2.1 Statistical Model

Let the financial position of interest X be a real-valued random variable, and F be its cumulative distribution function. In this article, we consider loss returns (i.e., minus log-returns), which implies that extreme losses correspond to the upper tail of F. Denote by U the associated tail quantile function defined as $U(t) = F ^{\leftarrow }(1 - 1/t)$ for all $t> 1$ , where $F ^{\leftarrow } (\cdot ) = \inf \{x \in \mathbb {R}, F(x) \geqslant \cdot \}$ stands for the generalized inverse of F. Estimating tail quantities, including expectiles and quantiles, from the perspective of extreme value statistics relies on the fundamental domain of attraction assumption that there exist a constant $\gamma \in \mathbb {R}$ and a positive scale function $a(\cdot )$ such that, for all $x>0,$

(2.1) $$ \begin{align} \lim_{t \to \infty}\frac{U(tx)-U(t)}{a(t)}=\int_1^{x} s^{\gamma-1}\,\mathrm{d}s. \end{align} $$

The parameter $\gamma $ is referred to as the extreme value index (EVI for short): a negative, zero, or positive EVI indicates, respectively, a distribution with short, light, or heavy right tail. In the case of heavy-tailed distributions (i.e., $\gamma>0$ ) that describe quite well the tail structure of most financial data, following Corollary 1.2.10 in de Haan and Ferreira (Reference de Haan and Ferreira2006), the condition (2.1) is equivalent to the Weissman extrapolation relationship

(2.2) $$ \begin{align} \lim_{t \to \infty}\frac{U(tx)}{U(t)}=x^\gamma,\quad \text{for}\: x>0. \end{align} $$

Equations (2.1) and (2.2) can, respectively, be rewritten as the GP approximation

(2.3) $$ \begin{align} \forall x>0, \ U(tx) \approx U(t) + a(t) \frac{x^{\gamma}-1}{\gamma} \text{ when } t \text{ is large} \end{align} $$

and the Weissman approximation

(2.4) $$ \begin{align} \forall x>0, \ U(tx) \approx x^{\gamma} U(t) \text{ when } t \text{ is large.} \end{align} $$

The asymptotic theory of extreme value estimators for tail quantities, including the EVI and extreme expectiles and quantiles, shows that the bias incurred in using these approximations is mainly determined by the speed of convergence in the domain of attraction condition (2.1) or (2.2). This is captured, for a general $\gamma \in \mathbb {R}$ , by the so-called extended second-order regular variation (SORV) assumption:

• $\mathcal {E}_2( \gamma , \rho , a, A)$ : There exist $\gamma \in \mathbb {R}$ , a second-order parameter $\rho \leqslant 0$ , a scale function $a(\cdot )>0,$ and a constant sign function $A(\cdot )$ which converges to $0$ at infinity such that

  $$\begin{align*}\forall x>0, \,\, \lim_{t \to \infty}\frac{1}{A(t)}\left(\frac{U(tx)-U(t)}{a(t)}-\frac{x^\gamma-1}{\gamma}\right)= \int_1^{x} s^{\gamma-1}\int_1^su^{\rho-1}\mathrm{d}u\,\mathrm{d}s.\end{align*}$$

This assumes that the right tail of the distribution of X can be approximated by a GPD tail at a known rate. As can be seen, for instance, from Sections 2.3 and 2.4 and Chapter 3 in de Haan and Ferreira (Reference de Haan and Ferreira2006), as well as Section 5.6 in Beirlant et al. (Reference Beirlant, Goegebeur, Segers and Teugels2004), this kind of condition is inevitable in order to carry out extreme value analysis when no parametric model structure is imposed on F. In the heavy-tailed case, it comes as a consequence of Theorem 3.1 in Fraga Alves et al. (Reference Fraga Alves, Gomes, de Haan and Neves2007) that this extended SORV condition reduces to the assumption:

• $\mathcal {C}_2(\gamma , \tilde \rho , \tilde {A})$ : There are $\gamma>0$ , a second-order parameter $\tilde \rho \leqslant 0,$ and a constant sign function $\tilde {A}(\cdot )$ which converges to $0$ at infinity such that

  $$ \begin{align*} \forall x>0, \,\, \lim_{t\to \infty} \frac{1}{\tilde{A}(t)}\left(\frac{U(tx)}{U(t)}-x^{\gamma}\right)=x^{\gamma}\int_1^{x} s^{\tilde\rho-1}\,\mathrm{d}s. \end{align*} $$

This condition can, for instance, be checked using goodness-of-fit statistics in the form of weighted integrals over the tail of the empirical quantile function, or over its distribution function counterpart. See Section 5.2 in de Haan and Ferreira (Reference de Haan and Ferreira2006) for several examples.

Theorem 3.1 by Fraga Alves et al. (Reference Fraga Alves, Gomes, de Haan and Neves2007) provides explicit expressions of the second-order parameters $\tilde \rho $ and $\tilde {A}(\cdot )$ in terms of $\gamma , \rho , a(\cdot )$ and $A(\cdot )$ . These parameters $(\tilde {\rho },\tilde {A})$ (resp. $(\rho ,A)$ in $\mathcal {E}_2$ ) control the bias incurred in using the Weissman (resp. GPD) approximation (2.4) (resp. (2.3)): the more negative $\tilde {\rho }$ (resp. $\rho $ ) is, the faster $\tilde {A}(t)$ (resp. $A(t)$ ) converges to 0 as $t\to \infty $ and the smaller the bias incurred is. Daouia et al. (Reference Daouia, Stupfler and Usseglio-Carleve2025) and Daouia et al. (Reference Daouia, Stupfler and Usseglio-Carleve2024b) provide a list of examples of commonly used families of continuous distributions satisfying both of these conditions, and with the explicit corresponding values of $\gamma , \rho , a(\cdot )$ and $A(\cdot )$ .

The crucial point is that it is not, in general, the case that $\rho =\tilde {\rho }$ and $A=\tilde {A}$ . As a matter of fact, it is a corollary of Theorem 3.1(i) by Fraga Alves et al. (Reference Fraga Alves, Gomes, de Haan and Neves2007) that all standard continuous distributions that satisfy both conditions have $\rho \leq \tilde {\rho }$ ; this means that the bias incurred in using the GPD approximation (2.3) is never qualitatively worse than the bias incurred in using the Weissman approximation (2.4). In fact, the former can be substantially smaller: it is seen from Table 4 in Daouia et al. (Reference Daouia, Stupfler and Usseglio-Carleve2024b) and Table 2 in Daouia et al. (Reference Daouia, Stupfler and Usseglio-Carleve2025) that the Fréchet distribution with $\alpha =1$ and the Inverse- $\Gamma $ distribution both have $\rho =2\tilde {\rho }$ . For pure GP observations, it is, of course, always better to rely on the GPD approach, since then condition $\mathcal {E}_2$ is in fact satisfied as an equality rather than a limit, while the GP distribution only satisfies condition $\mathcal {C}_2$ with $\tilde {\rho } =-\gamma $ . The fact that the GPD approximation carries lower bias is one justification for substituting it for the Weissman approximation.

