HdR

Arthur CHARPENTIER, HdR: Contribution à l’étude de la dépendance

Contributions to Dependence Modeling*A. Charpentier (Université de Rennes 1)

Habilitation à Diriger des Recherches,

Rennes, 2016.

@freakonometrics 1


2006-2016, Brief Summary

2006: PhD in Applied MathematicsDepedencies, with Applications in Insurance and FinanceSupervised by Jan Beirlant & Michel Denuit2006-2010: Maître de ConférencesUniversité de Rennes 12010-2014: ProfesseurUniversité du Québec à Montréal2014-2016: Maître de ConférencesUniversité de Rennes 1

@freakonometrics 2


‘Fondamental Results’ Dependence & Extremes

with Anne-Laure Fougères, Christian Genest & Johanna Nešlehová MultivariateArchimax Copulas (JMVA, 2014).Let ` be a d-variate stable tail dependence functionand φ be the generator of a d-variate Archimedeancopula. Then

Cφ,`(u1, · · · , ud) = φ−1 [`(φ(u1) + · · ·+ φ(ud))]

is a d-dimensional copula. Further, if 1 − ψ(1/s)is regularly varying (at ∞) with index α ∈ [0, 1],Cφ,` ∈MDA(C`?),

C`?(u1, · · · , ud) = exp[−`α

(| log(u1)| 1

α , · · · , | log(ud)|1α

)]see results obtained with Johan Segers Tails of Archimedean Copulas (JMVA).

@freakonometrics 3


‘Fondamental Results’ Nonparametric estimation (and Borders)

with Emmanuel Flachaire Transformed Kernel & Inequality and Risk Indices(Actualité Économique, 2015)

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@freakonometrics 4



with Emmanuel Flachaire Transformed Kernel & Inequality and Risk Indices(Actualité Économique, 2015)

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@freakonometrics 5



with Ewen Gallic? Density Estimation & Ripley Correction (Geoinformatica, 2015)

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fH,S(z) = 1det(H)

n∑i=1

ωH(zi)K(H−1[z − zi]) with ωH(zi) = A(Dzi,r)A(Dzi,r ∩ S)

@freakonometrics 6



with Gery Geenens & Davy Pandaveine Copula Density Estimation and ProbitTrransform, (Bernoulli, 2015)

From a n-i.i.d. samepl {xi, yi} define the nor-malized pseudo-sample {(si, ti)}

ui = Φ−1(ui) = Φ−1(FX(xi)) and vi = Φ−1(vi)

fST (s, t) = 1n|HST |1/2

n∑i=1

K

(H−1/2ST

(s− sit− ti

)).

c(τ)(u, v) = fST (Φ−1(u),Φ−1(v))φ(Φ−1(u))φ(Φ−1(v))

is the so-called naive estimator...

c~(τ2)

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@freakonometrics 7



with Gery Geenens & Davy Pandaveine Copula Density Estimation and ProbitTrransform, (Bernoulli, 2015)... with possible ameliorations.One can derive asymptotic normality√nh2

(c∗(τ,2)(u, v)− c(u, v)− h4B(u, v)

)L→ N

(0, σ(2)

2 (u, v))as n→∞,

where

B(u, v) =b(2)(u, v)

φ(Φ−1(u)) · φ(Φ−1(v))

Application: LOSS-Alae dataset.

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@freakonometrics 8


‘Fondamental Results’ Multivariate Risk Aversion

with Alfred Galichon and Marc Henry, Multivariate Local Utility, (MOR, 2015)

Machina (1982) R has a local utility representation if there is UP such that

R(P)−R(Pε) = −∫UP(x)d(P− Pε)(x) + o(‖P− Pε‖),

i.e. UP is Fréchet derivative of R in P.

Ex: Entropic measure, R(P) = − 1αEP(e−αX), then UP(x) = 1

α

e−αx

EP(e−αX) .

Ex: Distorted measure R(P) =∫ 1

0F−1X (u)ϕ(u)du, then UP(x) =

∫ x

ϕ(FX(z))dz.

R is Schur-concave if and only if UP is concave, ∀P ∈ L2.

Let X0 ∼ P and X1 ∼ Q such that E(X1|X0) = X0. There exists (Xt)t∈[0,1]

(martingale interpolation) such that X0 = X0, X1 = X1, with dXt = ΣtdBt,and R(Xs) ≤ R(Xt) for all s < t.

