Section of Statistics - KU Leuven · 2014. 4. 7. · General info Master of Mathematics - Option...

Section of Statistics

Department of Mathematics

wis.kuleuven.be/stat

31/03/2014

1 / 32

Outline

Outline

1 General information about research, courses, theses

2 Information about each research group

3 Questions

2 / 32

General info

Three major research themes:

Nonparametric

and

semiparametric

statistics

Robust

and

computational

statistics

Financial mathematics

and

actuarial statistics

3 / 32

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

4 / 32

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

2 Stochastic models (J. De Spiegeleer)

Becoming familiar with stochastic modelling of dependent stochasticvariables, 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 modellenbestudeerd, alsook de methoden om hiermee multivariate gegevens teanalyseren. Er wordt vooral aandacht besteed aan de praktische aspecten bijde 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 weightis flexible to some extent.

Theoretical parts: literature review, study of theoretical properties, workingout proofs.

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

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Research groups

Outline

1 General information about research, courses, theses

2 Information from each research group

3 Questions

8 / 32

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 statisticalconcepts and properties

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Examples

0 5 10 15 20 25

010

2030

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

10 / 32


Examples

modeling the dynamics of a dependence structureobservations (hourly measurements) of wind speeds at two (or more) weatherstations

Apr Mai Jun Jul

010

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4050

60

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mph

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010

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Butler Grade − 2013ho

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ave

rage

win

d sp

eed

in m

ph

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

other related topics: portfolio selection in financial applications, study of thevalue-at-risk

11 / 32


Examples

flexible regression modeling and variable selectionpredict the level of atmospheric ozone concentration from daily meteorologicalmeasurements

determine important pollutants, and their effects

20 40 60 80

010

2030

humidity

func

tiona

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hum

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dataP−splinesOLS

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010

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temp

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tem

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dataP−splinesOLS

12 / 32


Examples

flexible regression modeling and quantile regressiondescribe the evolution in time for an infected patient (for patients with differentconditions)

conditional quantile regression

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−1 0 1 2 3 4 5 6

010

2030

4050

60

time since infection

CD

4 pe

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tage

afte

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−1 0 1 2 3 4 5 6

010

2030

4050

6070

time since infection

CD

4 pe

rcen

tage

afte

r in

fect

ion

τ = 0.5

τ = 0.1

τ = 0.9

13 / 32


Thesis topics and promotors

Advisors: Irene 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

14 / 32



Statistics of Bernstein copula estimator

The Bernstein copula estimator builds upon the empirical copula estimator that isappropriately smoothed by Bernstein polynomials. This estimator was introducedby Sancetta and Satchell (2004), and studied by Janssen et al. (2012, 2014).

This project studies the asymptotic statistical properties (e.g. asymptoticdistributional behavior) of the Bernstein copula estimator. The interest also goesto the applicability of this copula estimator and requires designing and conductinga simulation study to check the finite sample performance against its maincompetitors. Use of the R software and packages.

Advisor: Dominik Sznajder

15 / 32

Research groups Robust and computational statistics

Robust and computational statistics


high-breakdown estimators and algorithms for high-dimensional andfunctional 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

16 / 32


Examples

−10 −5 0 5 10 15

−5

05

1015

Classical and robust tolerance ellipse

log(body)

log(

brai

n)

MCDClassical

3.6 3.8 4.0 4.2 4.4 4.64.

04.

55.

05.

56.

0

log.Te

log.

light LS (all)

LS (reduced)

LTS

17 / 32



Inference for robust SUR

The SUR model (seemingly unrelated regression) consists of several sets of linearregression equations, which are related via a specific structure of the covariancematrix of the error terms. Recently a robust version has been proposed (Hubert,Verdonck and Yorulmaz, 2013). Its robustness is illustrated via simulations andvia data examples from actuarial statistics.

In this thesis this robust method will be studied in detail, and robust inferencevia robust bootstrap (Salibian-Barrera, Van Aelst, and Willems, 2008) will bedeveloped.

Possible advisors: M. Hubert, S. Van Aelst, T. Verdonck

18 / 32



Clustering of multivariate and functional data via depth functions

0

1

2

3

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time

Res

pons

0

10

20

30

0 5 10 15 20

Multivariate Distance to central curve group 1

Mul

tivar

iate

Dis

tanc

e to

cen

tral

cur

ve g

roup

2

Possible advisors: M. Hubert, P. Rousseeuw

19 / 32



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 actuallycontaminated, but most observations are contaminated in at least one of theircomponents. This is called elementwise contamination. Robustness measures forthis type of contamination have been proposed (Alqallaf, Van Aelst, Yohai, andZamar, 2009).

In this thesis methods that can handle elementwise contamination will beinvestigated and their properties will be compared.

Possible advisors: S. Van Aelst, P. Rousseeuw

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Robustness and sparsity

When analyzing high-dimensional data one often aims to obtain sparse modelsbecause they reveal the most relevant structure, are more stable, and are easierto 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 oftenobtained by using some form of penalization in the estimation procedure.

