Arthur CHARPENTIER - Rennes, SMART Workshop, 2014 Kernel Based Estimation of Inequality Indices and Risk Measures Arthur Charpentier [email protected]http://freakonometrics.hypotheses.org/ based on joint work with E. Flachaire initiated by some joint work with A. Oulidi, J.D. Fermanian, O. Scaillet, G. Geenens and D. Paindaveine (Université de Rennes 1, 2015) 1
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Kernel Based Estimationof Inequality Indices and Risk Measures
Or other parametric distribution. E.g. a lognormal distribution for losses
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Stochastic Dominance and Related Indices
• Non-parametric Model(s)
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Nonparametric estimation of the density
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Agenda
sample {y1, · · · , yn} −→ fn(·)↗↘
Fn(·) or F−1n (·)
R(fn)
• Estimating densities of copulas◦ Beta kernels◦ Transformed kernels• Combining transformed and Beta kernels• Moving around the Beta distribution◦ Mixtures of Beta distributions◦ Bernstein Polynomials• Some probit type transformations
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Non parametric estimation of copula densitysee C., Fermanian & Scaillet (2005), bias of kernel estimators at endpoints
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Kernel based estimation of the uniform density on [0,1]
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Kernel based estimation of the uniform density on [0,1]
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Non parametric estimation of copula density
e.g. E(c(0, 0, h)) = 14 · c(u, v)− 1
2 [c1(0, 0) + c2(0, 0)]∫ 1
0ωK(ω)dω · h+ o(h)
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Estimation of Frank copula
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Estimation of Frank copula
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with a symmetric kernel (here a Gaussian kernel).
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Non parametric estimation of copula density
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Standard Gaussian kernel estimator, n=100
Estimation of the density on the diagonal
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Standard Gaussian kernel estimator, n=1000
Estimation of the density on the diagonal
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Standard Gaussian kernel estimator, n=10000
Estimation of the density on the diagonal
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Nice asymptotic properties, see Fermanian et al. (2005)... but still: on finitesample, bad behavior on borders.
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Beta kernel idea (for copulas)see Chen (1999, 2000), Bouezmarni & Rolin (2003),
see Geenens, C. & Paindaveine (2014) for more details on probit transformationfor copulas.
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Combining the two approachesSee Devroye & Györfi (1985), and Devroye & Lugosi (2001)
... use the transformed kernel the other way, R→ [0, 1]→ R
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Devroye & Györfi (1985) - Devroye & Lugosi (2001)Interesting point, the optimal T should be F ,
thus, T can be Fθ
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Heavy Tailed distributionLet X denote a (heavy-tailed) random variable with tail index α ∈ (0,∞), i.e.
P(X > x) = x−αL1(x)
where L1 is some regularly varying function.
Let T denote a R→ [0, 1] function, such that 1− T is regularly varying atinfinity, with tail index β ∈ (0,∞).
Define Q(x) = T−1(1− x−1) the associated tail quantile function, thenQ(x) = x1/βL?2(1/x), where L?2 is some regularly varying function (the de Bruynconjugate of the regular variation function associated with T ). Assume here thatQ(x) = bx1/β
Let U = T (X). Then, as u→ 1
P(U > u) ∼ (1− u)α/β .
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Heavy Tailed distributionsee C. & Oulidi (2007), α = 0.75−1 , T0.75−1 , T0.65−1︸ ︷︷ ︸
lighter
, T0.85−1︸ ︷︷ ︸heavier
and Tα
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Heavy Tailed distributionsee C. & Oulidi (2007), impact on quantile estimation ?
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Heavy Tailed distributionsee C. & Oulidi (2007), impact on quantile estimation ? bias ? m.s.e. ?
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Which transformation ?
GB2 : t(y; a, b, p, q) = |a|yap−1
bapB(p, q)[1 + (y/b)a]p+q , for y > 0,
GB2q→∞
��a=1
��p=1
##
q=1
((GG
a→0
��a=1
��
p=1
��
Beta2
q→∞ww
SM
q→∞xx
q=1''
Dagum
p=1
��Lognormal Gamma Weibull Champernowne
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Estimating a density on R+
• Stay on R+ : xi’s• Get on [0, 1] : ui = T
θ(xi) (distribution as uniform as possible)
◦ Use Beta Kernels on ui’s◦ Mixtures of Beta distributions on ui’s◦ Bernstein Polynomials on ui’s• Get on R : use standard kernels (e.g. Gaussian)◦ On x?i = log(xi)◦ On x?i = BoxCox
λ(xi)
◦ On x?i = Φ−1[Tθ(xi)]
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Arthur CHARPENTIER - Rennes, SMART Workshop, 2014
Beta kernel
g(u) =n∑i=1
1n· b(u; Ui
h,
1− Uih
)u ∈ [0, 1].
with some possible boundary correction, as suggested in Chen (1999),u