Abdelaati Daouia, Simone A. Padoan and Gilles Stupfler “Extreme expectile estimation for short-tailed data, with an application to market risk assessment” March 2023 1414

1414

“Extreme expectile estimation for short-tailed data, with an application to market risk assessment”

Abdelaati Daouia, Simone A. Padoan and Gilles Stupfler

March 2023

Extreme expectile estimation for short-tailed data, with an application to market risk

assessment

Abdelaati Daouia

Toulouse School of Economics, University of Toulouse Capitole, France Simone A. Padoan

Department of Decision Sciences, Bocconi University, Italy and

Gilles Stupfler

Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers, France March 6, 2023

Abstract

The use of expectiles in risk management has recently gathered remarkable mo- mentum due to their excellent axiomatic and probabilistic properties. In particular, the class of elicitable law-invariant coherent risk measures only consists of expectiles.

While the theory of expectile estimation at central levels is substantial, tail estima- tion at extreme levels has so far only been considered when the tail of the underlying distribution is heavy. This article is the first work to handle the short-tailed setting where the loss (e.g. negative log-returns) distribution of interest is bounded to the right and the corresponding extreme value index is negative. We derive an asymptotic expansion of tail expectiles in this challenging context under a general second-order extreme value condition, which allows to come up with two semiparametric estima- tors of extreme expectiles, and with their asymptotic properties in a general model of strictly stationary but weakly dependent observations. A simulation study and a real data analysis from a forecasting perspective are performed to verify and compare the proposed competing estimation procedures.

Keywords: Expectiles, Extreme values, Second-order condition, Short tails, Weak depen- dence

1 Introduction

The class of expectiles, introduced by Newey and Powell(1987), defines useful descriptors ξ_τ of the higher (τ ≥ ¹₂) and lower (τ ≤ ¹₂) regions of the distribution of a random variableX through the asymmetric least squares (ALS) minimization problem

ξ_τ = arg min

θ∈R

E

η_τ(X−θ)−η_τ(X)

, (1)

where η_τ(x) = |τ −1{x ≤ 0}|x², with τ ∈ (0,1) and 1{·} being the indicator function.

Expectiles are well-defined, finite and uniquely determined as soon the first moment of X is finite. They generalize the mean ξ_1/2 =E(X) in the same way quantiles generalize the median, thus defining an ALS analog to quantiles. Indeed, Koenker and Bassett (1978) showed that the τth quantile q_τ of X solves the asymmetricL¹ minimization problem

q_τ ∈arg min

θ∈R

E(%_τ(X−θ)−%_τ(X)),

where %τ(x) = |τ −1{x ≤ 0}| |x|. Expectiles have received renewed attention for their ability to quantify tail risk at least since the contribution of Taylor (2008). They de- pend on the tail realizations of X and their probability, while quantiles only depend on the frequency of tail realizations, see Kuan et al. (2009). Most importantly, Ziegel (2016) showed that expectiles are the sole coherent law-invariant measure of risk which is also elicitable in the sense of Gneiting (2011), meaning that they abide by the intuitive diversi- fication principle (Bellini et al.,2014) and that their prediction can be performed through a straightforward principled backtesting methodology. These merits have motivated the development of procedures for expectile estimation and inference over the last decade. A key, but difficult, question in any risk management setup is the estimation of the expectile ξ_τ at extreme levels, which grow to 1 as the sample size increases. This question was first tackled inDaouia et al.(2018,2020) under the assumption that the underlying distribution

is heavy-tailed, that is, its distribution function tends to 1 algebraically fast. The latest developments under this assumption have focused on, among others, bias reduction (Girard et al., 2022), accurate inference (Padoan and Stupfler, 2022), and handling more complex data in regression (Girard et al., 2021, 2022) or time series (Davison et al., 2022) setups.

The problem of estimating extreme expectiles outside of the set of heavy-tailed models is substantially more complicated from a statistical standpoint. The contribution of the present paper is precisely to build and analyze semiparametric extreme expectile estima- tors in the challenging short-tailed model, in which the extreme value index (EVI) of the underlying distributions is known to be negative. This requires employing a dedicated extrapolation relationship for population extreme expectiles. Only Mao et al. (2015) have initiated such a study at the population level when X belongs to the domain of attraction of a Generalized Extreme Value distribution (GEV). Differently to Mao et al. (2015), we work in the general semiparametric Generalized Pareto (GP) setting through a standard second-order condition, which makes it possible to derive an asymptotic expansion of ex- treme expectiles without resorting to an unnecessary restriction about the link between the EVI and second-order parameter that featured in Mao et al. (2015). Based on this asymptotic expansion, we present and study two different extreme value estimators of tail expectiles. The first one builds upon the Least Asymmetrically Weighted Squares (LAWS) estimator of expectiles, namely the empirical counterpart ofξ_τ in (1), obtained at interme- diate levels τ = τ_n → 1 with n(1−τ_n) → ∞ as the sample size n → ∞. The short-tail model assumption allows then to come up with an expectile estimator extrapolated to the far tail at arbitrarily extreme levels τ = 1 −p_n such that (1−τ_n)/p_n → ∞ as n → ∞, in a semiparametric way reminiscent of how extreme quantiles are fitted in Section 4.3 of de Haan and Ferreira (2006). The second extrapolating estimator directly relies on the

asymptotic expansion of ξ_τ that involves its quantile analog q_τ, the endpoint q₁ ≡ ξ₁ and the EVI, by plugging in the GP quantile-based estimators of these tail quantities. Our esti- mation theory is valid in a general setting of strictly stationary and weakly dependent data satisfying reasonable mixing and tail dependence conditions. We explore various theoretical and practical features of extreme expectile estimation in this setting, and explain why this problem is statistically more difficult than extreme quantile estimation. In particular, an extreme expectile ξ_τ is intrinsically less spread than its quantile analog q_τ, even at asym- metry levels τ ≈1 where it remains much closer to the center of the distribution than q_τ. Consequently, any semiparametric procedure for extreme expectile estimation should be expected to suffer at least from a worse bias than for extreme quantile estimation.

