Section of Statistics Department of Mathematics - - PowerPoint PPT Presentation

Section of Statistics

Department of Mathematics

wis.kuleuven.be/stat

31/03/2014

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Outline

Outline

1

General information about research, courses, theses

2

Information about each research group

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Questions

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General info

Three major research themes:

Nonparametric and semiparametric statistics Robust and computational statistics Financial mathematics and actuarial statistics

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General info

Courses offered by our section

We offer courses within the

1

Master of Mathematics

2

Master of Statistics

3

Master of Financial and Actuarial Engineering

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General info

Master of Mathematics - Option Applied Mathematics

Core courses:

1

Advanced statistical inference (S. Van Aelst) In this course modern statistical models and procedures such as e.g. resampling techniques (bootstrap, jackknife), robust statistical methodology, nonparametric association measures, censoring and survival analysis are

• introduced. The practical use of these methods will be discussed as well.

2

Stochastic models (J. De Spiegeleer) Becoming familiar with stochastic modelling of dependent stochastic variables, practicing examples of stochastic models.

3

Statistische modellen en data-analyse (M. Hubert) From the bachelor Mathematics, can still be chosen. In deze cursus worden de voornaamste multivariate statistische modellen bestudeerd, alsook de methoden om hiermee multivariate gegevens te

• analyseren. Er wordt vooral aandacht besteed aan de praktische aspecten bij

de analyse van concrete data-voorbeelden.

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General info

Master of Mathematics - Option Applied Mathematics

In-depth courses:

1

Advanced nonparametric statistics (I. Gijbels) [2014-2015]

2

Robust statistics (P. Rousseeuw, M. Hubert) [2015-2016]

3

Statistics of extremes (J. Beirlant) [probably 2015-2016]

4

Statistics for finance and insurance (T. Verdonck)

5

Selected topics in Mathematics (D. Paindaveine, ULB) [2014-2015]

6

Fundamentals of financial mathematics (W. Schoutens, P. Leoni)

7

Financial engineering (W. Schoutens, P. Leoni)

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General info

Master thesis

Diversity of topics, all driven by a real world problem. A typical thesis contains theoretical and applied parts. Their relative weight is flexible to some extent. Theoretical parts: literature review, study of theoretical properties, working

• ut proofs.

Applied parts: simulation study to verify theoretical properties and to study finite-sample behavior, real data analysis, use of R/Matlab, some programming.

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Research groups

Outline

1

General information about research, courses, theses

2

Information from each research group

3

Questions

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Research groups Nonparametric and semiparametric statistics

Nonparametric and semiparametric statistics

Key research topics: flexible regression models and modeling of complex data variable selection and sparse estimation methods investigating dependencies (conditional and unconditional) via copulas; modeling the dependence dynamics statistical analysis of functional data: core probabilistic and statistical concepts and properties

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Research groups Nonparametric and semiparametric statistics

Examples

5 10 15 20 25 10 20 30 40 Y1 Y2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 Y1 Y2

Kendall’s tau = 0.667 same dependence structure; different marginals estimation of the dependence structure via copulas, two or more dimensions

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Research groups Nonparametric and semiparametric statistics

Examples

modeling the dynamics of a dependence structure

• bservations (hourly measurements) of wind speeds at two (or more) weather

stations

Apr Mai Jun Jul 10 20 30 40 50 60 Kennewick − 2013 hourly average wind speed in mph Apr Mai Jun Jul 10 20 30 40 50 Butler Grade − 2013 hourly average wind speed in mph

forecast the wind speed in the region looking at both time series jointly

• ther related topics: portfolio selection in financial applications, study of the

value-at-risk

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Research groups Nonparametric and semiparametric statistics

Examples

flexible regression modeling and variable selection predict the level of atmospheric ozone concentration from daily meteorological measurements determine important pollutants, and their effects

20 40 60 80 10 20 30 humidity functional contribution of humidity data P−splines OLS 30 40 50 60 70 80 90 10 20 30 temp functional contribution of temp data P−splines OLS 12 / 32

Research groups Nonparametric and semiparametric statistics

Examples

flexible regression modeling and quantile regression describe the evolution in time for an infected patient (for patients with different conditions) conditional quantile regression

• −1

1 2 3 4 5 6 10 20 30 40 50 60 time since infection CD4 percentage after infection

τ = 0.5 τ = 0.1 τ = 0.9

• −1

1 2 3 4 5 6 10 20 30 40 50 60 70 time since infection CD4 percentage after infection

τ = 0.5 τ = 0.1 τ = 0.9 13 / 32

Research groups Nonparametric and semiparametric statistics

Thesis topics and promotors

Advisors: Ir` ene Gijbels, Dominik Sznajder and collaborators Semi-and nonparametric estimation of copulas and applications Investigating fundamental concepts when modeling dependencies with copulas Non-and semiparametric estimation of copulas Modeling Value-At-Risk, Portfolio selection, applications of copula modeling Exploiting the copula modeling for functional data Flexible regression models and variance/dispersion estimation Investigating the heteroscedasticity issues in flexible regression models