We focus here on the estimation of extreme expectiles when F is attracted to the maximum domain of Pareto-type distributions with tail index $0<\gamma <1$ , using the GPD approximation. Together with condition $\mathbb {E}|X_{-}|<\infty $ , with $X_{-}=\min (X,0)$ the negative part of X, the assumption $\gamma <1$ ensures that the first moment of X exists, and hence expectiles of X are well-defined (note that the first moment of X and hence its expectiles do not exist when $\gamma>1$ ). Under this model assumption, Bellini and Di Bernardino (Reference Bellini and Di Bernardino2017) have established in their Proposition 2.3 an interesting asymptotic connection between extreme expectiles and their quantile analogs, namely, the asymptotic equivalence

(2.5) $$ \begin{align} \xi_{\tau}\sim (\gamma^{-1}-1)^{-\gamma}{q_{\tau}} \ \text{ as } \ \tau\to 1. \end{align} $$

It follows that tail expectiles $\xi _{\tau }$ are more spread than tail quantiles $q_{\tau }$ when ${\gamma>1/2}$ , whereas $\xi _{\tau }<q_{\tau }$ for all large $\tau $ when $\gamma <1/2$ . The asymptotic equivalence (2.5) is fundamental in the construction of Weissman-type estimators for extreme expectiles, which we review in Appendix A.

Our objective is, rather, to construct and study below two new classes of extreme expectile estimators: the first one in Section 2.2 is built upon asymmetric least squares extrapolation, while the second one in Section 2.3 relies on extreme quantile estimation. We build here a general theory based on generic estimators of the scale and shape parameters in conjunction with approximation (2.3); the application to GPML estimators and moment estimators of the scale and shape parameters is discussed in Section 2.4.

2.2 Extremal Asymmetric Least Squares Estimation

From now on, let $X_1, \ldots , X_n$ be a sample of observations drawn from F, with corresponding survival function $\overline {F}=1-F$ , and let $X_{1,n}\leqslant \cdots \leqslant X_{n,n}$ be their corresponding order statistics. In the intermediate case when $\tau = \tau _n \uparrow 1$ such that $n \overline {F}(\xi _{\tau _n})\to \infty $ as $n \to \infty $ , the population expectile $\xi _{\tau _n}$ is extreme but well inside the sample so that it can be consistently estimated by its empirical version

(2.6) $$ \begin{align} \widehat{\xi}_{\tau_n}=\operatorname*{\mbox{arg min}}_{u\in\mathbb{R}}\frac{1}{n}\sum_{i=1}^{n}\eta_{\tau_n} (X_i-u), \end{align} $$

where $\eta _{\tau }(y)=|\tau -\mathbb {1} \{ y\leqslant 0 \}|y^2$ , as described in Appendix A. In the more challenging extreme case when $\tau = \tau ^{\prime }_n>\tau _n $ such that $(1-\tau _n)/(1-\tau ^{\prime }_n)\to \infty $ as $n \to \infty $ , the population expectile $\xi _{\tau ^{\prime }_n}$ lies at the far tail of the underlying distribution, possibly beyond the sample maximum $X_{n,n}$ in the data. Using (2.3) with $t=1/\overline {F}(\xi _{\tau _n})$ and $x=\overline {F}(\xi _{\tau _n})/\overline {F}(\xi _{\tau ^{\prime }_n})$ , and recalling that $\lim _{t\to \infty } (U(1/\overline {F}(t))-t)/a(1/\overline {F}(t)) = 0$ (see, e.g., the proof of Theorem B.3.19 in de Haan and Ferreira, Reference de Haan and Ferreira2006), we obtain the expectile-specific approximation

$$\begin{align*}\xi_{\tau^{\prime}_n} \approx \xi_{\tau_n} + a(1/\overline{F}(\xi_{\tau_n})) \frac{(\overline{F}(\xi_{\tau_n})/\overline{F}(\xi_{\tau^{\prime}_n}))^{\gamma}-1}{\gamma} \end{align*}$$

for n large enough. Since, in the heavy-tailed case with $0<\gamma <1$ , we have $\overline {F}(\xi _{\tau })/(1-\tau ) \sim \gamma ^{-1}-1$ as $\tau \to 1$ (Bellini et al., Reference Bellini, Klar, Müller and Gianin2014, Thm. 11), it follows that

$$ \begin{align*}\frac{\overline{F}(\xi_{\tau_n})}{\overline{F}(\xi_{\tau^{\prime}_n})} \approx \frac{1-\tau_n}{1-\tau^{\prime}_n} \end{align*} $$

as $n\to \infty $ , which in turn suggests the extrapolating approximation

$$\begin{align*}\xi_{\tau^{\prime}_n} \approx \xi_{\tau_n} + a(1/\overline{F}(\xi_{\tau_n})) \frac{((1-\tau_n)/(1-\tau^{\prime}_n))^{\gamma}-1}{\gamma}. \end{align*}$$

Therefore, given estimators $\breve {\sigma }_n$ and $\breve {\gamma }_n$ of $a(1/\overline {F}(\xi _{\tau _n}))$ and $\gamma $ , respectively, this motivates the extreme expectile estimator

(2.7) $$ \begin{align} \breve{\xi}_{\tau^{\prime}_n}^{\star} = \breve{\xi}_{\tau^{\prime}_n}^{\star}(\breve{\gamma}_n,\breve{\sigma}_n) :=\widehat{\xi}_{\tau_n} + \breve{\sigma}_n \frac{((1-\tau_n)/(1-\tau^{\prime}_n))^{\breve{\gamma}_n}-1}{\breve{\gamma}_n}. \end{align} $$

Various estimators of the scale function $\sigma (\cdot )\equiv a(1/\overline {F}(\cdot ))$ and shape parameter $\gamma $ are available in the literature, including the (pseudo-)GPML estimators and the Moment-type estimators: if $k=k_n\to \infty $ is a sequence of integers such that $k/n\to 0$ , as $n\to \infty $ , then

• • The GPML estimators (Smith, Reference Smith1987; Drees et al., Reference Drees, Ferreira and de Haan2004) of $(a(n/k_n),\gamma )$ are given by

  $$\begin{align*}(\widehat{a}^{\mathrm{ML}}(n/k_n),\widehat{\gamma}_n^{\mathrm{ML}})=\operatorname*{\mbox{arg max}}_{\sigma,\gamma>0} \prod_{i=1}^{k_n} h(X_{n-i+1,n}-X_{n-k_n,n}|\sigma,\gamma), \end{align*}$$

  where the Generalized Pareto density function is defined as $h(\cdot |\sigma ,\gamma )=H'(\cdot |\sigma ,\gamma )$ .