Our focus on the problem of estimating extreme expectiles for bounded distributions is motivated by the perhaps somewhat surprising finding that weekly returns of equities, used in applications to circumvent the non-synchronicity of daily data, may have short- tailed distributions. This is illustrated in Figure 1 for 14 major companies and financial institutions, where the data consists of the loss returns (i.e. negative log-returns) on their weekly equity price from 21st September 2014 to 12th June 2022, corresponding to 403 trading weeks. The representative price is constructed by averaging daily closing prices within the corresponding week. The nature of the upper tail of these loss returns is reflected by the EVI of their distribution whose negative, zero or positive values indicate respectively a distribution with short, light or Pareto-type tail. None of these three scenarios can be excluded in practice for these 14 data examples, where the EVI is estimated on successive rolling windows of length n = 150 using the GP distribution fitted to exceedances over a high threshold by means of the Maximum Likelihood (ML) method, with the optimal threshold being chosen by the path stability procedure as described below in Section4. It is

−0.6

−0.4

−0.2 0.0 0.2

0 50 100 150 200 250

Rolling window

EVI estimates

variable Alibaba Alphabet Amazon Apple

Bank of America Goldman Sachs Intel

JPMorgan Chase Meta

Morgan Stanley NASDAQ Netflix T. Rowe Price Walmart

Figure 1: Maximum Likelihood estimates of the extreme value index over the resulting 253 successive rolling windows of 150 stationary data, obtained from 14 time series of weekly logarithmic (loss) returns between 21st September 2014, and 12th June 2022.

therefore important to construct an appropriate and fully data-driven estimation procedure for the challenging scenario of short-tailed data. This problem also appears naturally in production econometrics when analyzing the productivity of firms (Kokic et al.,1997). All our methods and data have been incorporated into the R package ExtremeRisks.

In Section2, we explain in detail the short tail distributional assumption onX, state our asymptotic expansion linking extreme expectiles and quantiles, construct our two classes of extreme expectile estimators and study their asymptotic properties. A simulation study examines their finite-sample performance in Section 3, and a time series of Bitcoin data is analyzed in Section 4. The online Supplementary Material contains all the proofs in Section A and further simulation results in Section B.

2 Main results

2.1 Connection between extreme expectiles and quantiles

Let F : x 7→ P(X ≤ x) be the distribution function of the random variable of interest X and F = 1−F be its survival function. Define the associated quantile function by q_τ = inf{x ∈ R|F(x) ≥ τ} and the tail quantile function U by U(s) = q_1−s⁻¹, s > 1.

Differently from existing literature on extreme expectile estimation, we focus on the case when the distribution of X is short-tailed, or equivalently, when its EVI γ is negative.

According to Theorem 1.1.6 on p.10 of de Haan and Ferreira (2006), this corresponds to assuming that there is a positive function a such that

∀z >0, lim

s→∞

U(sz)−U(s)

a(s) = z^γ−1

γ , with γ <0.

This assumption can be informally rewritten as

∀z >0, U(sz)≈U(s) +a(s)z^γ−1

γ whens is large. (2)

This means that extreme values ofX at the far tail (represented byU(sz)) can be achieved by extrapolating in-sample large values (represented by U(s)) if the scale function a(s) and the shape parameter γ can be consistently estimated. The theory of the resulting extreme value estimators is usually developed under the following second-order refinement of the short-tailed model assumption above, which will be our main condition throughout (see de Haan and Ferreira, 2006, Equation (2.3.13) on p.45):

ConditionC₂(γ, a, ρ, A) There existγ <0,ρ≤0, a positive functiona(·) and a measurable function A(·) having constant sign and converging to 0 at infinity such that, for all z >0,

s→∞lim 1 A(s)

U(sz)−U(s)

a(s) −z^γ−1 γ

= Z z

1

v^γ−1 Z v

1

u^ρ−1du

dv.

This condition enables one to control the bias incurred by using the approximation (2) and represented by the function A. Under this condition, the right endpoint x^? = sup{x ∈ R|F(x) < 1} of X is necessarily finite (see de Haan and Ferreira, 2006, Theorem 1.2.1 on p.19). This justifies calling this model a short-tailed (or bounded) model.

Suppose now that E|min(X,0)| < ∞ and that condition C₂(γ, a, ρ, A) is satisfied, so that E|X| < ∞ and expectiles of X are well-defined and finite. First, we motivate an asymptotic expansion of extreme expectiles that will be instrumental in our subsequent theory of extreme expectile estimation. Recall that the τth expectile ξ_τ satisfies

ξ_τ −E(X) = 2τ−1

1−τ E((X−ξ_τ)1{X > ξ_τ}), (3) see Equation (12) in Bellini et al. (2014). Writing E((X−x)1{X > x}) as an integral of the quantiles of X abovex and using condition C₂(γ, a, ρ, A) justifies the approximation

E((X−ξ_τ)1{X > ξ_τ})≈ F(ξ_τ)a(1/F(ξ_τ))

1−γ asτ ↑1, and therefore

limτ↑1

a(1/F(ξ_τ))F(ξ_τ)

1−τ = (x^?−E(X))(1−γ). (4)

The convergence a(s)/(x^? − U(s)) → −γ as s → ∞ (see de Haan and Ferreira, 2006, Lemma 1.2.9 on p.22) then suggests

limτ↑1

(x^?−ξ_τ)F(ξ_τ)

1−τ = (x^?−E(X))(1−γ⁻¹). (5)

The approximations F(ξ_τ)/(1−τ)≈F(ξ_τ)/F(q_τ)≈(x^?−ξ_τ)^−1/γ/(x^?−q_τ)^−1/γ motivated by the regular variation property of x 7→ F(x^? −1/x) (see de Haan and Ferreira, 2006, Theorem 1.2.1.2 on p.19) finally entail

limτ↑1

x^?−ξ_τ

(x^? −q_τ)^1/(1−γ) = [(x^?−E(X))(1−γ⁻¹)]^{−γ/(1−γ)}. (6)

Consequently, extreme expectiles can be extrapolated from their quantile analogs in con- junction with endpoint and EVI estimation. Analyzing the asymptotic properties of the estimators built in this way will require quantifying the difference between the ratio (x^?− ξ_τ)/(x^?−q_τ)^1/(1−γ) and its limit in (6). This is the focus of our first main result below.