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Research groups Nonparametric and semiparametric statistics

Thesis topics and promotors

Statistics of Bernstein copula estimator The Bernstein copula estimator builds upon the empirical copula estimator that is appropriately smoothed by Bernstein polynomials. This estimator was introduced by Sancetta and Satchell (2004), and studied by Janssen et al. (2012, 2014). This project studies the asymptotic statistical properties (e.g. asymptotic distributional behavior) of the Bernstein copula estimator. The interest also goes to the applicability of this copula estimator and requires designing and conducting a simulation study to check the finite sample performance against its main

• competitors. Use of the R software and packages.

Advisor: Dominik Sznajder

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Research groups Robust and computational statistics

Robust and computational statistics

Key research topics: high-breakdown estimators and algorithms for high-dimensional and functional data robust inference, model selection, robust estimation for non i.i.d. data depth functions for multivariate and functional data classification and clustering of multivariate and functional data

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Research groups Robust and computational statistics

Examples

−10 −5 5 10 15 −5 5 10 15

Classical and robust tolerance ellipse

log(body) log(brain) MCD Classical

3.6 3.8 4.0 4.2 4.4 4.6 4.0 4.5 5.0 5.5 6.0 log.Te LS (all) LS (reduced) LTS

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Research groups Robust and computational statistics

Thesis topics and promotors

Inference for robust SUR The SUR model (seemingly unrelated regression) consists of several sets of linear regression equations, which are related via a specific structure of the covariance matrix of the error terms. Recently a robust version has been proposed (Hubert, Verdonck and Yorulmaz, 2013). Its robustness is illustrated via simulations and via data examples from actuarial statistics. In this thesis this robust method will be studied in detail, and robust inference via robust bootstrap (Salibian-Barrera, Van Aelst, and Willems, 2008) will be developed. Possible advisors: M. Hubert, S. Van Aelst, T. Verdonck

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Research groups Robust and computational statistics

Thesis topics and promotors

Clustering of multivariate and functional data via depth functions

1 2 3 4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time Respons

10 20 30 5 10 15 20

Multivariate Distance to central curve group 1 Multivariate Distance to central curve group 2

Possible advisors: M. Hubert, P. Rousseeuw

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Research groups Robust and computational statistics

Thesis topics and promotors

High dimensional data and elementwise contamination Standard robust methods consider an entire observation as regular or outlying. In high dimensions often only a few components of an observation are actually contaminated, but most observations are contaminated in at least one of their

• components. This is called elementwise contamination. Robustness measures for

this type of contamination have been proposed (Alqallaf, Van Aelst, Yohai, and Zamar, 2009). In this thesis methods that can handle elementwise contamination will be investigated and their properties will be compared. Possible advisors: S. Van Aelst, P. Rousseeuw

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Research groups Robust and computational statistics

Thesis topics and promotors

Robustness and sparsity When analyzing high-dimensional data one often aims to obtain sparse models because they reveal the most relevant structure, are more stable, and are easier to interpret. For regression this means that only few coefficients should be different from 0. For PCA it means that many loadings become zero. This sparsity is often

• btained by using some form of penalization in the estimation procedure.

In this thesis, methods will be studied for regression or PCA that yield sparse solutions and can also handle outliers (elementwise contamination). Possible advisors: S. Van Aelst, P. Rousseeuw

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Research groups Financial mathematics and actuarial statistics

Financial mathematics

Key research topics: Hybrid securities and capital solutions for the financial industry Advanced equity models Systemic and systematic risk measurement Pricing commodities

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Research groups Financial mathematics and actuarial statistics 23 / 32

Research groups Financial mathematics and actuarial statistics

Thesis topics and advisors

Advisors: Wim Schoutens, Jan De Spiegeleer, Peter Leoni Contingent Debt under stochastic credit spreads This works aims to study the impact of stochastic credit spreads on the valuation and price dynamics of contingent capital. For the moment CoCo bonds are valued with a single factor equity-based model. Using a 2-dimensional trinomial tree, stochastic credit spreads are going to combined with the geometric Brownian motion for the stock price. Stochastic credit spreads are going to be useful when dealing with extension risk. The student(s) will implement the model in MatLab.

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Research groups Financial mathematics and actuarial statistics

Thesis topics and advisors

Pricing Convertible bonds using Finite Elements Convertible bonds are priced in practice using either finite differences or finite

• elements. In this study the student will apply the finite element method on the

valuation of convertible bonds. Reference : Financial Engineering with Finite Elements (The Wiley Finance Series) by Juergen Topper. Monte Carlo methods for Convertible Bonds A popular method to value a convertible bond using Monte Carlo is the so-called Longstaff Schwartz method where convertible bonds are priced using a regression in each of the time steps. This method delivers a lower boundary of the price. The student will investigate other approaches where a lower and upper boundary

• f the price can be calculated. The model needs to be implemented in MatLab.