• • The Moment estimators (Dekkers et al., Reference Dekkers, Einmahl and de Haan1989) of $(a(n/k_n),\gamma )$ are defined as

  $$\begin{align*}\big(\widehat{a}^{\mathrm{Mom}}(n/k_n),\widehat{\gamma}_n^{\mathrm{Mom}}\big)&=\big(X_{n-k_n,n} M_{k_n}^{(1)} (1-\widehat{\gamma}_n^{(-)}), M_{k_n}^{(1)}+\widehat{\gamma}_n^{(-)}\big), \\ \text{where } \ \widehat{\gamma}_n^{(-)}&=1 - \frac{1}{2} \left( 1-\frac{(M_{k_n}^{(1)})^2}{M_{k_n}^{(2)}} \right)^{-1}\\ \text{and } \ M_{k_n}^{(j)}&=\frac{1}{{k_n}} \sum_{i=1}^{k_n} (\log X_{n-i+1,n}-\log X_{n-{k_n},n})^j, \ \text{ for } j=1,2. \end{align*} $$

Such estimators of the scale parameter $a(1/\overline {F}(u_n))$ typically converge on the relative scale at the rate $1/\sqrt {n \overline {F}(u_n)}$ (see Sections 3.4 and 4.2 in de Haan and Ferreira, Reference de Haan and Ferreira2006, for the i.i.d. case, and Section 6 in Drees, Reference Drees2003, for the case of dependent data). For $u_n=\xi _{\tau _n}$ , estimators $\breve {\sigma }_n$ of the parameter $a(1/\overline {F}(\xi _{\tau _n}))$ are therefore expected to converge on the relative scale at the rate $1/\sqrt {n \overline {F}(\xi _{\tau _n})}$ , which we have seen to be asymptotically proportional to the rate of convergence $1/\sqrt {n(1-\tau _n)}$ of the intermediate expectile estimator $\widehat {\xi }_{\tau _n}$ (see Theorem 2 below). It is also customary to use estimators $\breve {\gamma }_n$ that rely on the top $k_n=n(1-\tau _n)$ observations and converge at the same rate $1/\sqrt {n(1-\tau _n)}$ (see Sections 3.4–3.6 in de Haan and Ferreira, Reference de Haan and Ferreira2006, in the i.i.d. case, and Section 6 in Drees, Reference Drees2003, in the serially dependent case). A noteworthy difference between the GPML and Moment estimators is that the former are location- and scale-invariant, while the latter are only scale-invariant. This may be of importance in practical applications where the data are standardized and de-trended.

The next result shows that among the three estimators $\widehat {\xi }_{\tau _n}$ , $\breve {\gamma }_n$ , and $\breve {\sigma }_n$ on which $\breve {\xi }_{\tau ^{\prime }_n}^{\star }$ hinges in (2.7), it is the asymptotic behavior of $\breve {\gamma }_n$ which dominates that of $\breve {\xi }_{\tau ^{\prime }_n}^{\star }$ .

Theorem 1. Assume that

1. 1. $\mathbb {E}|X_{-}|^2<\infty $ and $\mathcal {E}_2(\gamma , \rho , a, A)$ holds with $0 <\gamma <1/2$ and $\rho < 0$ ;

2. 2. $\tau _n=1-k_n/n \to 1$ such that $n(1-\tau _n)=k_n \to \infty $ , $\sqrt {n (1-\tau _n)} A((1-\tau _n)^{-1})\to \lambda _1\in \mathbb {R}$ , $\sqrt {n (1-\tau _n)}/q_{\tau _n}\to \lambda _2\in \mathbb {R}$ , and $\sqrt {n(1-\tau _n)} (a((1-\tau _n)^{-1})/q_{\tau _n} - \gamma ) \to \mu \in \mathbb {R}$ , $n \to \infty $ ;

3. 3. $\tau ^{\prime }_n \to 1$ such that $(1-\tau _n)/(1-\tau ^{\prime }_n)\to \infty $ and $\sqrt {n(1-\tau _n)}/\log ((1-\tau _n)/(1-\tau ^{\prime }_n)) \to \infty $ , $n \to \infty $ ;

4. 4. $\sqrt {n(1-\tau _n)}\left (\frac {\widehat {\xi }_{\tau _n}}{\xi _{\tau _n}}-1\right ) = \mathrm {O}_{\mathbb {P}}(1)$ and for suitable estimators $\breve {\gamma }_n$ and $\breve {\sigma }_n$ ,

   $$\begin{align*}\sqrt{n(1-\tau_n)} \left( \frac{\breve{\sigma}_n}{a(1/\overline{F}(\xi_{\tau_n}))} - 1 \right) = \mathrm{O}_{\mathbb{P}}(1) \ \text{ and } \ \sqrt{n(1-\tau_n)} (\breve{\gamma}_n-\gamma)\stackrel{\mathrm{d}}{\longrightarrow} \Gamma, \end{align*}$$

   where $\Gamma $ is a nondegenerate limit.

Then

$$\begin{align*}\sqrt{n(1-\tau_n)} \frac{\breve{\xi}_{\tau^{\prime}_n}^{\star} - \xi_{\tau^{\prime}_n}}{a(1/\overline{F}(\xi_{\tau_n})) \int_1^{(1-\tau_n)/(1-\tau^{\prime}_n)} s^{\gamma-1} \log(s) \mathrm{d}s} \stackrel{\mathrm{d}}{\longrightarrow} \Gamma. \end{align*}$$

