapters. Here, in advance, are the slides.

Slides for Chapter 1.

Slides for Chapter 2.

Slides for Chapter 3.

Slides for Chapter 4.

Slides for Chapter 5.

Slides for Chapter 6.

Here are the articles by Efron, Berger, and Bayarri and Berger.

Here is a version of the problems with solutions. These are certainly subject to correction: please inform me of any errors etc.

Recommended study/reading:

Week 1

It would be worthwhile studying the in-depth treatment of finite decision problems in Young and Smith, or elsewhere, as this gives insight to the different ideas of decision theory.

Problems labelled 2.n can be tackled: let me know of any problem that there would be a collective (rather than individual) wish for me to work through in due course. Let me know also if you would like the solutions to the problems to be posted now, rather than later in the course.

Chapter 1 of Cox’s book `Principles of Statistical Inference’ would be worth reading. Chapter 5 of that book is very nice, but it might make sense to read it in a week or two, when we have covered further material.

The Efron article is quite accessible, even at this stage. I particularly like the `statistical triangle’: where does your Ph.D. work lie on that?

The Fisher papers I mentioned in the lecture as being especially influential are: `On the mathematical foundations of theoretical statistics’, Phil. Trans. Roy. Soc. A, 222, 309-68 (1922); `Theory of statistical estimation’, Proc. Camb. Phil. Soc., 22, 700-25 (1925); `Two new properties of mathematical likelihood’, Proc. R. Soc. Lond. A, 144, 285-307 (1934). It is very interesting to read them, to see how differently some of the ideas are expressed from the way we do now.

Week 2

I enjoyed putting together the data-analytic examples in Young and Smith. They flesh out shrinkage and empirical Bayes ideas, and might be found illuminating.

Problems labelled 3.n can be tackled. Let me know if there are particular problems you would like me to speak about next week, by e-mail (alastair.young at imperial.ac.uk), as I may have a little time during that session.

This might be a good time to tackle Cox’s Chapter 5, and the Bayarri and Berger article, which is very thought provoking. You might have fun reading Bayes, T. `An essay towards solving a problem in the doctrine of chances’, Phil. Trans. Roy. Soc., 53, 370-418 (1763).

I will post the problems with solutions in the next week or so, unless there are objections.

Week 3

Problems 5.1-5.6 and 6.1-6.2 cover material discussed this week: all are quite short and direct. I intend to work through key aspects of 5.2, 5.4, 6.1 and 7.2 next week. Let me know if there are other problems you would like me to discuss.

Next week I also intend to complete the discussion of Fisherian ideas, and cover at least the preliminaries for the final Chapter 6. Chapters 2 to 4 of Cox would make very useful preliminary reading.

Week 4

Next week I intend to complete the material on frequentist theory, of which there is quite a bit remaining.

We have set in place a sound enough description of frequentist approaches to testing for both Berger articles to be fully comprehensible.

We have yet to cover material for problems 4.n and 7.m, and 6.3 requires a result we will discuss next week.

Week 5

I hope people might consider reading Chapters 7 and 8 of Young and Smith, which demonstrate how Fisherian ideas lie at the heart of commonly used, likelihood-based, inference procedures, which are probably more important in practice than optimal frequentist methods.

I am happy to receive e-mails with any queries about the material. The whole problem sheet can now be tackled! The calculations I went through at the end of the lecture suggested that for problem 7.4 the power of the unconditional test is 0.4497, while that of the conditional test is only fractionally less, 0.4488. Please check this.

The 2011 exam question is here, with its solution. This year’s assessment will be similar.