Proposition 1. Suppose that E|min(X,0)| < ∞ and condition C2(γ, a, ρ, A) holds with ρ <0, and let x^? be the finite right endpoint of F. Then

x^?−ξ_τ = (x^?−q_τ)^1−γ1

[(x^?−E(X))(1−γ⁻¹)]^1−γγ 1−

x^?−q_τ

(x^?−E(X))(1−γ⁻¹) _1−γ¹

(1 + o(1)) + γ[(x^?−E(X))(1−γ⁻¹)]^−1−γρ

ρ(γ+ρ)(1−γ−ρ) A (x^?−q_τ)^1−γ1 1−τ

!

(1 + o(1))

!

as τ ↑1. In particular

x^? − ξ_τ = (x^?−q_τ)^1−γ1

[(x^?−E(X))(1−γ⁻¹)]^1−γγ

1 + O((1−τ)^−1−γγ ) + O(|A((1−τ)^−1−γ1 )|) .

The additional conditionρ <0 in Proposition1is very mild and satisfied in all standard short-tailed models, see Beirlant et al. (2004, Table 2.2 on p.68).

Remark 1 (An equivalent asymptotic expansion). An equivalent version of the asymptotic expansion in Proposition 1, where the error terms rely solely on the expectile ξ_τ, is

x^?−ξ_τ = [(x^?−E(X))(1−γ⁻¹)]^{−γ/(1−γ)}(x^?−q_τ)^1/(1−γ)

×

1− x^?−ξτ

(x^? −E(X))(1−γ⁻¹)(1 + o(1)) + γ

ρ(γ+ρ)(1−γ−ρ)A(1/F(ξ_τ))(1 + o(1))

.

This is immediate from Proposition1, (5) and (6), since|A|is regularly varying with indexρ.

Remark 2 (Comparison with Proposition 3.4 in Mao et al. (2015)). Proposition 1 is an extension, tailored to our general semiparametric GP setting and extended second-order regular variation assumption C₂(γ, a, ρ, A), of Proposition 3.4 in Mao et al. (2015). The latter result is formulated under a different, nonstandard second-order regular variation

condition on F when X belongs to the domain of attraction of a Generalized Extreme Value distribution. It is readily checked by straightforward but tedious calculations that their quantitiesc,γ,ρandA(s) respectively correspond to (with the notation of Lemma A.3 in Section A.1) C^1/γ, −1/γ, −ρ/γ and −C^−ρ/γA(s^−1/γ)/(γ(γ+ρ)) of the present paper.

In particular, when their asymptotic expansion applies, it coincides with ours, but we lift an unnecessary restriction on the second-order parameter ρ that features in their result.

Remark 3 (Expectiles are less extreme than quantiles). An immediate consequence of Equation (6) is that (x^? −ξ_τ)/(x^? −q_τ) → ∞ as τ ↑ 1, that is, extreme quantiles are closer to the endpoint of a short-tailed distribution than extreme expectiles. It is therefore unsurprising that the bias due to the approximation of tail expectiles by their quantile analogs under the second-order framework, which is asymptotically proportional to A((1− τ)⁻¹(x^?−q_τ)^1/(1−γ)), converges more slowly to 0 than the corresponding bias term in the heavy-tailed setting, whose order isA((1−τ)⁻¹), see Proposition 1(i) inDaouia et al.(2020).

As a second consequence, at least as far as handling bias is concerned, estimating extreme expectiles under short-tailed models using a semiparametric extreme value methodology should be expected to be much harder than under heavy-tailed models.

Remark 4 (On remainder terms in the asymptotic expansion). The quantityx^? −E(X), which is a measure of the spread of the distribution tail, appears in the asymptotic equiva- lent of (x^?−ξ_τ)/(x^?−q_τ) and in both the remainder terms of the asymptotic expansion for x^?−ξ_τ. By contrast, no measures of spread appear in the asymptotic connection between extreme expectiles and quantiles of heavy-tailed distributions, although the expectation E(X), which can be understood as a location parameter, appears in an error term propor- tional to 1/qτ, as can be seen from Proposition 1 inDaouia et al. (2020).

With Proposition 1 at our disposal, we can now construct and study two classes of

extreme expectile estimators. The first one, in Section 2.2 below, is built upon asymmetric least squares minimization, while the second one, in Section 2.3, is directly obtained by plugging in Equation (6) estimators of E(X) and of the tail quantities γ, x^? and qτ.

2.2 Asymmetric least squares estimation

Suppose that the available data has been generated from the random variables X₁, . . . , X_n with common distribution function F, and let τ_n ↑ 1 (as n → ∞) be a high asymmetry level at which the target unknown expectile ξ_τn is to be estimated. A first solution is to construct the estimator minimizing the empirical counterpart of problem (1). This produces the Least Asymmetrically Weighted Squares (LAWS) estimator

ξb_τn = arg min

θ∈R

1 n

n

X

t=1

η_τn(X_t−θ)−η_τn(X_t) = arg min

θ∈R n

X

t=1

η_τn(X_t−θ). (7) A possible way to derive the asymptotic normality of ξb_τn would be to find a quadratic approximation to Pn

t=1η_τn(X_t−θ) and then to use stochastic convex optimization theo- rems such as those of Hjort and Pollard (1993). However, such developments already re- quire lengthy technical arguments in the better-known heavy-tailed model, as can be seen fromDaouia et al.(2018). Instead, we propose to use here an alternative technique leading to a much simpler proof, which is based on the following observation made byJones(1994):

the τth expectile of F is the τth quantile of the distribution functionE = 1−E, where E(x) = E(|X−x|1{X > x})

E(|X−x|) . This survival function can equivalently be rewritten as

E(x) = ϕ⁽¹⁾(x)

2ϕ⁽¹⁾(x) +x−E(X), with ϕ^(κ)(x) =E((X−x)^κ1{X > x}).

Since ξb_τ is the τth expectile of the empirical distribution function Fb_n(x) = 1

n

n

X

t=1

1{X_t> x},

it must therefore be the τth quantile of the distribution functionEb_n = 1−Eb_n defined as Eb_n(x) = ϕb⁽¹⁾n (x)

2ϕb⁽¹⁾n (x) +x−X_n, where ϕb^(κ)_n (x) = 1 n

n

X

t=1

(X_t−x)^κ1{X_t > x},

with Xn being the sample mean. Intuitively, to derive now the asymptotic behavior of ξb_τn −ξ_τn, it suffices to obtain the asymptotic behavior of Eb_n(x)/E(x) at a level x = x_n close to ξ_τn in an appropriate sense and to apply a suitable inversion argument.