@freakonometrics 9


‘Applied Mathematics’ Causality and Time Series

with David Sibaï? Hurst/Gumbel and Floods, (Environmentrics 2008) or HeatWaves, (CC 2011) or with Marilou Durand ? Dynamics of Earthquakes, (JS 2015)

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juil. 02 juil. 12 juil. 22 août 01 août 11 août 21 août 31

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@freakonometrics 10


‘Applied Mathematics’ Causality and Time Series

with Mathieu Boudrault Multivariate INAR, (2012)Inspired by Steutel & van Harn (1979) define a mul-tivariate thinning operator,

[P ◦N ]i =d∑j=1

pi,j ◦Nj , with p ◦N =N∑k=1

Yk

where Y1, Y2, · · · are i.id. B(p)’s. A MINAR is

Xt = P ◦Xt−1 + εt

where εt are i.id. Poisson random vectors.

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See also joint work with M Toledo Bastos & Dan Mercea Onsite & Online ProtestActivity, (JC, 2015).

@freakonometrics 11


‘Applied Mathematics’ Applications of Game Theory

with Benoit Le Maux Natural Catastrophes and Cooperation, (JPE, 2014)

E[u(ω −X)]︸︷︷︸no insurance

≤ E[u(ω − α−l + I)]︸︷︷︸insurance=V

with indeminty I(·) can be function of the propor-tion of the population claiming a loss.With limited liability

V = U(−α)−∫ 1

0x[U(−α)−U(−α−`+I(x))]f(x)dx

See also work with Stéphane Mussard Income Inequality Games (JEI, 2011) andwith Romuald Élie .

@freakonometrics 12


‘Applied Mathematics’ Actuarial Science

Books on Mathématiques de l’AssuranceNon-Vie with Michel Denuit and bookon Computational Actuarial Science.

Articles on insurance models, Insurability of Climate Risk (GP 2009), on ClaimsReserving, micro vs. macro with Mathieu Pigeon (Risks, 2016) or on Bonus-MalusSystems with Arthur David & Romuald Élie (2016).

@freakonometrics 13


Popular Writing / Articles en Français

@freakonometrics 14


On-going work

Enora Betz?, Pierre-Yves Geoffard & Julien Tomas Bodily Injury Claims in France:Court or Negociated Settlement ? »*

Emmanuel Flachaire & Magali Fromont Machine Learning & Econometrics

Alfred Galichon & Lucas Vernet Min-Cost Flows Models in Economics »*

Amadou Barry? & Karim Oualkacha Quantile and Expectile Regression for randomeffects model »*

Antoine Ly? Classification with Unbalanced Samples »*

Ewen Gallic? & Olivier Cabrignac Mortality in France and Familal Dependencies,from Genealogical Data »*

Arnaud Goussebaile? Insurance of Natural Catastrophes, Risk and Ambiguity »*

Ndéné Ka, Stéphane Mussard & Oumar Ndiaye Gini Regression andHeteroskedasticity »*

@freakonometrics 15


« Min-Cost Flow Models in Economics*

(source Church (2009))

@freakonometrics 16

http://www.narsc.org/newsite/wp-content/uploads/2009/03/Professor-Richard-Church-Lecture.pdf


« Min-Cost Flow Models in Economics*

@freakonometrics 17


« Quantile and Expectile Regression for Random Effects Models*

Quantile, q(α, Y ) = argminθ ∈ R

{E(rQα (Y − θ))

}with rQα (u) = |α− 1(u ≤ 0)| · |u|.

Empirical version q(α, Y ) = argminθ ∈ R

{1n

n∑i=1

rQα (yi − θ)}

Quantile Regression βQ

(α, y,x) = argminβ ∈ Rp

{1n

n∑i=1

rQα (yi − xiTβ)}

see Koenker (2005). Following Newey & Powell (1987) define expectiles as

µ(τ, Y ) = argminθ ∈ R

E(rEτ (Y − θ)) with rEτ (u) = |τ − 1(u ≤ 0)| · u2.

Expectile Regression βE

(τ, y,x) = argminβ ∈ Rp

{1n

n∑i=1

rEτ (yi − xiTβ)}.

Properties of estimators in the context of panel data, (yi,t,xi,t).

@freakonometrics 18

https://www.cambridge.org/core/books/quantile-regression/C18AE7BCF3EC43C16937390D44A328B1

https://www.jstor.org/stable/1911031


« Quantile and Expectile Regression for Random Effects Models

@freakonometrics 19


« Conclusion (?)

Work in ‘fundamental’ resultsas well as applications(insurance, finance, economics, climate).

Work with researchers in appliedmathematics and economicsinvolving students(undergraduate, graduate, PhD, post-doc)

Currently involved in projects• ANR, multivariate inequalities• ACTINFO research chair

@freakonometrics 20

Education

HdR