slides smart-2015

Arthur CHARPENTIER - Rennes, SMART Workshop, 2014

Kernel Based Estimationof Inequality Indices and Risk Measures

Arthur Charpentier

[email protected]

http://freakonometrics.hypotheses.org/

based on joint work with E. Flachaireinitiated by some joint work with

A. Oulidi, J.D. Fermanian, O. Scaillet, G. Geenens and D. Paindaveine

(Université de Rennes 1, 2015)

1


Stochastic Dominance and Related Indices

• First Order Stochastic Dominance (cf standard stochastic order, �st)

X �1 Y ⇐⇒ FX(t) ≥ FY (t),∀t⇐⇒ VaRX(u) ≤ VaRY (u),∀u

• Convex Stochastic Dominance (cf martingale property)

X �cx Y ⇐⇒ E[Y |X] = X ⇐⇒ ESX(u) ≤ ESY (u),∀u and E(X) = E(Y )

• Second Order Stochastic Dominance (cf submartingale property,stop-loss order, �icx)

X �2 Y ⇐⇒ E[Y |X] ≥ X ⇐⇒ ESX(u) ≤ ESY (u),∀u

• Lorenz Stochastic Dominance (cf dilatation order)

X �L Y ⇐⇒X

E[X] �cxX

E[Y ] ⇐⇒ LX(u) ≤ LY (u),∀u

2



• Parametric Model(s)E.g. N (µX , σ2

X) �1 N (µY , σ2Y )⇐⇒ µX ≤ µY and σ2

X = σ2Y

E.g. N (µX , σ2X) �cx N (µY , σ2

Y )⇐⇒ µX = µY and σ2X ≤ σ2

Y

E.g. N (µX , σ2X) �2 N (µY , σ2

Y )⇐⇒ µX ≤ µY and σ2X ≤ σ2

Y

Or other parametric distribution. E.g. a lognormal distribution for losses

3



• Non-parametric Model(s)

4


Nonparametric estimation of the density

5


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

6


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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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)

0.2 0.4 0.6 0.8

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Estimation of Frank copula

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with a symmetric kernel (here a Gaussian kernel).

8


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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Nice asymptotic properties, see Fermanian et al. (2005)... but still: on finitesample, bad behavior on borders.

9


Beta kernel idea (for copulas)see Chen (1999, 2000), Bouezmarni & Rolin (2003),

Kxi(u) ∝ exp(− (u− xi)2

h2

)vs. Kxi(u) ∝

(u

x1,ib

1 [1− u1]x1,i

b

)·(u

x2,ib

2 [1− u2]x2,i

b

)

10


Beta kernel idea (for copulas)Beta (independent) bivariate kernel , x=0.0, y=0.0 Beta (independent) bivariate kernel , x=0.2, y=0.0 Beta (independent) bivariate kernel , x=0.5, y=0.0

Beta (independent) bivariate kernel , x=0.0, y=0.2 Beta (independent) bivariate kernel , x=0.2, y=0.2 Beta (independent) bivariate kernel , x=0.5, y=0.2

Beta (independent) bivariate kernel , x=0.0, y=0.5 Beta (independent) bivariate kernel , x=0.2, y=0.5 Beta (independent) bivariate kernel , x=0.5, y=0.5

11


Beta kernel idea (for copulas)

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Estimation of the copula density (Beta kernel, b=0.1)

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Estimation of the copula density (Beta kernel, b=0.05)

12


Beta kernel idea (for copulas)

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01

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4

Beta kernel estimator, b=0.05, n=100


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13


Transformed kernel idea (for copulas)[0, 1]× [0, 1]→ R× R→ [0, 1]× [0, 1]

14


Transformed kernel idea (for copulas)

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15


Transformed kernel idea (for copulas)

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01

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4

Transformed kernel estimator (Gaussian), n=100


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see Geenens, C. & Paindaveine (2014) for more details on probit transformationfor copulas.

16


Combining the two approachesSee Devroye & Györfi (1985), and Devroye & Lugosi (2001)

... use the transformed kernel the other way, R→ [0, 1]→ R

17


Devroye & Györfi (1985) - Devroye & Lugosi (2001)Interesting point, the optimal T should be F ,

thus, T can be Fθ

18


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 .

19


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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sity

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Heavy Tailed distributionsee C. & Oulidi (2007), impact on quantile estimation ?