We do so in a general framework of strictly stationary, weakly dependent random vari- ables. Recall that a strictly stationary sequence (X_t)t≥1is said to beα−mixing (or strongly mixing) if α(l) = sup_m≥1α_m(l)→0 when l→ ∞, where

α_m(l) = sup

A∈F_1,m B∈F_m+l,∞

|P(A∩B)−P(A)P(B)|

with F_1,m = σ(X₁, . . . , X_m) and Fm+l,∞ = σ(X_m+l, X_m+l+1, . . .) being past and future σ−algebras. The α−mixing condition is one of the weakest dependence assumptions in

the mixing time series literature: more restrictive conditions include β−, ρ−, φ− and ψ−mixing, see Bradley(2005). We make the following assumption about the mixing rate.

Condition M There exist sequences of positive integers (l_n) and (r_n), both tending to infinity, such that ln/rn →0,rn/n →0 andn α(ln)/rn →0, as n→ ∞.

The sequences (ln) and (rn) are respectively interpreted as “small-block” and “big-block”

sequences, and are used to develop a big-block/small-block argument as a prerequisite to evaluating the asymptotic variance of ξb_τn. Condition M has already been used in the literature on the extreme values of time series, seee.g.Rootz´en et al.(1998, Equation (2.1)).

We also require the following tail dependence condition on the joint extreme behavior of (X_t)_t≥1 at different time points.

Condition D For any integer t≥1, there is a function R_t on [0,∞]²\ {(∞,∞)} such that

∀(x, y)∈[0,∞]²\ {(∞,∞)}, lim

s→∞sP(F(X₁)≤x/s, F(X_t+1)≤y/s) =R_t(x, y),

and there exist a constant K ≥ 0 and a nonnegative summable sequence (ρ(t))t≥1 such that, for s large enough,

∀t ≥1, ∀x, y ∈(0,1], sP(F(X₁)≤x/s, F(X_t+1)≤y/s)≤ρ(t)√

xy+K s xy.

The function R_t, called the tail copula of (X₁, X_t+1) (seeSchmidt and Stadtm¨uller,2006), finely quantifies the degree of asymptotic dependence between X₁ and X_t+1. Condition D ensures that the probability of a joint extreme value ofX1 andXt+1 is of the same order of magnitude as the probability of an extreme value of X₁, meaning that clusters of extreme values across time cannot form too often. A similar anti-clustering assumption is made in Drees(2003), see conditions (C2) and (C3) therein.

Under these temporal dependence assumptions and using our insight about the link between the LAWS estimator and the empirical estimator ofE, we can prove the following result on the joint asymptotic normality of the LAWS estimator and an empirical quantile having the same order of magnitude.

Theorem 1. Assume that X satisfies condition C₂(γ, a, ρ, A). Let τ_n, α_n ↑ 1 be such that nF(ξ_τn)→ ∞, F(ξ_τn)/(1−α_n)→1 and

q

nF(ξ_τn)A(1/F(ξ_τn)) = O(1).

(i) Suppose that (X_t)t≥1 is a strictly stationary sequence of copies of X, whose dis- tribution function F is continuous, satisfying conditions M and D. Assume that rnF(ξτn)→0, and that there is δ >0 such that

E(|min(X,0)|^2+δ)<∞, X

l≥1

l^2/δα(l)<∞ and r_n



 r_n q

nF(ξ_τn)





δ

→0.

Then q

nF(ξ_τn)

a(1/F(ξ_τn))(bξ_τn−ξ_τn,qb_αn−q_αn)−→ N^d (0,V(γ) + 2C(γ, R))

where the 2×2 symmetric matrices V(γ) and C(γ, R) are defined elementwise as V₁₁(γ) = 2/[(1−γ)(1−2γ)], V₁₂(γ) = 1/(1−γ) and V₂₂(γ) = 1,

C₁₁(γ, R) = 1 γ²

Z Z

(0,1]²

∞

X

t=1

R_t(x^−1/γ, y^−1/γ) dxdy C₁₂(γ, R) = − 1

2γ Z 1

0

∞

X

t=1

[R_t(x^−1/γ,1) +R_t(1, x^−1/γ)] dx and C₂₂(γ, R) =

∞

X

t=1

R_t(1,1).

(ii) If the X_i are i.i.d. copies of X and E(|min(X,0)|²)<∞, then the above asymptotic normality result holds with R_t ≡0 for any t ≥1, that is,

q

nF(ξ_τn)

a(1/F(ξ_τn))(bξ_τn−ξ_τn,bq_αn−q_αn)−→ N^d (0,V(γ)).

If X is bounded, then assumptionP

l≥1l^2/δα(l)<∞ in (i) can be weakened toP

l≥1α(l)<

∞, and no integrability assumption on X is necessary.

In Theorem 1, condition nF(ξ_τn) → ∞ requires that τ_n be intermediate, i.e. not too large. Assumption

q

nF(ξτn)A(1/F(ξτn)) = O(1) is a bias condition which corresponds exactly to the usual bias condition p

n(1−τ_n)A((1−τ_n)⁻¹) = O(1) in extreme quantile estimation when replacing ξ_τn by q_τn, see Theorem 2.4.1 on p.50 of de Haan and Ferreira (2006). The conditions onr_nin Theorem1(i) are similar to those of Theorem 3.1 inDavison et al. (2022) under heavy-tailed models, taking into account that F(ξ_τn) is asymptotically proportional to 1−τ_n in the latter setting. The integrability assumption on X and the condition on the mixing rateα(l) ensure that a central limit theorem applies toXn, as part of the proof of the asymptotic normality of Eb_n(x)/E(x) at high levels x=x_n close toξ_τn. Remark 5(On bias conditions).An inspection of the proof of Theorem1(i) reveals that the bias condition

q

nF(ξ_τn)A(1/F(ξ_τn)) = O(1) is only needed for the asymptotic normality of qbαn−qαn, and is unnecessary for the validity of the asymptotic normality ofξbτn−ξτn alone.

Remark 6 (Comparison with the i.i.d. case). It is natural to compare the asymptotic normality of the LAWS estimator for short-tailed data with the corresponding result one obtains for heavy-tailed data. We restrict the comparison to the i.i.d. setting for the sake of simplicity. If X has a heavy right tail, that is, U(sz)/U(s) → z^γ as s → ∞, for any z > 0, where 0 < γ < 1/2, and under the assumptions that E(|min(X,0)|^2+δ) < ∞ for some δ >0, τ_n ↑1 and n(1−τ_n)→ ∞, one has, by Theorem 2 in Daouia et al. (2018),

pn(1−τn) ξbτn

ξ_τn −1

!

