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Hello , I am a student who's gonna start honours in mathematics . Currently , I am at the stage of finding a suitable honours thesis topic . I've chosen my supervisor , who's research interest is on statistical inference and probability theory , but more on inference , I suppose.

My undergraduate coursework consists mostly of pure mathematics , and I have not really much knowledge in statistical science , but I am really interested in the theory of statistics , that's why I chose a supervisor doing stat.

But since my honours is in mathematics , my thesis should be sufficiently pure mathmatical. So my supervisor suggest that I could do something in shrinkage estimation , but he is even struggling on finding the right paper on it . And I guess I'd better conduct some research myself .

I have no suffcient background in statistics , as I mentioned above , but I've done a lot of courses on mathematics( group ,rings ,field extensions , galois theory , representation theory , analysis,probability theory, general topology , algebraic topology ,moduli space etc. ) and small project also . Now I really wanna combine theses things with theory of statists, but dont know where to start with.

I've heard that it is possible to put geometries into statistical theory , and this looks very attractive :D . I am particularly interested in geometry and topology .

I am also thinking doing something on statistical inference and stochastic analysis.

However , as my supervisor suggest , shrinkage estimation is a very good choice . So maybe it would be better to follow his suggestion ?

Anyway , I am really excited , but also struggling on choosing the right topic. Could anyone give me some advice on it ? Both reference to good papers on shrinkage estimation or perhaps suggestions on other related good topics would be both great . Thanks in advance !

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Since you have done representation theory, you should have a look at Persi Diaconis wonderful:

Link

(which can be downloaded freely from that link). It should contain a lot of possible projects!, P Diaconis often uses the frase " somebody should look into this ... " There is a lot which is undone there!

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The intersection between computability theory and statistics is pretty interesting. From this paper by Vovk (2009): "It is widely accepted that advances in computing have brought about deep changes in the theory and practice of statistics. However, the use of the theory of computing, and, in particular, of its core notion of computability, has been very limited in the classical areas of statistics, such as parameter estimation and hypothesis testing."

Relatedly, Ackerman, et al. (2011) demonstrate a computable random variable $(X,Y)$ with non-computable conditional distribution $P(Y \mid X)$.

Certainly this area is pretty "mathy"; it remains to be seen if it has implications for statistical practice.

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