20 30 40 50 60 70 80

0.00

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sity

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Heavy Tailed distributionsee C. & Oulidi (2007), impact on quantile estimation ? bias ? m.s.e. ?

22


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

23


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)]

24


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

h→ ρ(u, h) = 2h2 + 2.5− (4h4 + 6h2 + 2.25− u2 − u/h)1/2

Problem : choice of the bandwidth h? ? Standard loss function

L(h) =∫

[gn(u)− g(u)]2du =∫

[gn(u)]2du− 2∫gn(u) · g(u)du︸︷︷︸

CV (h)

+∫

[g(u)]2du

where

CV (h) =(∫

gn(u)du)2− 2n

n∑i=1

g(−i)(Ui)

25


Mixture of Beta distributions

g(u) =k∑j=1

πj · b(u; αj , βj

)u ∈ [0, 1].

Problem : choice the number of components k (and estimation...). Use ofstochastic EM algorithm (or sort of) see Celeux & Diebolt (1985).

Bernstein approximation

g(u) =m∑k=1

[mωk] · b (u; k,m− k) u ∈ [0, 1].

where ωk = G

(k

m

)− G

(k − 1m

).

26


On the log-transformWith a standard Gaussian kernel

fX(x) = 1n

n∑i=1

φ(x;xi, h)

A Gaussian kernel on a log transform,

fX(x) = 1xfX?(log x) = 1

n

n∑i=1

λ(x; log xi, h)

where λ(·;µ, σ) is the density of the log-normal distribution. Here, in 0,

bias[fX(x)] ∼ h2

2 [fX(x) + 3x · f ′X(x) + x2 · f ′′X(x)]

andVar[fX(x)] ∼ fX(x)

xnh

27


On the Box-Cox-transformMore generally, instead of transformed sample Yi = log[Xi], consider

Yi = Xλi − 1λ

when λ 6= 0.

Find the optimal transformation using standard regression techniques (leastsquares)

X?i = Xλ?

i − 1λ?

when λ? 6= 0

and X?i = log[Xi] if λ? = 0. The density estimation is here

fX(x) = xλ?−1fX?

(xλ

? − 1λ?

)

28


Illustration with Log-normal samplesStandard kernel (− Silvermans’s rule h?)

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Illustration with Log-normal samplesLog transform, x?i = log xi + Gaussian kernel

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Illustration with Log-normal samplesProbit-type transform, x?i = Φ−1[T

θ(xi)] + Gaussian kernel

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Illustration with Log-normal samplesBox-Cox transform, x?i = BoxCox

λ(xi) + Gaussian kernel

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Illustration with Log-normal samplesui = T

θ(xi) + Mixture of Beta distributionsl

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Illustration with Log-normal samplesui = T

θ(xi) + Beta kernel estimation

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Quantities of interestStandard statistical quantities

• miae,(∫ ∞

0

∣∣∣fn(x)− f(x)∣∣∣ dx

)

• mise,(∫ ∞

0

[fn(x)− f(x)

]2dx)

• miaew,(∫ ∞

0

∣∣∣fn(x)− f(x)∣∣∣ |x|dx)

• misew,(∫ ∞

0

[fn(x)− f(x)

]2x2 dx

)

35


Quantities of interest

Inequality indices and risk measures, based on F (x) =∫ x

0f(t)dt,

• Gini, 1µ

∫ ∞0

F (t)[1− F (t)]dt

• Theil,∫ ∞

0

t

µlog(t

µ

)f(t)dt

• Entropy −∫ ∞

0f(t) log[f(t)]dt

• VaR-quantile, x such that F (x) = P(X ≤ x) = α, i.e. F−1(α)

• TVaR-expected shorfall, E[X|X > F−1(α)]

where µ =∫ ∞

0[1− F (x)]dx.

