On automatic bias reduction for extreme expectile estimation

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On automatic bias reduction for extreme expectile

estimation

Antoine Usseglio-Carleve, Stéphane Girard, Gilles Stupfler

To cite this version:

Antoine Usseglio-Carleve, Stéphane Girard, Gilles Stupfler. On automatic bias reduction for extreme expectile estimation. CMStatistics 2020 - 13th International Conference of the ERCIM WG on Compu-tational and Methodological Statistics, Dec 2020, London / Virtual, United Kingdom. �hal-03087164�

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ENSAI Rennes

On automatic bias reduction for extreme expectile

estimation

Joint work with St´ephane Girard

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Introduction

Expectile estimation

Bias reduction

Simulation study and real data example

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Antoine UsseglioCarleve

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Expectiles

Quantiles [Koenker and Bassett, 1978] have been recently criticized

[Acerbi, 2002], [Artzner et al., 1999] for not being a coherent risk measure.

q(α) ∈ arg min

t∈R E [ρα

(Y − t) − ρα(Y )] ,

where ρα(y ) = |α − 1{y ≤0}| |y |. Some authors thus proposed

expectiles [Newey and Powell, 1987] as an alternative :

e(α) = arg min

t∈R E [ηα

(Y − t) − ηα(Y )] ,

where ηα(y ) = |α − 1{y ≤0}| y2.

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Expectiles

Figure: Quantile (red) and expectile (blue) loss functions for α = 0.05, 0.5 and 0.95.

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Expectiles

• According to [Jones, 1994], e(α) is solution of

E (y ) = E(Y − y )1{Y >y }

2E(Y − y )1{Y >y } + (y − E[Y ])

= 1 − α.

• According to [Bellini et al., 2014], if F (y ) = y−1/γ`(y ), then

lim α→1 F (e(α)) 1 − α = γ −1_{− 1 ; lim} α→1 e(α) q(α) = γ −1_{− 1}−γ , for γ < 1. 5 of 25 Antoine UsseglioCarleve

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Introduction

Bias reduction

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Intermediate expectiles estimation

Y1, . . . , Yn are i.i.d. realizations of Y . If αn<< 1 − 1/n (or

equivalently n(1 − αn) → ∞ as n → ∞) is an intermediate sequence,

two approaches have been considered for expectile estimation:

• The first one, used in [Daouia et al., 2018], directly derives from the definition of expectiles:

b en(αn) = arg min θ∈R n X i =1 ηαn(Yi− θ).

• The second one, introduced in [Girard et al., 2020], uses the

property of [Jones, 1994]: b en(αn) = inf n y ∈ R | bEn(y ) ≤ 1 − αn o , with b En(y ) = Pn i =1(Yi− y )1{Yi>y } Pn i =1|Yi− y |

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-Extreme expectiles estimation

Let us assume lim t→∞ F (ty ) F (t) = y −1/γ .

In this context, extreme quantiles may be estimated using the Weissman estimator. If βn>> 1 − 1/n and αn<< 1 − 1/n,

q(βn) q(αn) ≈ 1 − βn 1 − αn −γ ⇒q_b_n∗(βn) =qbn(αn) 1 − βn 1 − αn −_bγ . Since quantiles and expectiles are asymptotically proportional, the same approximation holds for extreme expectiles, and

[Daouia et al., 2018] introduced

b e_n∗(βn) =ebn(αn) 1 − βn 1 − αn −bγ or ee ∗ n(βn) =qb ∗ n(βn) bγ −1_{− 1}−bγ . 8 of 25 Antoine UsseglioCarleve

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Tail index estimation

Let us consider the second order assumption (C2):

∀y > 0, lim t→∞ 1 A(1/F (t)) F (ty ) F (t) − y −1/γ = y−1/γy ρ/γ _{− 1} γρ .