Reference : Monte Carlo Methods in Financial Engineering (Stochastic Modelling and Applied Probability) by Paul Glasserman.

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Research groups Financial mathematics and actuarial statistics

Thesis topics and advisors

Advisor: P. Leoni Stack building of gas storages Gas storages are common physical facilities that are used to balance and trade the gas complex. They provide flexibility to time market moves and inject when prices are low and withdraw when prices are higher. The thesis will build an aggregation model to study the portfolio effects of such

• products. The stack typically leads to non-linear optimisation problems for which

the classical linear programs fail to find the intrinsic injection-withdrawal plan.

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Research groups Financial mathematics and actuarial statistics

Thesis topics and advisors

Skew functions in commodities Parametric skew functions often exhibit arbitrage in the wings and a lot of proposals have been formulated over the last decades to resolve this. The thesis will first extend the classical vanna-Volga method to use 5 options as input rather than 3. The concept will be the same as the idea originally introduced in FX markets: use liquid options’ market information to decompose the inter/extrapolated option into by minimizing higher order Greeks. The thesis will compare various hedging choices and furthermore study the SABR model to provide a thorough comparison of performance between both methods.

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Research groups Financial mathematics and actuarial statistics

Thesis topics and advisors

Comparison of hedging performance of spread option models In commodities most problems are related to spread options. The thesis will study, implement and then compare models such as Margrabe, Kirk, Bachelier or on models directly applied to the spread itself. This comparison will focus on the delta hedging of such products and in the case of Bachelier the thesis will adjust the Greeks to compensate for cross effects between the underlying price and the volatility. Furthermore, a thorough numerical study will be performed on mean-reverting spreads to understand the performance of delta hedging in this case. In case of physical options, the volatility input is usually a blend between implied volatility and intra month volatility. For 1 or 2 of such models, the thesis will derive the decomposition of spread options into plain vanilla options and identify the residual risk.

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Research groups Financial mathematics and actuarial statistics

Extreme value and actuarial statistics

Key research topics: Bias reduction techniques in extreme value analysis: risk measures, incomplete data, multivariate data Statistics in the reinsurance business Dimension reduction techniques in multivariate extreme value analysis Loss reserving models Stochastic mortality models

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Research groups Financial mathematics and actuarial statistics

Examples

• 1990

1995 2000 2005 2010 −40 20 40 60 80 negative weekly returns in %

• ●●●●●● ● ● ● ● ● ●
• 1

2 3 4 5 6 −2 2 4 quantiles of standard exponential log(Xn−k,n) 100 200 300 400 500 −0.5 0.0 0.5 1.0 k estimate of gamma (a) Hill (b) bias reduced Hill (c) POT 100 200 300 400 −1.0 −0.5 0.0 0.5 1.0 k estimate of gamma (a) Hill (b) bias reduced Hill (c) POT

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Research groups Financial mathematics and actuarial statistics

Thesis topics and advisors

Extreme Value Analysis for incomplete data Large claims in insurance are hardly ever exact. They are left truncated and/or right censored, or even interval censored. In this thesis the adaptation of extreme value methods to such situations will be

• studied. One can start with recent work on randomly right censored data methods

developed in literature and by the advisors. Advisors: J. Beirlant and I. Gijbels

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Research groups Financial mathematics and actuarial statistics

Thesis topics and advisors

Analysis of a Belgian reinsurance data base In collaboration with H. Aelbrecher a monograph on statistical methods in reinsurance is in preparation. We use a data base from an international reinsurance company to illustrate this work. The methods involved contain a large variety of statistical methods in risk management, ranging from copula modelling, extreme value analysis, estimation of claims that incurred but that are not completely paid among others. Advisor: J. Beirlant in cooperation with H. Aelbrecher (Univ. Lausanne)

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Research groups Financial mathematics and actuarial statistics

Thesis topics and advisors

Inference for robust chain-ladder method The chain-ladder method is a widely used technique to forecast the reserves that have to be kept regarding claims that are known to exist, but for which the actual size is unknown at the time the reserves have to be set. In practice it can be easily seen that even one outlier can lead to a huge over- or underestimation of the

• verall reserve when using the chain-ladder method. Therefore Verdonck and

Debruyne (2011) have proposed a robust alternative. Besides the reserve estimates, it is also important to obtain an approximation to the estimation error

• f a fitted model in a statistical context.

In this thesis some robust bootstrapping techniques will be adopted and compared

• n real data. Work will be done in R.

Reference: T. Verdonck and M. Debruyne. The influence of individual claims on the chain-ladder estimates: analysis and diagnostic tool. Insurance: Mathematics and Economics, 48(1), 85-98, 2011. Possible advisors: T. Verdonck, S. Van Aelst

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Questions

QUESTIONS?

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