36


Computations AspectsHere, for each method, we return two functions,

• function fn(·)

• a random generator for distribution fn(·)

◦ H-transform and Gaussian kernel

draw i ∈ {1, · · · , n} and X = H−1(Z) where Z ∼ N (H(xi), b2)

◦ H-transform and Beta kernel

draw i ∈ {1, · · · , n} and X = H−1(U) where U ∼ B(H(xi)/h, [1−H(xi)]/h)

◦ H-transform and Beta mixture

draw k ∈ {1, · · · ,K} and X = H−1(U) where U ∼ B(αk,βk)

37


◦ ‘standard’ Gaussian kernel (benchmark)

draw i ∈ {1, · · · , n}, and X ∼ N (xi, b2) (almost)

up to some normalization, · 7→ fn(·)∫∞0 fn(x)dx

38


MISE

∫ ∞0

[f (s)n (x)− f(x)

]2dx

●

Singh−Maddala

MISE

●●● ●●

●●●●●● ●

● ●●● ●● ●●● ●● ●

●●●● ●● ●

● ●●● ●● ●●● ●● ●

●●●● ●● ● ●

●●● ●●

●● ●● ● ●● ●●● ●

standard kernellog kernel

log mixboxcox kernel

boxcox mixprobit kernelbeta kernel

beta mix

0.00

0

0.00

5

0.01

0

0.01

5

●

Mixed Singh−Maddala

MISE

●●

● ●●● ●●● ●●● ●

● ●● ●●● ●● ●●● ●

● ●● ●● ●●● ●● ●●●● ●

●●● ●●

●●●




beta mix

0.00

0.05

0.10

0.15

39


MIAE

∫ ∞0|f (s)n (x)− f(x)|dx

●

Singh−Maddala

MIAE

● ●●● ●

●● ●● ●●

● ●●

●●●● ●●

● ●● ●●

●●●● ●

●● ●●● ●

● ●●●




beta mix

0.00

0.05

0.10

0.15

0.20

●


MIAE

●

●● ●

● ●●●●● ●

● ●




beta mix

0.05

0.10

0.15

0.20

0.25

0.30

40


Gini Index

1µ

(s)n

∫ ∞0

F (s)n (t)[1− F (s)

n (t)]dt

Singh−Maddala

Gini Index

● ●

●●

● ●

●●

●

●●●

● ●




beta mix

0.45

0.50

0.55

0.60


Gini Index

●

●●

●

●

● ●●

●●




beta mix

0.55

0.60

0.65

0.70

41


Value-at-Risk, 95%

Q(s)n (α) = inf{x, α ≤ F (s)

n (x)}

●

Singh−Maddala

Quantile 95%

●● ●●●● ●

● ● ●● ●

●●

● ●●●●● ●

● ●●●●●● ●●

●● ●●●●●●

●●● ●

●●




beta mix

10 20 30 40 50


Quantile 95%

●●

● ●

●

●

● ●

●




beta mix

2.0

2.5

3.0

3.5

4.0

4.5

42


Value-at-Risk, 99%

Q(s)n (α) = inf{x, α ≤ F (s)

n (x)}

Singh−Maddala

Quantile 99%

● ●●●

●● ● ●●●

●●

●●● ●● ●●●●

●● ●●●●

●● ●●●●

●● ●

●●




beta mix

6 8 10 12 14 16 18

●


Quantile 99%

● ●

● ●●●●●● ●●●●

●●● ●●●

● ●●●●●

● ● ●●

●● ●




beta mix

2 4 6 8 10

43


Tail Value-at-Risk, 95%

E[X|X > Q(s)n (α)]

Singh−Maddala

Expected Shortfall 95%

●

●

●●

●

●●

●




beta mix

4.0

4.5

5.0

5.5

6.0

6.5

●


Expected Shortfall 95%

● ●●●

● ●●●

● ●●●

● ●●●

●




beta mix

4 6 8 10 12

44


Possible conclusion ?

• estimating densities on transformated data is definitively a good idea

• but we need to find a good transformationX parametric + betaX parametric + probitX log-transformX Box-Cox

0.719

0.719

0.7190.719

0.719

0.719

0.719

0.7191.428

2.137

47.75

48.00

48.25

48.50

48.75

−4.8 −4.4 −4.0 −3.6longitude

latit

ude

0.0110.7191.4282.1372.845

0.719 0.719

0.7190.719

0.719

0.719

0.7190.719

0.719

1.428

1.428

1.428

1.428

1.428

1.4281.428

1.428

1.428

1.428

1.428

1.428

1.428

1.428

1.428

1.428

2.137

2.137

47.75

48.00

48.25

48.50

48.75

−4.8 −4.4 −4.0 −3.6longitude

latit

ude

0.0110.7191.4282.1372.845

(joint work with E. Gallic)

45

slides smart-2015

Documents

smart workshop

y ey x x esxu esy u

u2arthur charpentier

cx n y

nonparametric estimation

symmetric kernel

beta kernel idea

estimation of frank