Under the i.i.d. model assumption, the property $\sqrt {n(1-\tau _n)}\left (\frac {\widehat {\xi }_{\tau _n}}{\xi _{\tau _n}}-1\right ) = \mathrm {O}_{\mathbb {P}}(1)$ follows immediately from the asymptotic normality of $\widehat {\xi }_{\tau _n}$ derived in Theorem 1 of Daouia et al. (Reference Daouia, Girard and Stupfler2020) when $0<\gamma <1/2$ and $\mathbb {E}|X_{-}|^2<\infty $ . Note also that by Corollary 4.3.2 on p. 135 in de Haan and Ferreira (Reference de Haan and Ferreira2006), the conditions of our Theorem 1 imply

$$\begin{align*}\frac{\int_1^{(1-\tau_n)/(1-\tau^{\prime}_n)} s^{\breve{\gamma}_n-1} \log(s) \mathrm{d}s}{\int_1^{(1-\tau_n)/(1-\tau^{\prime}_n)} s^{\gamma-1} \log(s) \mathrm{d}s} \stackrel{\mathbb{P}}{\longrightarrow} 1, \ \text{ as } n\to\infty, \end{align*}$$

which is useful for constructing an asymptotic confidence interval for $\xi _{\tau ^{\prime }_n}$ . For this same purpose and for the calculation of $\breve {\xi }_{\tau ^{\prime }_n}^{\star }$ in (2.7), the estimator $\breve {\sigma }_n$ of $a(1/\overline {F}(\xi _{\tau _n}))$ can be obtained in three different ways by setting

$$ \begin{align*} & \breve{\sigma}^{(1)}_n = \widehat{a}(1/\widehat{\overline{F}}_n(\widehat{\xi}_{\tau_n})), \quad \breve{\sigma}^{(2)}_n = \widehat{a}((1-\tau_n)^{-1}) \times ((1-\tau_n)/\widehat{\overline{F}}_n(\widehat{\xi}_{\tau_n}))^{\breve{\gamma}_n}, \\ & \hspace{0cm} \text{or }\ \breve{\sigma}^{(3)}_n = \widehat{a}((1-\tau_n)^{-1}) \times (\breve{\gamma}_n^{-1}-1)^{-\breve{\gamma}_n}. \end{align*} $$

The first estimator $\breve {\sigma }^{(1)}_n$ is simply the empirical counterpart of $a(1/\overline {F}(\xi _{\tau _n}))$ . The second estimator $\breve {\sigma }^{(2)}_n$ is obtained in view of the approximation $a(1/\overline {F}(\xi _{\tau _n})) \approx a((1-\tau _n)^{-1}) \times ((1-\tau _n)/\overline {F}(\xi _{\tau _n}))^{\gamma }$ that follows from the regular variation property $\lim _{s\to \infty }a(sz)/a(s)=z^{\gamma }$ , for any $z>0$ (see Lemma 1.2.9 on page 22 in de Haan and Ferreira, Reference de Haan and Ferreira2006), when taking $s=1/\overline {F}(q_{\tau _n})$ and $z=\overline {F}(q_{\tau _n})/\overline {F}(\xi _{\tau _n})$ . The third estimator $\breve {\sigma }^{(3)}_n$ is obtained from the same approximation above by replacing $\overline {F}(\xi _{\tau _n})/(1-\tau _n)$ with its limit $\gamma ^{-1}-1$ . These three versions of $\breve {\sigma }_n$ will be denoted in the sequel as $\breve {\sigma }_n^{\mathrm {ML},i},\, i=1,2,3$ , when using the GPML method and as $\breve {\sigma }_n^{\mathrm {Mom},i},\, i=1,2,3$ , when using the Moment method. This will in turn result in the associated GPML-based expectile estimators $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {ML}},\breve {\sigma }_n^{\mathrm {ML},i})$ and Moment-type estimators $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {Mom}},\breve {\sigma }_n^{\mathrm {Mom},i})$ . These estimators can be calculated by using our function gpdExpect from the R package Expectrem with the respective methods "direct_ML1" $,\ldots ,$ "direct_ML3", and "direct_MOM1" $,\ldots ,$ "direct_MOM3".

It is also natural and instructive to compare Theorem 1 with the results for the competing Weissman-type estimators reviewed in Appendix A. We have from the proof of Theorem 1 the following equivalent statement:

(2.8) $$ \begin{align} \frac{\sqrt{n(1-\tau_n)}}{\log((1-\tau_n)/(1-\tau^{\prime}_n))}\left(\frac{\breve{\xi}_{\tau^{\prime}_n}^{\star}}{ \xi_{\tau^{\prime}_n}} -1\right) \stackrel{\mathrm{d}}{\longrightarrow} \Gamma. \end{align} $$

Therefore, $\breve {\xi }_{\tau ^{\prime }_n}^{\star }$ has the same rate of convergence as $\widetilde {\xi }^\star _{\tau ^{\prime }_n}$ in (A.6) and $\widehat {\xi }^\star _{\tau ^{\prime }_n}$ in (A.7), as well as their bias-corrected versions $\widehat {\xi }_{\tau ^{\prime }_n}^{\star ,\mathrm {BR}}$ in (B.5) and $\widetilde {\xi }_{\tau ^{\prime }_n}^{\star ,\mathrm {BR}}$ in (B.6). While the asymptotic distribution $\Gamma $ of these Weissman-type estimators is typically that of the Hill estimator $\widehat {\gamma }_{n}^{\mathrm {H}}$ or the expectile-based estimator $\widehat {\gamma }_{n}^{\mathrm {E}}$ in (A.3), or their bias-reduced versions $\widehat {\gamma }_{n}^{\mathrm {H},\mathrm {BR}}$ and $\widehat {\gamma }_{n}^{\mathrm {E},\mathrm {BR}}$ in (A.10) and (A.11), it would naturally correspond in our generalized Pareto approach to the limit distribution of the GPML estimator $\widehat {\gamma }_n^{\mathrm {ML}}$ or Moment estimator $\widehat {\gamma }_n^{\mathrm {Mom}}$ . Interestingly, the GPML and Moment methods often correct for bias and can be more robust in small and moderate samples or when the optimal $k_n$ selection is unclear from Hill plots. Selection of the threshold $k_n$ is known to be a difficult task in general. Schneider, Krajina, and Krivobokova (Reference Schneider, Krajina and Krivobokova2021) develop two threshold selection methods under the assumption that $\gamma>0$ and compare them with eight competitors, including bootstrap techniques. Approaches that are specifically tailored to GP models are the changepoint tests of Northrop and Coleman (Reference Northrop and Coleman2014) and the simultaneous threshold stability plots of Wadsworth (Reference Wadsworth2016) based on point process calculations. The latter two methods are implemented in the R package mev developed by Belzile et al. (Reference Belzile, Wadsworth, Northrop, Huser, Grimshaw, Zhang, Stephens and Owen2025). In our simulation study (Section 3), we shall examine the performance of our estimators as functions of the threshold, so as to find the best performers across a large range of threshold values; our financial real data analysis (Section 4) will use a simple selection rule based on finding the best extreme expectile forecast across time.