The most widespread estimator of the tail index γ is the Hill estimator: b γ_kH_n = 1 kn kn X i =1 logYn−i +1,n Yn−kn,n ,

where kn→ ∞ and kn/n → 0 as n → ∞. Under (C2), and if

√ knA(n/kn) → λ ∈ R, then p kn γb H kn− γ → N λ 1 − ρ, γ 2 . 9 of 25 Antoine UsseglioCarleve

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Tail index estimation

Using the asymptotic relationship between quantiles and expectiles, we can introduce the following tail index estimator (see

[Girard et al., 2020]): b γ_kE_n = 1 +n bFn(ebn(1 − kn/n)) kn !−1 . Under (C2) with 0 < γ < 1/2, and if

√ knA(n/kn) → λ1 ∈ R and √ knq(1 − kn/n)−1 → λ2 ∈ R, then √ kn b γkEn− γ → N γ γ −1_{− 1}1−ρ 1 − ρ − γ λ1+ γ 2 γ−1− 1γ+1E[Y ]λ2, γ3(1 − γ) 1 − 2γ ! . 10 of 25 Antoine UsseglioCarleve

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Tail index estimation

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Asymptotic variances gamma V ar iance

Figure: Asymptotic variances ofγ_e_kH

n (black curve) andeγ

E

kn (red curve) as

functions of γ ∈ (0, 1).

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Back to extreme expectiles estimation

0 20 40 60 80 100 3 4 5 6 7 8

Extreme expectile estimation

k

Estimate

Figure:Mean estimates of 1, 000 estimatese_b_n∗(βn) using_eγkHn (black) andeγ

E kn

(blue) for kn= 1, ..., 100 in the case of a Burr distribution with γ = 0.25,

ρ = −5, n = 1, 000 and βn= 1 − 5/n = 0.995.

Why so much bias ?

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Introduction

Bias reduction

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Bias reduction of the tail index estimators

By doing the assumption that A(t) = bγtρ_{, the following}

bias-reduced version ofbγ

H

kn is proposed in [Gomes and Martins, 2002]:

e γ_kH_n =_bγ_kH_n 1 − b 1 − ρ n kn ρ! .

We thus propose a similar approach for eγ

E

kn. For that purpose, we

notice F (e(α))/(1 − α) = γ−1− 1 (1 + r (α)), where 1 + r (α) =

1 −E[Y ] e(α) 1 2α − 1 1 + A 1 F (e(α)) 1 γ(1 − γ − ρ)(1 + o(1)) −1 as α ↑ 1. 14 of 25 Antoine UsseglioCarleve

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Bias reduction of the tail index estimators

We thus introduce the following bias-reduced estimator:

e γE_k_n = 1 +n bFn(ebn(1 − kn/n)) kn 1 1 + r (1 − kn/n) !−1 , where 1 + r (1 − kn/n) = 1 − Yn b en(1 − kn/n) 1 1 − 2kn/n 1 +b[bFn(ebn(1 − kn/n))] −ρ 1 − γ − ρ !−1 . Under some conditions concerning ρ and b, we can prove

p kn eγ E kn− γ → N 0,γ 3_{(1 − γ)} 1 − 2γ . 15 of 25 Antoine UsseglioCarleve

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Bias reduction of the extrapolation step

We can find some bias reduction approaches for extreme quantile

estimators (see for instance [Gomes and Pestana, 2007]). The

second order condition C2 giving

q(βn) = q 1 −kn n n(1 − βn) kn −γ   1 + n(1−βn) kn −ρ − 1 ρ A n kn (1 + o(1))   ,

we easily deduce, with A(t) = bγtρ,

b q∗,RB_n (βn) =qb ∗ n(βn) 1 +[n(1 − βn)/kn] −ρ_{− 1} ρ bγ(n/kn) ρ . 16 of 25 Antoine UsseglioCarleve

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Bias reduction of the extrapolation step

The bias reduction of extreme expectiles is less obvious, and 3 bias terms have to be eliminated, hence

( b en∗,RB(βn) =eb ∗ n(βn)(1 + B1,n)(1 + B2,n)(1 + B3,n) e en∗,RB(βn) =ee ∗ n(βn)(1 + B1,n)(1 + B3,n) , where              1 + B1,n= 1 +[n(1−βn)/kn] −ρ₋₁ ρ bγ(n/kn) ρ 1 + B2,n= 1 + r 1 −k_nn γ 1 + (γ −1₋₁₎−ρ (1+r (1−kn_n))−ρ−1 ρ bγ( n kn) ρ −1 1 + B3,n= (1 + r?(βn))−γ 1 +(γ −1₋₁₎−ρ (1+r?_(β n))−ρ−1 ρ bγ(1 − βn) −ρ .