−→ Nd

0, 2γ³ 1−2γ

.

In this same setting, (U(sz)−U(s))/a(s) → (z^γ −1)/γ as s → ∞, with a(s) = γU(s), and F(ξ_τn)/(1−τ_n) → γ⁻¹ −1 = (1−γ)/γ (this was first shown by Bellini et al., 2014, Theorem 11). Therefore, when X has a heavy right tail,

q

nF(ξ_τn)

a(1/F(ξ_τn))(bξ_τn−ξ_τn)≈

√1−γ γ^3/2 ×p

n(1−τ_n) ξb_τn ξ_τn −1

!

−→ Nd

0,2(1−γ) 1−2γ

.

It follows that the rates of convergence of the LAWS estimator look similar in both heavy and bounded tail settings, but there is a phase transition in terms of asymptotic variance:

the term 1−γ appears in its numerator for heavy tails, while it appears in the denominator for short tails, as established in Theorem 1. Interestingly, the two asymptotic variances in the heavy and short-tailed settings converge to 2, and therefore exactly match together in the light-tailed middle scenario, when γ →0.

Remark 7 (Comparison with the asymptotic normality of intermediate quantiles). It is instructive to compare Theorem 1 with the asymptotic normality result for the direct in- termediate quantile estimator at level τn, namely, the empirical counterpartqbτn =Xdnτne,n. According to Theorem 2.4.1 on p.50 of de Haan and Ferreira(2006), when the X_i are i.i.d.,

q

nF(q_τn)

a(1/F(q_τn))(qb_τn−q_τn)−→ N^d (0,1).

Observe that, by a combination of Lemma 1.2.9 in de Haan and Ferreira (2006) and Lemma A.1 in Section A.1, a(1/F(x))/(x^?−x)→ −γ as x↑x^?, and therefore

q

nF(ξτn) a(1/F(ξ_τn))

, q

nF(qτn) a(1/F(q_τn)) =

s

F(ξ_τn)

F(q_τn) ×x^? −q_τn x^? −ξτn

(1 + o(1)).

By (5) and (6) this ratio is asymptotically proportional to (x^? −q_τn)−(γ+1/2)/(1−γ) under the mild further condition ρ < 0. In other words, the intermediate LAWS estimator ξb_τn converges faster than qbτn when γ >−1/2, has the same rate of convergence if γ = −1/2, and converges at a slower rate if γ <−1/2.

As a corollary of Theorem 1, we obtain the asymptotic normality of the empirical estimator Fb_n(bξ_τn) of F(ξ_τn), on which the rate of convergence of ξb_τn crucially depends.

Corollary 1. Work under the conditions of Theorem 1. Then q

nF(ξτn)

Fb_n(bξ_τn) F(ξ_τn) −1

!

−→ Nd

0, 2γ²+γ+ 1 (1−γ)(1−2γ) +2

Z Z

(0,1]²

∞

X

t=1

1

γ²R_t(x^−1/γ, y^−1/γ) + 1

γ[R_t(x^−1/γ,1) +R_t(1, x^−1/γ)] +R_t(1,1)

dxdy

! .

Remark 8 (On rates of convergence). The rate of convergence ofFb_n(bξ_τn) is rather natural:

for a sequence (u_n) tending tox^? such that nF(u_n)→ ∞, Lemma A.5 states that q

nF(u_n) Fb_n(u_n) F(u_n) −1

!

−→ Nd 0,1 + 2

∞

X

t=1

R_t(1,1)

! .

It is worth noticing that the asymptotic variance of Fb_n(bξ_τn) does not coincide with the variance that would be obtained if ξτn were known, namely, if Fbn(ξτn) were considered instead. This is due to the asymptotic dependence existing between ξb_τn and high order statistics (and therefore between ξb_τn and Fb_n), see Theorem1and the proof of Corollary1.

We now have the tools necessary to construct an extreme value estimator of a properly extreme expectile ξ1−pn, where pn ↓ 0 at any possible rate as n → ∞, by extrapolating

the intermediate LAWS estimator ξb_τn to the right place at the far tail. Using (2) with s = 1/F(ξ_τn) and z =F(ξ_τn)/F(ξ_1−pn) motivates the approximation

ξ1−p_n ≈ξ_τn +a(1/F(ξ_τn))(F(ξ_τn)/F(ξ1−p_n))^γ−1

γ .

By Theorem1,ξ_τn is estimated by the LAWS estimator ξb_τn at ratea(1/F(ξ_τn))/

q

nF(ξ_τn).

The scale parameter a(1/F(ξ_τn)) and shape parameter γ can be estimated by a variety of techniques, such as the GP-pseudo-ML estimator and Moment-type estimator, see Sec- tions 3.4 and 3.5 in de Haan and Ferreira (2006). Typical such estimators of the scale function a(1/F(u_n)), when u_n↑ x^? such that nF(u_n)→ ∞, converge on the relative scale at the rate 1/

q

nF(u_n); see Sections 3.4 and 4.2 in de Haan and Ferreira (2006) in the i.i.d. case, and Section 6 in Drees (2003) in the dependent data setup. Since, by Corol- lary 1, the (unknown) quantity F(u_n) = F(ξ_τn) can be consistently estimated at the rate 1/

q

nF(ξτn), we therefore expect to be able to estimatea(1/F(ξτn)) at this rate on the rel- ative scale. Finally, given an intermediate levelτ_n, it is customary to estimate the extreme value index γ at the rate 1/p

n(1−τ_n) when the top k = bn(1−τ_n)c values in the data are used, see Sections 3.4, 3.5 and 3.6 in de Haan and Ferreira(2006) in the i.i.d. case, and again Section 6 in Drees(2003) when the data points are serially dependent. It remains to find a way to estimate F(ξ_τn)/F(ξ_1−pn), which depends on the target quantityξ_1−pn itself.