When the data follow a GPD, $\widehat {\gamma }_n^{\mathrm {ML}}$ is the estimator that tends to achieve the lowest mean squared error (MSE) in moderate to large samples, but can be unstable for small samples. The estimator $\widehat {\gamma }_n^{\mathrm {Mom}}$ is more robust when $\widehat {\gamma }_n^{\mathrm {ML}}$ becomes unstable, while still being competitive in terms of variance (see, e.g., Beirlant et al., Reference Beirlant, Goegebeur, Segers and Teugels2004, and Yılmaz, Kara, and Özdemir, Reference Yılmaz, Kara and Özdemir2021, for a detailed comparison and review on estimation procedures for the EVI). The preference for $\breve {\xi }_{\tau ^{\prime }_n}^{\star }$ over the Weissman-type competitors stems not only from such practical advantages of the EVI estimators $\widehat {\gamma }_n^{\mathrm {ML}}$ and $\widehat {\gamma }_n^{\mathrm {Mom}}$ over the Hill-based alternatives, but also from its reliance on additional location parameter estimates that contribute to capturing uncertainty more effectively. Our experience with real and simulated heavy-tailed profit–loss data shows that, across a wide range of $k_n$ values and without finite-sample correction, $\breve {\xi }_{\tau ^{\prime }_n}^{\star }$ stands out as the best performer in terms of smoothness and stability for moderate-sized datasets, but also in terms of relative bias and mean-squared error, with its Gaussian asymptotic confidence intervals attaining coverage closer to the nominal level (because it fits the exceedances over a threshold using a parametric model, whereas the nonparametric Weissman estimators are more data-driven and require large samples to provide accurate estimates). We will specialize the discussion of our generic Theorem 1 to both shape parameter estimators $\breve {\gamma }_n\in \{\widehat {\gamma }_n^{\mathrm {ML}}, \widehat {\gamma }_n^{\mathrm {Mom}}\}$ in Section 2.4 when the scale parameter estimator takes the form $\breve {\sigma }_n:=\breve {\sigma }^{(3)}_n = \widehat {a}((1-\tau _n)^{-1}) \, (\breve {\gamma }_n^{-1}-1)^{-\breve {\gamma }_n}$ , with $\widehat {a}(\cdot )\in \{\widehat {a}^{\mathrm {ML}}(\cdot ), \widehat {a}^{\mathrm {Mom}}(\cdot )\}$ . This choice of $\breve {\sigma }_n$ is motivated by its good performance in our numerical illustrations; one could, analogously, state a result for $\breve {\sigma }_n:=\breve {\sigma }^{(2)}_n$ given the $\sqrt {n(1-\tau _n)}$ -convergence of $\widehat {\overline {F}}_n(\widehat {\xi }_{\tau _n})$ (which, in the i.i.d. setting, follows from a Gaussian approximation to the tail empirical process, see, e.g., Theorem 5.1.2 on page 159 in de Haan and Ferreira, Reference de Haan and Ferreira2006), and for $\breve {\sigma }_n:=\breve {\sigma }^{(1)}_n$ given a specific, explicit form of the estimator $\widehat {a}(\cdot )$ . Next, we show that its associated assumption in Theorem 1 holds under convenient conditions on both $\breve {\gamma }_n$ and $\widehat {a}((1-\tau _n)^{-1})$ .

Theorem 2. Assume that

1. 1. $\mathbb {E}|X_{-}|<\infty $ and $\mathcal {E}_2(\gamma , \rho , a, A)$ holds with $0 <\gamma <1$ and $\rho < 0$ ;

2. 2. $\tau _n\uparrow 1$ such that $n(1-\tau _n) \to \infty $ , $\sqrt {n (1-\tau _n)} A((1-\tau _n)^{-1})=\mathrm {O}(1)$ , $\sqrt {n (1-\tau _n)}/ q_{\tau _n}=\mathrm {O}(1)$ , and $\sqrt {n(1-\tau _n)} (a((1-\tau _n)^{-1})/q_{\tau _n} - \gamma ) = \mathrm {O}(1)$ , $n \to \infty $ .

Take $\breve {\sigma }_n = \widehat {a}((1-\tau _n)^{-1}) \times (\breve {\gamma }_n^{-1}-1)^{-\breve {\gamma }_n}$ for suitable estimators $\widehat {a}((1-\tau _n)^{-1})$ and $\breve {\gamma }_n$ such that

$$\begin{align*}\sqrt{n (1-\tau_n)} \left( \frac{\widehat{a}((1-\tau_n)^{-1})}{a((1-\tau_n)^{-1})} - 1 \right) = \mathrm{O}_{\mathbb{P}}(1) \ \text{ and } \ \sqrt{n(1-\tau_n)} (\breve{\gamma}_n-\gamma)\stackrel{\mathrm{d}}{\longrightarrow} \Gamma, \end{align*}$$

where $\Gamma $ is a nondegenerate limit. Then

$$\begin{align*}\sqrt{n (1-\tau_n)} \left( \frac{\breve{\sigma}_n}{a(1/\overline{F}(\xi_{\tau_n}))} - 1 \right) = \mathrm{O}_{\mathbb{P}}(1). \end{align*}$$

In the financial context of real-valued profit–loss heavy-tailed distributions, we provide Monte Carlo evidence in Section 3 that, at least, our GPML estimator $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {ML}},\breve {\sigma }_n^{\mathrm {ML},2})$ outperforms the best-known Weissman competitors among those described in Appendix A, both in terms of relative bias and mean-squared error, with its associated asymptotic confidence interval having slightly better coverage at the expense of being wider.

2.3 Extremal Quantile-Based Estimation

Using now (2.3) with $t=(1-\tau _n)^{-1}$ and $x=(1-\tau _n)/(1-\tau ^{\prime }_n)$ leads to the well-known Generalized Pareto quantile approximation

$$\begin{align*}q_{\tau^{\prime}_n} \approx q_{\tau_n} + a((1-\tau_n)^{-1}) \frac{((1-\tau_n)/(1-\tau^{\prime}_n))^{\gamma}-1}{\gamma} \end{align*}$$

for all n large enough, which results in the extrapolated extreme quantile estimator

$$ \begin{align*} \check{q}_{\tau^{\prime}_n}^\star = \widehat{q}_{\tau_n}+\frac{\widehat{\sigma}_n}{\widehat{\gamma}_n}\left(\left(\frac{1-\tau^{\prime}_n}{1-\tau_n}\right)^{-\widehat{\gamma}_n}-1\right) \end{align*} $$

for suitable estimators $\widehat {\sigma }_n$ and $\widehat {\gamma }_n$ of the scale and shape parameters $a((1-\tau _n)^{-1})$ and $\gamma $ , respectively. In our heavy-tailed model ( $\gamma>0$ ), the asymptotic connection $\xi _{\tau ^{\prime }_n}\sim (\gamma ^{-1}-1)^{-\gamma }{q_{\tau ^{\prime }_n}}$ from (2.5) motivates then the quantile-based estimator