Note that these bias reduced estimators are proposed in the R package Expectrem.

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Introduction

Bias reduction

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Simulation study

• We simulate n = 1, 000 independent realizations Y1, . . . , Yn of a

Burr distribution:

F (y ) = (1 + y−ρ/γ)1/ρ.

• We consider ρ = −5, −1 and −0.5, and γ = 0.1, 0.2, 0.3 and 0.4.

• For each case, we estimate the expectile of level

βn= 1 − 5/n = 0.995.

• How to choose kn ?

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Simulation study

For the Hill based estimators, we choose the kn which minimizes the

following AMSE with A(t) = bγtρ:

An kn 2 (1 − ρ)2 + γ2 kn hence k_nH=     (1 − ρ)2 −2ρb2 !1/(1−2ρ) n−2ρ/(1−2ρ)    . For _bγ_kE

n, we minimize the following Partial AMSE:

γ(γ−1_{− 1)}1−ρ 1 − γ − ρ A(n/kn) 2 +γ 3_{(1 − γ)} 1 − 2γ × 1 kn , hence b k_nE= min       (γ−1− 1)2ρ−1_{(1 − γ − ρ)}2 −2ρb2(1 − 2γ) !1/(1−2ρ) n−2ρ/(1−2ρ)    , jn 2 k − 1  . 20 of 25 Antoine UsseglioCarleve

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Simulation study

0 100 200 300 400 1.3 1.4 1.5 1.6 1.7 1.8 0 100 200 300 400 2.0 2.5 3.0 0 100 200 300 400 3 4 5 6 0 100 200 300 400 2 4 6 8 10 12 14 0 100 200 300 400 1.3 1.4 1.5 1.6 1.7 0 100 200 300 400 2.0 2.5 3.0 0 100 200 300 400 3 4 5 6 0 100 200 300 400 2 4 6 8 10 12 14 0 100 200 300 400 1.2 1.3 1.4 1.5 1.6 1.7 0 100 200 300 400 1.5 2.0 2.5 3.0 0 100 200 300 400 2 3 4 5 6 0 100 200 300 400 2 4 6 8 10 12 0 100 200 300 400 4 6 8 10 12 0 100 200 300 400 6 8 10 12 14 0 100 200 300 400 10 15 20 25 0 100 200 300 400 10 15 20 25 30 35 40 Figure: 21 of 25 Antoine UsseglioCarleve

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Reinsurance example

Let Y be a claim amount, and R a retention level.

• Insurer pays min(Y , R).

• _{Reinsurer pays max(Y − R, 0).}

The reinsurance premium Π(R) may be calculated using the distortion principle:

Πg(R) =

Z ∞

R

g (F (x ))dx ,

where g : [0, 1] → [0, 1] is a nondecreasing concave function.

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Reinsurance example

If Y is heavy-tailed and g (1/.) regularly varying with index δ < −γ, then lim β↑1 Πg(e(β)) e(β) g (1 − β) = (γ−1− 1)−δ −δ/γ − 1 ,

by taking an extreme expectile as retention level. We thus use our expectile estimators to approximate the reinsurance premium, and compare our results with those obtained with the data set secura in [Vandewalle and Beirlant, 2006], with two distortion functions:

g (x ) = x (Net premium principle)

g (x ) = 1 − (1 − x )κ (Dual-power principle) .

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Reinsurance example

4e+06 5e+06 6e+06 7e+06 8e+06 9e+06 1e+07

0

50000

100000

150000

Net premium principle

Retention level

Premium

4e+06 5e+06 6e+06 7e+06 8e+06 9e+06 1e+07

0 50000 100000 150000 Dual−power principle Retention level Premium

Figure:Πg(R) as function of R = e(β) for β ranging from 1 − 10/n ≈ 0.973

to 1 − 1/(8n) ≈ 0.9997 (here n = 370). The premiums are estimated using e

γ_kH

n (solid blue curve), eγ

E

kn (solid red curve) and the na¨ıve estimator (dotted

blue curve). The black curve is constructed by linear interpolation using the

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-Materials

• R package Expectrem, available at

https://github.com/AntoineUC/Expectrem.

• _{Girard, S., Stupfler, G. and Usseglio-Carleve, A. (2020) On}

automatic bias reduction for extreme expectile estimation, preprint.

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