Combining (5) and (6) with the fact that the function s7→ x^?−U(s) is regularly varying with index γ (de Haan and Ferreira, 2006, Corollary 1.2.10 on p.23) suggests that

F(ξτn)

F(ξ1−pn) ≈ 1−τn

p_n ×x^?−ξ1−pn

x^?−ξ_τn ≈ 1−τn

p_n

x^?−q1−pn

x^?−q_τn

1/(1−γ)

≈ 1−τ_n p_n

1−τ_n p_n

γ/(1−γ)

=

1−τ_n p_n

1/(1−γ)

(8) which in turn leads to the expectile-specific approximation

ξ1−pn ≈ξ_τn +a(1/F(ξ_τn))((1−τn)/pn)^γ/(1−γ)−1

γ .

Consequently, like extreme quantiles, extreme expectiles can be extrapolated from their values at lower levels. More specifically, given estimators bσ_n and bγ_n of a(1/F(ξ_τn)) and γ, respectively, one can then construct the ξ1−pn estimator

ξb_1−p^?

n =ξb_τn+bσ_n((1−τ_n)/p_n)^{bγn/(1−bγn)}−1

bγ_n . (9)

Since (1−τ_n)/F(ξ_τn) → 0, the parameter γ is estimated at a slower rate than the other quantities, so we expect the asymptotic behavior of bγn to govern that of ξb_1−p^? _n. The last theorem of this section makes this intuition rigorous. Its proof crucially relies on Theorem1 and on Proposition 1 in order to quantify the bias in the approximation (8).

Theorem 2. Work under the conditions of Theorem1. If moreover ρ <0, n(1−τ_n)→ ∞, (1−τ_n)/p_n→ ∞, p

n(1−τ_n)/log((1−τ_n)/p_n)→ ∞, p

n(1−τ_n)(x^?−q_τn)^1/(1−γ) = O(1), pn(1−τ_n)A((1−τ_n)⁻¹) = O(1), bσ_n and bγ_n are such that

q

nF(ξ_τn)

bσ_n

a(1/F(ξ_τn))−1

= O_P(1) and p

n(1−τ_n)(bγ_n−γ)−→^d Γ, where Γ is a nondegenerate limit, then

pn(1−τ_n)

a(1/F(ξ_τn))(bξ_1−p^? _n−ξ1−p_n)−→^d Γ γ².

2.3 Quantile-based estimation

We use here Proposition1to present an alternative estimator of extreme expectiles, purely based on quantiles, and to develop its asymptotic theory. Similarly to the setup of extreme quantile estimation in Section 4.3 ofde Haan and Ferreira(2006), assume thatk =k_n→ ∞ is a sequence of positive integers such that k/n → 0 and that respective estimators bγ_n, ba(n/k) and Xn−k,n of γ, a(n/k) and U(n/k) are given such that

√ k

γb_n−γ,ba(n/k)

a(n/k)−1,Xn−k,n−U(n/k) a(n/k)

d

−→(Γ,Λ, B) (10)

where (Γ,Λ, B) is a nontrivial trivariate weak limit. This assumption is satisfied by moment and GP-pseudo-ML estimators of the shape and scale parameters, among others, see an overview in Section 4.2 of de Haan and Ferreira (2006) in the case where the Xi are independent random variables. It is also satisfied when (X_t)t≥1 is a strictly stationary but serially dependent sequence: this is for example the case when the data points areβ−mixing and satisfy an anti-clustering condition similar to the tail dependence assumption D, as a consequence of the powerful results of Drees(2003).

Let p_n ↓ 0 with k/(np_n) → ∞, so that the level 1−p_n is much more extreme than 1−k/n. Following Section 4.3 in de Haan and Ferreira (2006), the extreme quantileq1−pn

and the right endpoint x^? can be estimated by

qb^?_1−pn =Xn−k,n+ba(n/k)(k/(np_n))^bγn−1

bγ_n and xb^? =Xn−k,n− ba(n/k)

bγ_n . (11) According to Proposition 1, an estimator of ξ1−pn follows then as

ξe_1−p^? _n =xb^?−[(xb^?−X_n)(1−bγ_n⁻¹)]^{−bγn/(1−bγn)}(bx^?−qb^?_1−pn)^1/(1−γbn). (12) The next result provides its asymptotic properties, where two sequences (u_n) and (v_n) are said to be asymptotically proportional if (u_n/v_n) tends to a finite positive limit asn→ ∞.

Theorem 3. Suppose that E|min(X,0)|<∞and condition C₂(γ, a, ρ, A)holds withρ <0, and let x^? be the finite right endpoint of F. Assume that condition (10) holds true and that √

k(X_n −E(X)) −→^P 0 with k = k_n being asymptotically proportional to n^α, for some α ∈ (0,1). Let p_n be asymptotically proportional to n^−β where β > 0 is such that α+β−1>0. If moreover √

kA(n/k)→λ∈R, then we have, up to changing probability

spaces and with appropriate versions of the estimators involved,

ξe_1−p^? _n−ξ1−p_n = a(ⁿ_k)

√k 1 γ²

Γ +γ²B−γΛ− λγ

γ+ρ + o_P(1)

−

an k

k npn

γ_1−γ¹

(1−γ)^−1−γ1 γ(x^?−E(X))^1−γγ

log(^{np1/(1−γ)n}_k )

√k Γ + o_P

logn

√k !

+ O(n^{βγ/(1−γ)}(n^{βγ/(1−γ)}+|A(n^β/(1−γ))|)).

Remark 9 (Mixing and the central limit theorem). Condition √

k(X_n− E(X)) −→^P 0 is satisfied in practice if √

n(Xn −E(X)) = O_P(1), which is in particular true when a central limit theorem applies. As already highlighted below Theorem 1, this will be the case if E(|min(X,0)|²) < ∞ when the X_i are independent, or if there is δ > 0 such that E(|min(X,0)|^2+δ) <∞ and P

l≥1l^2/δα(l) < ∞ when (X_t)t≥1 is α−mixing. In particular, when the data mixes geometrically fast, then √

n(X_n−E(X)) = O_P(1) as soon as X has a finite moment of order 2 +δ, for some δ >0.

Remark 10 (Our assumptions on k). The assumption that k = kn is asymptotically equivalent to a positive and finite multiple of n^α, is only very slightly stronger than the usual pair of extreme value conditions k → ∞ and k/n → 0 made throughout Section 4 inde Haan and Ferreira(2006). The only difference is that our assumption does not allow to takek growing to infinity logarithmically fast; such sequences produce, however, very small values of k in practice and would therefore yield estimators having very large variances.

