Seminar (SS 2017)

Non-parametric minimax-test theory

Preliminary discussion:

Thursday, April 27th, 2017, 13:30, MΛTHEMΛTIKON, INF 205, 4th floor, room 4.200

Time and location of the seminar:

The seminar will be held as a Blockseminar on:

July 10th, 2017, 09-19, MΛTHEMΛTIKON, INF 205, 4th floor, r4.414

July 11th, 2017, 09-19, MΛTHEMΛTIKON, INF 205, 4th floor, r4.200

Contact:

Sandra Schluttenhofer <schluttenhofer[at]math.uni-heidelberg.de>

Jan JOHANNES <johannes[at]math.uni-heidelberg.de>

Questions, please directly by email or by using the contact form.

Language:

The seminar will be in English if there is at least one non-German speaking participant. Otherwise the presentations will be in German.

Area:

Applied Mathematics, Stochastics

Please register for the seminar by using MÜSLI.

Description of the seminar:

This seminar will be an introduction into the minimax approach in hypothesis testing. We start with a classical Gaussian sequence model, in which we would like to be able to detect a signal, i.e. we want to test a zero mean against certain (possibly nonparametric) alternatives. The challenge is to find a test that performs best in some sense. For a simple alternative the form of a best test is given by the Neyman-Pearson lemma, for composite alternative hypothesis the situation is more difficult. An important question is how close the alternative can be to the null hypothesis such that the problem is still testable for prescribed error probabilities. Furthermore, how fast the alternative is allowed to approach the null hypothesis if the noise level tends to zero.

The main reference for this seminar will be the first three chapters of the book [1]. Depending on the number of participants it might also be possible to look at how this theory can be applied to other models, e.g. to a functional Gaussian model ([1]), to inverse problems ([2]) or to a sparse signal detection problem ([3]) .

Possible presentation topics are:

1. How do we compare tests? And how can we find optimal tests?
   ([1], Section 2.2, (the Neyman-Pearson lemma) lemma 2.1. + proof, Ex. 2.1)
2. The Bayesian approach in hypothesis testing and its connection to the minimax approach
   ([1], Section 2.3 + 2.4.2, Thm. 2.1 or Thm. 2.3 + proof)
3. Distances on the space of measures
   ([1], Section 2.2.4)
4. Minimax rate of testing for detecting non-zero coordinates
   ([3], Prop. 1 + proof, Prop. 2 + proof)
5. Asymptotics in Hypothesis testing
   ([1], Section 2.5)
6. Minimax rate of testing for ill-posed inverse problems
   ([2], Thm. 4.1 + proof or Thm. 4.2 + proof)

Requirements:

Statistics I,Statistics I, Probability theory I

Reference:

[1] Ingster, Yu.I. and Suslina, I.A. (2003). Nonparametric goodness-of-fit testing under Gaussian models, Lecture Notes in Statistics 169, New York, NY: Springer.

[2] Ingster, Yu.I., Sapatinas T. And Suslina, I.A. (2012). Minimax signal detection in ill-posed inverse problems, The Annals of Statistics, 40(3):1524–1549.

[3] Baraud, Y. (2002). Non-asymptotic minimax rates of testing in signal detection, Bernoulli, 8(5):577–606.