(2.9) $$ \begin{align} \check{\xi}^\star_{\tau^{\prime}_n}=\check{\xi}^\star_{\tau^{\prime}_n}(\widehat{\gamma}_n,\widehat{\sigma}_n)&:=(\widehat{\gamma}_n^{-1}-1)^{-\widehat{\gamma}_n}\, \check{q}_{\tau^{\prime}_n}^\star \\ &=(\widehat{\gamma}_n^{-1}-1)^{-\widehat{\gamma}_n}\left(\widehat{q}_{\tau_n}+\frac{\widehat{\sigma}_n}{\widehat{\gamma}_n}\left[\left(\frac{1-\tau^{\prime}_n}{1-\tau_n}\right)^{-\widehat{\gamma}_n}-1\right]\right) \nonumber \end{align} $$

for $\xi _{\tau ^{\prime }_n}$ , which extrapolates the intermediate expectile estimator $\check {\xi }_{\tau _n}:=(\widehat {\gamma }_n^{-1}-1)^{-\widehat {\gamma }_n}\widehat {q}_{\tau _n}$ to the far tail at $\tau ^{\prime }_n$ . Next, we establish the asymptotic distribution of this new GPD-type estimator for generic estimators $\widehat {\gamma }_n$ and $\widehat {\sigma }_n$ of the shape and scale parameters.

Theorem 3. Assume that

1. 1. $\mathbb {E}|X_{-}|<\infty $ and $\mathcal {E}_2(\gamma , \rho , a, A)$ holds with $0 <\gamma <1$ and $\rho < 0$ ;

2. 2. $\tau _n=1-k_n/n \to 1$ such that $n(1-\tau _n)=k_n \to \infty $ , $\sqrt {n (1-\tau _n)} A((1-\tau _n)^{-1})\to \lambda _1\in \mathbb {R}$ , $\sqrt {n (1-\tau _n)}/q_{\tau _n}\to \lambda _2\in \mathbb {R}$ , and $\sqrt {n(1-\tau _n)} (a((1-\tau _n)^{-1})/q_{\tau _n} - \gamma ) \to \mu \in \mathbb {R}$ , $n \to \infty $ ;

3. 3. $\tau ^{\prime }_n \to 1$ such that $(1-\tau _n)/(1-\tau ^{\prime }_n)\to \infty $ and $\sqrt {n(1-\tau _n)}/\log ((1-\tau _n)/(1-\tau ^{\prime }_n)) \to \infty $ , $n \to \infty $ ;

4. 4. for suitable estimators $\widehat {\gamma }_n$ of $\gamma $ and $\widehat {\sigma }_n$ of $\sigma (q_{\tau _n})= a((1-\tau _n)^{-1})$ ,

   (2.10) $$ \begin{align} \sqrt{n(1-\tau_n)}\left( \widehat{\gamma}_n-\gamma, \frac{\widehat{\sigma}_n}{\sigma(q_{\tau_n})}-1, \frac{\widehat{q}_{\tau_n}-q_{\tau_n}}{\sigma(q_{\tau_n})} \right) \stackrel{\mathrm{d}}{\longrightarrow} (\Gamma, \Lambda, B), \end{align} $$

   where $(\Gamma , \Lambda , B)$ is a nontrivial trivariate weak limit.

Then

$$\begin{align*}\sqrt{n(1-\tau_n)} \frac{\check{\xi}_{\tau^{\prime}_n}^{\star} - \xi_{\tau^{\prime}_n}}{a((1-\tau_n)^{-1}) \int_1^{(1-\tau_n)/(1-\tau^{\prime}_n)} s^{\gamma-1} \log(s) \mathrm{d}s} \stackrel{\mathrm{d}}{\longrightarrow} \frac{\Gamma}{(\gamma^{-1}-1)^{\gamma}}. \end{align*}$$

As can be seen from the proof of Theorem 3, an equivalent statement of this convergence is

(2.11) $$ \begin{align} \frac{\sqrt{n(1-\tau_n)}}{\log((1-\tau_n)/(1-\tau^{\prime}_n))} \left(\frac {\check{\xi}_{\tau^{\prime}_n}^\star}{ \xi_{\tau^{\prime}_n}}-1\right) \stackrel{\mathrm{d}}{\longrightarrow} \Gamma, \end{align} $$

which yields exactly the same rate of convergence as the competing Weissman-type estimators seen in Appendix A. Similarly to these estimators, it is the asymptotic distribution $\Gamma $ of the chosen tail index estimator $\widehat {\gamma }_n$ which dominates the limit distribution of our extreme expectile estimator $\check {\xi }^\star _{\tau ^{\prime }_n}(\widehat {\gamma }_n,\widehat {\sigma }_n)$ . In Section 2.4 below, we shall specialize the discussion of Theorem 3 to the GPML and Moment-type estimators $\check {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {ML}},\widehat {\sigma }_n^{\mathrm {ML}})$ and $\check {\xi }_{\tau ^{\prime }_n}^\star (\widehat {\gamma }_n^{\mathrm {Mom}},\widehat {\sigma }_n^{\mathrm {Mom}})$ obtained by taking the estimators $\widehat {\gamma }_n$ and $\widehat {\sigma }_n$ in (2.9) for the shape and scale parameters $\gamma $ and $\sigma (q_{\tau _n})\equiv a(n/k_n)$ to be either $\widehat {\gamma }_n^{\mathrm {ML}}$ and $\widehat {\sigma }_n^{\mathrm {ML}}$ or $\widehat {\gamma }_n^{\mathrm {Mom}}$ and $\widehat {\sigma }_n^{\mathrm {Mom}}$ , respectively, where

$$\begin{align*}\widehat{\sigma}_n^{\mathrm{ML}}:=\widehat{a}^{\mathrm{ML}}(n/k_n) \ \text{ and } \ \widehat{\sigma}_n^{\mathrm{Mom}}:=\widehat{a}^{\mathrm{Mom}}(n/k_n). \end{align*}$$

The two extrapolated estimators $\check {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {ML}},\widehat {\sigma }_n^{\mathrm {ML}})$ and $\check {\xi }_{\tau ^{\prime }_n}^\star (\widehat {\gamma }_n^{\mathrm {Mom}},\widehat {\sigma }_n^{\mathrm {Mom}})$ can be calculated by using the function gpdExpect from the R package Expectrem, with method="indirect_ML" and method="indirect_MOM", respectively. Our simulation study in Section 3 indicates that their finite-sample performance is very similar to that of the third direct GPML and Moment estimators $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {ML}},\breve {\sigma }_n^{\mathrm {ML},3})$ and $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {Mom}},\breve {\sigma }_n^{\mathrm {Mom},3})$ , respectively (as the only difference lies, by construction, between their related intermediate direct and indirect expectile estimators). Moreover, in the case of nonnegative loss distributions, the Moment-type estimator $\check {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {Mom}},\widehat {\sigma }_n^{\mathrm {Mom}})$ shows competitive relative mean-squared errors with remarkably lower relative bias estimates than those of the bias-corrected Weissman estimators. In the case of real-valued profit–loss distributions, this indirect Moment estimator still posts respectable results for low values of $\gamma $ , but the direct GPML estimator $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {ML}},\breve {\sigma }_n^{\mathrm {ML},2})$ stands out in terms of relative mean-squared errors and coverage probabilities of its associated confidence interval, exhibiting consistent results over all values of $\gamma $ .