We also note that in standard settings such as those of Beirlant et al. (2004, Table 2.2 on p.68), A(s) is asymptotically proportional to s^ρ, in which case the optimal choices of k satisfying the usual bias-variance tradeoff for extreme value index estimation would fulfill

√kA(n/k) → λ ∈ R\ {0}, that is, k = O(n^{−2ρ/(1−2ρ)}). In other words, it is reasonable to expect that optimal choices of k in practice have to be asymptotically equivalent to a positive and finite multiple of a fractional power of n.

It follows from Theorem 3 that the asymptotic behavior of the extreme expectile es- timator ξe_1−p^?

n is more complex than that of the extreme quantile estimator bq_1−p^?

n: while, from Theorem 4.3.1 on p.134 and Theorem 4.5.1 on p.146 of de Haan and Ferreira(2006), qb_1−p^? _n −q1−pn converges to the same distribution _γ¹2

Γ +γ²B−γΛ−λ_γ+ρ^γ

as xb^? −x^? at the rate a(n/k)/√

k for γ < 0, the asymptotic distribution of ξe_1−p^? _n −ξ1−pn may be a nonstandard mixture of the two limiting distributions of bx^?−x^? and bγ_n−γ. In particular, Corollary 2 shows that when, for example, β = 1 and α (and hence k) is chosen small enough, it is in fact the asymptotic distribution Γ ofbγ_n−γ that dominates inξe_1−p^?

n−ξ_1−pn, while Corollary 3examines what can otherwise be said.

Corollary 2. Work under Theorem3. Ifα <1−β/(1−γ)andα <2βmin(−γ,−ρ)/(1−γ),

√k log(np^1/(1−γ)n /k)

ξe_1−p^? _n−ξ1−pn

[a(ⁿ_k)(_np^k

n)^γ]^1/(1−γ)

−→ −γd ⁻¹(1 − γ)^{−1/(1−γ)}(x^? − E(X))^{−γ/(1−γ)}Γ.

Remark 11 (Link between bias terms and short-tailedness). The closer γ is to 0, the stronger the constraint on α through the condition α < 2βmin(−γ,−ρ)/(1−γ). This is analogous to what happens in extreme expectile estimation for heavy-tailed distributions, where the condition √

k/q_1−k/n = O(1) (see e.g. Daouia et al., 2020, Theorem 5) becomes a strong restriction as the tail gets less heavy, i.e. when γ approaches 0.

Remark 12 (Comparison between the LAWS and quantile-based estimators). One may compare the rates of convergence of ξb_1−p^? _n and ξe_1−p^? _n by setting τ_n = 1−k/n. Using the convergence a(s)/(x^? − U(s)) → −γ as s → ∞ and Equation (6), one finds under the assumptions of Corollary 2that

√ k a(1/F(ξ1−k/n))

, √

k

log(np^1/(1−γ)n /k)[a(n/k)(k/(np_n))^γ]^1/(1−γ)

∝log(n)[a(n/k)]^1/(1−γ)

a(1/F(ξ1−k/n))(k/(np_n))^γ/(1−γ) ∝n(α+β−1)γ/(1−γ)

log(n)→0.

This means that ξe_1−p^? _n converges to ξ1−p_n faster than ξb_1−p^? _n when k (or 1−τ_n) is chosen small. We shall illustrate this finding below in our simulation study.

Conditionα <1−β/(1−γ) may not hold in a given example, especially whenβis large enough, or equivalently, p_n is small enough. Yet, interestingly this condition can always be satisfied for sufficiently small α in the standard setting β = 1 of extreme value analysis. If it is not satisfied, then ξe_1−p^? _n tends to inherit the asymptotic behavior of bx^?, rather thanbγ, as established in the following result.

Corollary 3. Under the assumptions of Theorem 3, if moreover α >1−β/(1−γ), then

ξe_1−p^? _n−ξ1−p_n = a(ⁿ_k)

√kγ²

Γ +γ²B−γΛ− λγ

γ+ρ + o_P(1)

+ O(n^1−γβγ (n^1−γβγ +|A(n^1−γβ )|)).

The condition α > 1−β/(1−γ) itself is not sufficient to ensure the convergence of ξe_1−p^? _n; in practice, the bias term may dominate the asymptotics depending on the choice of k. This is most easily seen when A(s) is asymptotically proportional to s^ρ and β = 1, corresponding to the standard extreme value situation wherep_n≈c/n. In this case, (i) one automatically hasα+β−1 = α >0, (ii) condition√

kA(n/k)→λ∈Ressentially amounts to α≤ −2ρ/(1−2ρ), and (iii) condition α >1−β/(1−γ) becomes α >−γ/(1−γ). For the bias term in Corollary 3to be negligible, one requires

√k

a(n/k) ×n^γ/(1−γ)(n^γ/(1−γ)+|A(n^1/(1−γ))|)→0.

Since a(n/k) is asymptotically proportional to (n/k)^γ by Lemma A.3(i), this is equivalent to assuming

α 1

2+γ

+ γ²−min(−γ,−ρ) 1−γ <0.

When γ > −1/2, as is often the case in applications, and 0 < −ρ < −γ, representing situations where the bias due to the second-order framework is high, this condition becomes

α <− 2(γ²+ρ) (1−γ)(1 + 2γ).

Depending on the value of ρ, this final condition may not be compatible withα >−γ/(1− γ): in fact, if ρ is close enough to 0, it may even be impossible to satisfy whatever the value of α (since the right-hand side of the above displayed inequality tends to a negative constant asρ→0, whenγ >−1/2). In this case, with the choicep_n =c/n, the asymptotic behavior of bx^?−x^? can never dominate inξe_1−p^? _n−ξ1−pn.

2.4 Selection of the expectile asymmetry level

In practical situations it is crucial to make an informed decision as to what the asymmetry level of the target expectile should be. In financial applications, where the dual interpre- tation of expectiles in terms of the gain-loss ratio is available (Bellini and Di Bernardino, 2017), it is sensible to set the expectile level so as to achieve a certain value of the gain-loss ratio. Otherwise, it has been proposed in the literature to select τ such that ξτ coincides with another pre-specified intuitive risk measure: Bellini and Di Bernardino(2017) suggest to choose the expectile level τ so that ξ_τ is identical to the Value-at-Risk (or quantile) q_α, where α is a high tail probability level specified by the statistician or the practitioner.

