General Information

Elective for Master students in mathematical statistics.
Language of instruction: The course will be given in English

Contents

Weak convergence on general function spaces. Non-measurable functionals (Hoffman-Jörgensen's theory). Characterisation
of tightness and convergence of finite dimensional distributions.
Empirical processes. Covering numbers and bracketing numbers. VC
classes (Vapnik–Chervonenki's classes) of functions.
Functional differentiability (smooth statistical functionals). Application to survival
analysis (Nelson-Aalen and Kaplan-Meier estimators).
Quantile estimators.
Bootstrap methods. Functional differentiability for bootstrap. Bootstrap
for empirical processes.
Nonp arametric estimation of densities. Limit distributions. Convergence rates.
The partial sum process. Donsker's theorem for this.
Non parametric estimations of regression functions.
M and Z estimators. Applications to maximum likelihood och least square
estimators.
Empirical processes and partial sum process results for
weakly and strongly dependent stationary data. Kernel estimation of
densities and regression functions.
The empirical spectral process. Non parametric estimation of spectral densities.

Examination details

The examination consists of an oral exam and a project seminar.
Students who fail the regular exam are offered a re-examination shortly afterwards.

Marking scale: Fail, Pass, Pass with distinction.
For a passing grade on the course the student should pass the oral exam as well
the project presentation.
The final grade of the course is given by the grade of the oral exam.

601 Examination, 7,5 hp
Grading scale: Fail, Pass, Pass with distinction

Admission requirements

For admission to the course knowledge equivalent to MASC01 Probabilty theory, 7.5
credits, is required. English B.

Reading list

The main course literature is Van der Vaart's book (recommended buying), complemented with research papers. The other books are complementary reading. All three/four books are excellent reference literature.

Van der Vaart (1998), "Asymptotic Statistics", Cambridge University Press. (main source).
Van der Vaart and Wellner (1996), "Weak convergence and empirical processes", Springer.
Billingsley (1968), "Convergence of Probability Measures", Wiley.
Billingsley (1999), "Convergence of Probability Measures", Wiley, 2nd edition.

We have electronic access to Van der Vaart's book and Billingsley's,

Contact and other information

Director of studies: studierektor@matstat.lu.se

Syllabus

Kursplan NF (SV)

Syllabus NF (EN)