2.4 GPML and Moment-Type Estimators

In this section, we focus the discussion of the asymptotic theory on the GPML and Moment estimators $(\widehat {a}^{\mathrm {ML}}(n/k_n),\widehat {\gamma }_n^{\mathrm {ML}})$ and $\big (\widehat {a}^{\mathrm {Mom}}(n/k_n),\widehat {\gamma }_n^{\mathrm {Mom}}\big )$ of the scale and shape parameters $(a(n/k_n),\gamma )$ , for the discretized intermediate level $\tau _n =1-k_n/n$ with $k_n \to \infty $ and $k_n/n \to 0$ as $n\to \infty $ . Although our generic Theorems 1–3 are valid in a general setup allowing for serial dependency in the data, we impose an i.i.d. assumption on the underlying sample.

We first consider the extrapolated asymmetric least squares estimators $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {ML}},\breve {\sigma }_n^{\mathrm {ML}})$ and $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {Mom}},\breve {\sigma }_n^{\mathrm {Mom}})$ defined in (2.7) by taking therein the estimators $\breve {\gamma }_n$ and $\breve {\sigma }_n$ of the shape and scale parameters $\gamma $ and $\sigma (\xi _{\tau _n})\equiv a(1/\overline {F}(\xi _{\tau _n}))$ to be either $\widehat {\gamma }_n^{\mathrm {ML}}$ and $\breve {\sigma }_n^{\mathrm {ML}}$ or $\widehat {\gamma }_n^{\mathrm {Mom}}$ and $\breve {\sigma }_n^{\mathrm {Mom}}$ , respectively, with

$$ \begin{align*} & \breve{\sigma}_n^{\mathrm{ML}} := \widehat{a}^{\mathrm{ML}}(n/k_n) \times ((1/\widehat{\gamma}_n^{\mathrm{ML}})-1)^{-\widehat{\gamma}_n^{\mathrm{ML}}} = \widehat{\sigma}_n^{\mathrm{ML}} \times ((1/\widehat{\gamma}_n^{\mathrm{ML}})-1)^{-\widehat{\gamma}_n^{\mathrm{ML}}} \\ \text{and } \ & \breve{\sigma}_n^{\mathrm{Mom}} := \widehat{a}^{\mathrm{Mom}}(n/k_n) \times ((1/\widehat{\gamma}_n^{\mathrm{Mom}})-1)^{-\widehat{\gamma}_n^{\mathrm{Mom}}} =\widehat{\sigma}_n^{\mathrm{Mom}} \times ((1/\widehat{\gamma}_n^{\mathrm{Mom}})-1)^{-\widehat{\gamma}_n^{\mathrm{Mom}}}. \end{align*} $$

Following Theorem 3.4.2 on p. 92 in de Haan and Ferreira (Reference de Haan and Ferreira2006) and the asymptotic independence between functions of the stochastic process $(X_{n-\lfloor k_ns \rfloor ,n}-X_{n-k_n,n})_{0<s\leq 1}$ and the random variable $X_{n-k_n,n}$ (see, e.g., Corollary 2.4.6 on page 52 in de Haan and Ferreira, Reference de Haan and Ferreira2006), the GPML estimators $\widehat {\gamma }_n^{\mathrm {ML}}$ and $\widehat {\sigma }_n^{\mathrm {ML}}$ satisfy, under the second-order condition $\mathcal {E}_2(\gamma , \rho , a, A)$ with $\gamma>0$ and the bias condition $\sqrt {k_n}A(n/k_n)\to \lambda _1\in \mathbb {R}$ , the joint convergence

(2.12) $$ \begin{align} \sqrt{k_n}\left( \widehat{\gamma}_n^{\mathrm{ML}}-\gamma, \frac{\widehat{\sigma}_n^{\mathrm{ML}}}{a(n/k_n)}-1, \frac{X_{n-k_n,n}-U(n/k_n)}{a(n/k_n)} \right) \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{N}\left(\lambda_1\, b^{\mathrm{ML}} ,\Sigma^{\mathrm{ML}}\right), \end{align} $$

where $b^{\mathrm {ML}}=\left ( \frac {\gamma +1}{(1-\rho )(1+\gamma -\rho )}, \frac {-\rho }{(1-\rho )(1+\gamma -\rho )}, 0 \right )$ and the matrix $\Sigma ^{\mathrm {ML}}$ is given by

$$\begin{align*}\Sigma^{\mathrm{ML}}=\begin{pmatrix} (1+\gamma)^2 && -(1+\gamma) && 0 \\ -(1+\gamma) && 1+(1+\gamma)^2 && 0 \\ 0 && 0 && 1 \end{pmatrix}. \end{align*}$$

Under the extra condition $\sqrt {n(1-\tau _n)} (a((1-\tau _n)^{-1})/q_{\tau _n} - \gamma ) \to \mu \in \mathbb {R}$ , it can be seen from Corollary 4.2.2 on p. 133 in de Haan and Ferreira (Reference de Haan and Ferreira2006) that the Moment estimators $\widehat {\gamma }_n^{\mathrm {Mom}}$ and $\widehat {\sigma }_n^{\mathrm {Mom}}$ satisfy the joint convergence

(2.13) $$ \begin{align} & \sqrt{k_n}\left( \widehat{\gamma}_n^{\mathrm{Mom}}-\gamma, \frac{\widehat{\sigma}_n^{\mathrm{Mom}}}{a(n/k_n)}-1, \frac{X_{n-k_n,n}-U(n/k_n)}{a(n/k_n)} \right) \nonumber \\ & \hspace{4cm} \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{N}\left(\lambda^{\mathrm{Mom}}\left(b_{\gamma,\rho}^{\widehat{\gamma}_n^{\mathrm{Mom}}},b_{\gamma,\rho}^{\widehat{\sigma}_n^{\mathrm{Mom}}},0\right),\Sigma^{\mathrm{Mom}}\right), \end{align} $$

