MA8703 Theoretical Statistics pilot

Exam

Oppgave som løses i løpet av tre timer. Besvarelsen diskuteres med sensorer og en presentasjon av temaet.

Tidsplan for eksamensdagen:

0900-1200: Kandidat arbeider med oppgave som leveres som PDF ved e-post. 1200-1300: Kandidat finpusser på presentasjon. Sensorer diskuterer besvarelsen og oppgaven. 1300-1315: Digital oppkopling av møte og presentasjon. 1315-1335: Kandidaten gir en presentasjon med tittel: ‘The Kolmogorov definition of conditional probability and expectation’. 1335-1345: Spørsmål fra sensorer. 1345-1400: Kandidat og sensorer diskuterer besvarelsen og oppgaven.

For presentasjonen forventes det at definisjonen gitt i

Kolmogorov, A. Foundations of the Theory of Probability. Chelsea (1956), 1933. (Se spesielt sidene v, 12-13, 24, 37, 41, 47-56)

sammenlignes med definisjonene gitt i lærebøkene av Shao og Keener.

Shao, J. Mathematical Statistics. 2nd ed. Springer Texts in Statistics. New York: Springer-Verlag, 2003. https://doi.org/10.1007/b97553. Keener, Robert W. 2010. Theoretical Statistics: Topics for a Core Course. Springer Texts in Statistics. New York: Springer-Verlag. https://doi.org/10.1007/978-0-387-93839-4.

I alle tilfeller brukes Radon-Nikodym teoremet. Det forventes at det forklares at den elementære definisjonen ved tettheter forklart i grunnkurs er et spesialtilfelle av den generelle definisjonen på sidene 47-56 gitt av Kolmogorov.

Course content

The course is taught only if enough students register. If too few students register, then the course is only given as a guided self-study. The course gives a broad introduction to classical theoretical statistics motivated by applications in statistical inference and statistical machine learning. The topics of the course expands on the contents of the course TMA4295 - Statistical Inference or equivalent. Together with the course MA8704 - Probability Theory it provides a theoretical basis for PhD students in statistics. The choice of topics will be decided based on the background and wishes from the students. Relevant possible topics are sufficiency, likelihood, conditionality, statistical manifolds, invariance, decision theory, admissibility, shrinkage, coherence, entropy and information, nonparametric Bayes, model selection, prediction versus estimation, data generating models, distribution estimators, and asymptotic theory.

Learning outcome

1.Knowledge. The student has a good theoretical knowledge about fundamental principles and concepts in statistical inference. The student knows about general methods for construction of methods for statistical inference. The student knows about cases where optimal methods are available including both Bayesian and frequentist perspectives. The student knows about methods and criteria for evaluation of methods for statistical inference and learning.

2.Skills.The student should understand and be able to use the classical methods, fundamental principles, and concepts in theoretical statistics as described above. They should be able to apply these methods to various problems in statistics and related mathematical modelling.

3. Competence. The student will be able to participate in scientific discussions, read research presented in theoretical statistical journals, and carry out research in statistics at high international level. They will be able to participate in applied projects and contribute by suggesting and analyzing methods for statistical inference and learning.

Learning methods and activities

Lectures or alternatively guided self-study and compulsory theoretical exercises. A final written or oral exam is the basis for the grade awarded in the exam. Specific exercises must be approved to be allowed to take the exam. The exam may be given only in English. Students are free to choose Norwegian or English for written assignments.

Compulsory assignments

Theoretical exercises.

Further on evaluation

See «Learning methods and activities».

Specific conditions

Compulsory activities from previous semester may be approved by the department.

Mandatory previous knowledge

TMA4295 - Statistical Inference or equivalent. The course demands some degree of maturity in both mathematics and statistics.

Course materials

Will be announced at the start of the semester. Keener Shao Miescke-Liese Wainwright

Exam and curriculum

The exam will be oral. Time and place will be announced via e-mail.

Curriculum:

The three mandatory exercises. Ch 6-9 (minus section on EM algorithm) in Casella, Berger (2002) (This overlaps with previous Stat.Inf course, but not completely.) Ch 1-7, 10, 12, 13 in Keener (2010) (Also overlap, but mathematical proofs with measure theory are new.)