All programmes > Advanced Nonparametric Statistics and Smoothing

Advanced Nonparametric Statistics and Smoothing (B-KUL-G0A23A)

6 ECTSEnglish39 First termCannot be taken as part of an examination contract
Gijbels Irène |  Sznajder Dominik (substitute)
POC Master in statistiek

Aims

This course presents to the students an overview of  recent nonparametric techniques in statistical analysis and the use of these techniques in a variety of disciplines. The discussed techniques form the basis of  modern nonparametric or so-called smoothing procedures. The idea of this course is to get the students acquainted with the fundamentals, basic properties and use of  the most important recent nonparametric techniques. One of these techniques will be explored in more detail.  A second aim is to get students acquainted to research questions in this domain. As such the students will be exposed to get insights in the usefulness of nonparametric techniques and to formulate questions related to these.

Previous knowledge

Students have good knowledge about the basic principles of Probability Theory and Statistics, and are acquainted with these principles. They are familiar with, among others: concepts of r.v. and r. Vector and their basic characteristics (joint, marginal and conditional distrubutions and expectations), estimators and their properties (bias, variance, consistency, ...), (exact and asymptotic) distribution of an estimator or random quantity, (asymptotic) normality, law of large numbers and central limit theorem and the use of these results, maximum likelihood methods. Furthermore, they have the necessary mathematical knowledge about, among others, functions and their properties, limits and series, differentials and integrals, Taylor expansion, function spaces.
Beginning conditions: Students have had a solid course in probability theory and statistics and have as well had a basic analysis course which has covered the topics mentioned above.

Is included in these courses of study

Activities

6 ects. Advanced Nonparametric Statistics and Smoothing (B-KUL-G0A23a)

6 ECTSEnglishFormat: Lecture39 First term
Gijbels Irène |  Sznajder Dominik (substitute)
POC Master in statistiek

Content

The course will treat fundamentals, basic properties and use of modern nonparametric techniques:
Kernel estimators, local polynomial estimators, penalized likelihood techniques, spline approximations and spline smoothing, orthogonal series and wavelet techniques, among others.
These so-called smoothing techniques are applied in a variety of application areas in medicine (e.g. in nonparametric estimation of the hazard or survival function), in engineering (kernel estimators, neural networks, classification and pattern recognition, unsupervised learning, image analysis,...), in econometrics and economics (e.g. nonparametric estimation of a trend or volatility, moving averages), in social sciences (e.g. non- and semiparametric models to describe heterogeneity).
 
A table of content for the course can read as follows:
1. overview of nonparametric methods for estimating a density: kernel estimation methods, nearest-neighbour methods, maximum-likelihood-based methods, orthogonal series method, wavelets, ...
2. kernel estimators of densities: basic properties (bias, variance, mean squared error), asymptotic properties, asymptotic normality, rates of convergence (and their meaning/interpretation), selection of smoothing parameters (via cross-validation, plug-in, bootstrap or resampling procedures, ...).
3. nonparametric estimation of a regression function: the  cases of fixed and random design, homoscedasticity and heteroscedasticity, Nadaraya-Watson estimator, Gasser-Müller estimator, weighted least-squares methods, local polynomial fitting, splines, P-splines, wavelets, ... The impact and choices of parameters in each of these techniques will be discussed.
4. nonparametric estimation of hazard functions and applications (e.g. in survival analysis).
5. multivariate regression models: additive modelling and backfitting algorithms, dimension reduction techniques.
6. nonparametric smoothing and deconvolution problems (e.g. measurement errors).
7. nonparametric estimation of boundaries and frontiers with applications in image analysis and econometrics (for example).
8. modelling dependencies and nonparametric techniques, for example, use of nonparametric techniques in time series context.
9. other applications of nonparametric techniques: classification techniques, neural networks, statistical learning and data mining, modelling dependencies, ....
 
Parts 1---3 are basic items and will  be treated each year.  A further selection of minimal 2 items from items 4—9 will be made and this selection can possibly alter from year to year.

Format: more information

The contents of the three basic items will be presented to the students. A selection  from the set Items 4---9 will be covered.
Since this course is preparing the students to a research-oriented direction, it is also required that the students get acquainted to the literature in this domain.  As such each student will be asked to give a presentation (seminar) during the semester. The topic of this presentation must be linked to the use of nonparametric methods (discussed in the course) in an specific problem or area of application. The topic, possibly proposed by the student, has to be discussed and approuved by the instructor. 
Students will also be asked to use available statistical software on modern nonparametric techniques (software available as packages of statistical software, e.g. R) to get acquainted with the discussed methods.  A possibility is to do this as part of the presentation.

Evaluation

Evaluation: Advanced Nonparametric Statistics and Smoothing (B-KUL-G2A23a)

Type : Partial or continuous assessment with (final) exam during the examination period