where

$$ \begin{align*} \lambda^{\mathrm{Mom}} &:= \left\{ \begin{array}{ll} -\mu & \text{if } \ \gamma=-\rho \ \text{ or } \ (0<\gamma<-\rho \text{ and } l\neq 0), \\[10pt] \dfrac{\lambda_1 \,\rho}{\gamma+\rho} & \text{if } \ 0<-\rho<\gamma \ \text{ or } \ (0<\gamma<-\rho \text{ and } l=0), \end{array} \right. \\ b_{\gamma,\rho}^{\widehat{\gamma}_n^{\mathrm{Mom}}} &:= \left\{ \begin{array}{ll} - \dfrac{\gamma}{(1+\gamma)^2} & \text{if } \ 0<\gamma<-\rho \text{ and } l\neq 0, \\[10pt] \dfrac{\gamma-\gamma\rho+\rho}{\rho(1-\rho)^2} & \text{if } \ 0<-\rho\leq \gamma \ \text{ or } \ (0<\gamma<-\rho \text{ and } l=0), \end{array} \right. \\ b_{\gamma,\rho}^{\widehat{\sigma}_n^{\mathrm{Mom}}} &:= \left\{ \begin{array}{ll} \dfrac{\gamma}{(1+\gamma)^2} & \text{if } \ 0<\gamma<-\rho \text{ and } l\neq 0, \\[10pt] \dfrac{-\rho}{(1-\rho)^2} & \text{if } \ 0<-\rho\leq \gamma \ \text{ or } \ (0<\gamma<-\rho \text{ and } l=0), \end{array} \right. \\ \text{and } \ \Sigma^{\mathrm{Mom}} &:= \begin{pmatrix} \gamma^2+1 & \gamma-1 & 0\\ \gamma-1 &\gamma^2+2 & \gamma\\ 0 & \gamma & 1 \end{pmatrix}. \end{align*} $$

Note that the asymptotic variance of $\widehat {\gamma }_n^{\mathrm {Mom}}$ is lower than that of $\widehat {\gamma }_n^{\mathrm {ML}}$ , but the analysis of the respective biases of these two estimators, and therefore a definitive comparison of their finite-sample performance, is more complex. With the convergence results (2.12) and (2.13) at hand, we show that the two direct extreme expectile estimators $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {ML}},\breve {\sigma }_n^{\mathrm {ML}})$ and $\breve {\xi }_{\tau ^{\prime }_n}^{\star }(\widehat {\gamma }_n^{\mathrm {Mom}},\breve {\sigma }_n^{\mathrm {Mom}})$ are asymptotically normal.

Corollary 1. Under the first three conditions of Theorem 1, and provided the data are i.i.d., we have

$$ \begin{align*} & \sqrt{k_n} \frac{\breve{\xi}_{\tau^{\prime}_n}^{\star}(\widehat{\gamma}_n^{\mathrm{ML}},\breve{\sigma}_n^{\mathrm{ML}}) - \xi_{\tau^{\prime}_n}}{\breve{\sigma}_n^{\mathrm{ML}} \int_1^{k_n/(n(1-\tau^{\prime}_n))} s^{\widehat{\gamma}_n^{\mathrm{ML}}-1} \log(s) \mathrm{d}s} \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{N}\left( \frac{\lambda_1(1+\gamma)}{(1-\rho)(1+\gamma-\rho)}, (1+\gamma)^2\right) \\ & \text{and } \ \sqrt{k_n} \frac{\breve{\xi}_{\tau^{\prime}_n}^{\star}(\widehat{\gamma}_n^{\mathrm{Mom}},\breve{\sigma}_n^{\mathrm{Mom}}) - \xi_{\tau^{\prime}_n}}{\breve{\sigma}_n^{\mathrm{Mom}} \int_1^{k_n/(n(1-\tau^{\prime}_n))} s^{\widehat{\gamma}_n^{\mathrm{Mom}}-1} \log(s) \mathrm{d}s} \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{N}\left( \lambda^{\mathrm{Mom}} b_{\gamma,\rho}^{\widehat{\gamma}_n^{\mathrm{Mom}}}, \gamma^2+1\right). \end{align*} $$

Turning to the indirect quantile-based estimators $\check {\xi }^\star _{\tau ^{\prime }_n}(\widehat {\gamma }_n^{\mathrm {ML}},\widehat {\sigma }_n^{\mathrm {ML}})$ and $\check {\xi }^\star _{\tau ^{\prime }_n}(\widehat {\gamma }_n^{\mathrm {Mom}},\widehat {\sigma }_n^{\mathrm {Mom}})$ defined in (2.9), we obtain the following asymptotic normality results.

Corollary 2. Under the first three conditions of Theorem 3, and provided the data are i.i.d., we have

$$ \begin{align*} & \sqrt{k_n} \frac{\check{\xi}_{\tau^{\prime}_n}^{\star}(\widehat{\gamma}_n^{\mathrm{ML}},\widehat{\sigma}_n^{\mathrm{ML}}) - \xi_{\tau^{\prime}_n}}{\widehat{\sigma}_n^{\mathrm{ML}} \int_1^{k_n/(n(1-\tau^{\prime}_n))} s^{\widehat{\gamma}_n^{\mathrm{ML}}-1} \log(s) \mathrm{d}s} \\ & \hspace{1cm} \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{N}\left( \frac{\lambda_1(\gamma+1)(\gamma^{-1}-1)^{-\gamma}}{(1-\rho)(1+\gamma-\rho)}, (1+\gamma)^2 (\gamma^{-1}-1)^{-2\gamma} \right) \\ & \text{and } \ \sqrt{k_n} \frac{\check{\xi}_{\tau^{\prime}_n}^{\star}(\widehat{\gamma}_n^{\mathrm{Mom}},\widehat{\sigma}_n^{\mathrm{Mom}}) - \xi_{\tau^{\prime}_n}}{\widehat{\sigma}_n^{\mathrm{Mom}} \int_1^{k_n/(n(1-\tau^{\prime}_n))} s^{\widehat{\gamma}_n^{\mathrm{Mom}}-1} \log(s) \mathrm{d}s} \\ & \hspace{1cm} \stackrel{\mathrm{d}}{\longrightarrow} \mathcal{N}\left( \lambda^{\mathrm{Mom}} b_{\gamma,\rho}^{\widehat{\gamma}_n^{\mathrm{Mom}}} (\gamma^{-1}-1)^{-\gamma}, (\gamma^2+1)(\gamma^{-1}-1)^{-2\gamma}\right). \end{align*} $$

In order to keep the article to manageable proportions, we do not discuss the application of our theorems and all the associated ramifications in the presence of dependence in the data. Similarly to extreme quantile estimation under mixing conditions (Drees, Reference Drees2003) or ARMA-GARCH models (He et al., Reference He, Peng, Zhang and Zhao2022), our methods may work under serial dependence with an increased asymptotic variance. Theoretical results along these lines are left for future research. In our case study of financial time series, we consider a practical solution to eliminate serial dependence by filtering the time series with an ARMA-GARCH model. The residuals from the fitted model are then treated as i.i.d. after performing a series of Ljung–Box independence tests on residuals and their squares.