In this thesis, methods will be studied for regression or PCA that yield sparsesolutions 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


Hybrid securities and capital solutions for the financial industry

Advanced equity models

Systemic and systematic risk measurement

Pricing commodities

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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 valuationand price dynamics of contingent capital. For the moment CoCo bonds are valuedwith a single factor equity-based model. Using a 2-dimensional trinomial tree,stochastic credit spreads are going to combined with the geometric Brownianmotion for the stock price. Stochastic credit spreads are going to be useful whendealing with extension risk. The student(s) will implement the model in MatLab.

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Pricing Convertible bonds using Finite Elements

Convertible bonds are priced in practice using either finite differences or finiteelements. In this study the student will apply the finite element method on thevaluation of convertible bonds.

Reference : Financial Engineering with Finite Elements (The Wiley FinanceSeries) by Juergen Topper.

Monte Carlo methods for Convertible Bonds

A popular method to value a convertible bond using Monte Carlo is the so-calledLongstaff Schwartz method where convertible bonds are priced using a regressionin 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 boundaryof the price can be calculated. The model needs to be implemented in MatLab.

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

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Advisor: P. Leoni

Stack building of gas storages

Gas storages are common physical facilities that are used to balance and trade thegas complex. They provide flexibility to time market moves and inject when pricesare low and withdraw when prices are higher.

The thesis will build an aggregation model to study the portfolio effects of suchproducts. The stack typically leads to non-linear optimisation problems for whichthe classical linear programs fail to find the intrinsic injection-withdrawal plan.

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Skew functions in commodities

Parametric skew functions often exhibit arbitrage in the wings and a lot ofproposals have been formulated over the last decades to resolve this.

The thesis will first extend the classical vanna-Volga method to use 5 options asinput rather than 3. The concept will be the same as the idea originallyintroduced in FX markets: use liquid options’ market information to decomposethe inter/extrapolated option into by minimizing higher order Greeks.

The thesis will compare various hedging choices and furthermore study the SABRmodel to provide a thorough comparison of performance between both methods.

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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 onmodels directly applied to the spread itself. This comparison will focus on thedelta hedging of such products and in the case of Bachelier the thesis will adjustthe Greeks to compensate for cross effects between the underlying price and thevolatility.

Furthermore, a thorough numerical study will be performed on mean-revertingspreads to understand the performance of delta hedging in this case. In case ofphysical options, the volatility input is usually a blend between implied volatilityand intra month volatility. For 1 or 2 of such models, the thesis will derive thedecomposition of spread options into plain vanilla options and identify the residualrisk.

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Extreme value and actuarial statistics


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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0 1 2 3 4 5 6

−2

02

4

quantiles of standard exponential

log(

Xn−

k,n)

0 100 200 300 400 500

−0.

50.

00.

51.

0

k

estim

ate

of g

amm

a

(a) Hill(b) bias reduced Hill(c) POT

0 100 200 300 400

−1.

0−

0.5

0.0

0.5

1.0

k

estim

ate

of g

amm

a(a) Hill(b) bias reduced Hill(c) POT

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Extreme Value Analysis for incomplete data

Large claims in insurance are hardly ever exact. They are left truncated and/orright censored, or even interval censored.

In this thesis the adaptation of extreme value methods to such situations will bestudied. One can start with recent work on randomly right censored data methodsdeveloped in literature and by the advisors.

Advisors: J. Beirlant and I. Gijbels

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Analysis of a Belgian reinsurance data base

In collaboration with H. Aelbrecher a monograph on statistical methods inreinsurance is in preparation. We use a data base from an internationalreinsurance company to illustrate this work. The methods involved contain a largevariety of statistical methods in risk management, ranging from copula modelling,extreme value analysis, estimation of claims that incurred but that are notcompletely paid among others.

Advisor: J. Beirlant in cooperation with H. Aelbrecher (Univ. Lausanne)

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Inference for robust chain-ladder method

The chain-ladder method is a widely used technique to forecast the reserves thathave to be kept regarding claims that are known to exist, but for which the actualsize is unknown at the time the reserves have to be set. In practice it can be easilyseen that even one outlier can lead to a huge over- or underestimation of theoverall reserve when using the chain-ladder method. Therefore Verdonck andDebruyne (2011) have proposed a robust alternative. Besides the reserveestimates, it is also important to obtain an approximation to the estimation errorof a fitted model in a statistical context.

In this thesis some robust bootstrapping techniques will be adopted and comparedon real data. Work will be done in R.

Reference: T. Verdonck and M. Debruyne. The influence of individual claims onthe chain-ladder estimates: analysis and diagnostic tool. Insurance: Mathematicsand Economics, 48(1), 85-98, 2011.

Possible advisors: T. Verdonck, S. Van Aelst

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Questions

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Questions

QUESTIONS?

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