The proposal of Bellini and Di Bernardino (2017) is valid only when the underlying loss distribution is Gaussian. Daouia et al. (2018) later extended this idea to the heavy- tailed setup. We examine here the short-tailed situation, hitherto unexplored from this perspective. Fix a large quantile level α=α_n= 1−p_n. Setting τ =τ_n= 1−π_n to be the expectile level such that ξ_τ =q_α, Equation (5) leads to

(x^?−q1−pn)p_n

π_n = (x^?−ξ_τ)F(ξ_τ)

1−τ ≈(x^? −E(X))(1−γ⁻¹).

In other words,

π_n≈ x^?−q1−p_n

(x^?−E(X))(1−γ⁻¹)p_n.

This approximation suggests to estimate the quantity π_n by

πb_n ≡bπ_n(p_n) = bx^?−bq^?_1−pn

(xb^?−X_n)(1−bγ_n⁻¹)p_n

with the notation of (11). Our final main result shows that this estimator is asymptotically normal in the framework of Section 2.3.

Proposition 2. Work under Theorem3. If moreover α <min(−2βγ,−2ρ/(1−2ρ)), then

√k log(k/(np_n))

πb_n π_n −1

d

−→Γ.

Remark 13 (Comparison with the heavy-tailed setting). In the heavy-tailed case, accord- ing to Section 5 in Daouia et al. (2018), π_n ≈ p_n/(γ⁻¹ −1). An estimator of π_n is then πb_n = p_n/(bγ_n⁻¹−1). In this setting, it is straightforward to obtain, under a suitable bias condition when √

k(bγ_n−γ)→Γ, that

√ k

bπ_n π_n −1

d

−→ Γ γ(1−γ).

The estimator bπ_n therefore converges at a slightly faster rate in the heavy-tailed model.

The slower speed of convergence in the short-tailed framework is due to the presence of the quantity bx^?−qb^?_1−pn = −ba(n/k)γb_n⁻¹(k/(np_n))^bγn in the numerator of bπ_n, whose rate of convergence to x^?−q1−pn is precisely log(k/(np_n))/√

k, as obtained in Proposition2.

3 Simulation study

The finite-sample performance of the proposed extreme expectile estimators is illustrated here through a simulation study. Our setup first considers three models for i.i.d. data:

(i) The Xt have a Beta distribution, whose density function is f(x|α, β) = x^α−1(1−x)^β−1

B(α, β) , 0≤x≤1.

Here B(α, β) is the Beta function and the shape parameters are set asα= 3 and β = 5/2.

In this model, the extreme value index is γ =−2/5 and the upper endpoint is x^∗ = 1.

(ii) The X_t have a short-tailed power-law distribution, whose distribution function is F(x|x^∗, K, α) = 1−K(x^∗−x)^α, x^∗−K^−1/α≤x≤x^∗.

Herex^?,K and α are the distribution endpoint, a positive constant and the shape param- eter, respectively, which have been set as x^∗ = 5, K = 1/3 and α= 3, so that γ =−1/3.

(iii) The X_t have a GEV distribution, whose distribution function is F(x|γ) = exp(−(1 +γx)^−1/γ), 1 +γx >0.

We set the extreme value index γ =−1/3, so that the upper endpoint isx^∗ =−1/γ = 3.

We then consider the following three time series models, in which Φ denotes the standard normal distribution function andY_tis the AR(1) process defined asY_t+1 =%Y_t+p

1−%²ε_t, with independent standard normal innovations ε_t, and where% ∈(−1,1):

(iv) X_t = q_X(Φ(Y_t)), where q_X is the quantile function corresponding to the Beta distri- bution defined in (i), and where the correlation parameter is %= 0.95.

(v) X_t = q_X(Φ(Y_t)), where q_X is the quantile function corresponding to the short-tailed power-law distribution defined in (ii), and where the correlation parameter is %= 0.5.

(vi) Xt =qX(Φ(Yt)), where qX is the quantile function corresponding to the GEV distri- bution defined in (iii), and where the correlation parameter is %= 0.8.

The EVI and endpoints of models (iv)-(vi) are those of models (i)-(iii), respectively, and the time series models (iv)-(vi) are geometrically β-mixing (and in particular geomet- rically α−mixing) since the linear AR(1) process (Yt) is so. We consider sample sizes

n = 150,300,500 and aim to predict expectiles of extreme level τ_n⁰ = 1−p_n = 1−1/n = 0.9933,0.9967,0.9980. As the true expectile values cannot be given in closed form, they have been computed by intensive Monte Carlo simulations and are reported in Table 1.

We simulate M = 10,000 samples of n observations from each model and compare the purely empirical (LAWS) estimator ξb_τn⁰ in (7), the extrapolating LAWS estimators ξb_τ^?0

n

in (9) obtained by setting bσ_n = ba(1/Fb_n(bξ_τn)), its alternative version ξ^?_τ0

n obtained with σb_n = ba((1 −τ_n)⁻¹)×((1 −τ_n)/Fb_n(bξ_τn))^bγn in view of the approximation a(1/F(ξ_τn)) ≈ ((1 − τ_n)/F(ξ_τn))^γa((1 −τ_n)⁻¹) that follows from the regular variation property of the scale function a, and the extrapolating quantile-based (QB) estimator ξe_τ^?0

n in (12). In these last three estimators, (ba(n/k),bγ_n) are either the pair of GP-pseudo-ML estimators of (a(n/k), γ) (see de Haan and Ferreira, 2006, Section 3.4) based on the top k observations in the sample, or their versions based on the Dekkers et al. (1989) Moment estimator (see de Haan and Ferreira, 2006, Section 3.5 and 4.2). We set throughout τ_n = 1−k/n, let the effective sample size k range from 1% up to 25% of the total sample sizen, and record Monte Carlo approximations of the relative bias, variance and Mean Squared Error (MSE) of the estimators as a function of k.

Results are reported in Figures B.1–B.6 in Section B.1. In each figure the relative

Model τ_n⁰ = 0.9933 τ_n⁰ = 0.9967 τ_n⁰ = 0.9980

(i), (iv) 0.8571 0.8814 0.8968

(ii), (v) 4.5284 4.5939 4.6372

(iii), (vi) 1.9523 2.1020 2.2000

Table 1: Values of the expectile ξ_τn⁰ obtained through intensive Monte Carlo simulations for τ_n⁰ = 1−1/n, withn = 150,300,500.