Tail and Quantile Estimation for Real-Valued $\boldsymbol{\beta}$-Mixing Spatial Data

Abstract

This paper deals with extreme-value index estimation of a heavy-tailed distribution of a spatial dependent process. We are particularly interested in spatial rare events of a $\beta$-mixing process. Given a stationary real-valued multidimensional spatial process $\left\{X_{\mathbf{i}},\mathbf{i}\in{\mathbb{Z}}^{N}\right\}$, we investigate its heavy-tail index estimation. Asymptotic properties of the corresponding estimator are established under mild mixing conditions. The particularity of the tail proposed estimator is based on the spatial nature of the sample and its unbiased and reduced variance properties compared to well known tail index estimators. Extreme quantile estimation is also deduced. A numerical study on synthetic and real datasets is conducted to assess the finite-sample behaviour of the proposed estimators.

Appendices

APPENDIX

Proofs of the Main Results

To establish the proofs of the main results, we adopt [32]’s notation of the spatial locations (for seek of simplicity). That is the process $\left\{X_{\mathbf{i}},\mathbf{i}\in\mathbb{Z}^{N}\right\}$ is written as $\left\{X_{i},1\leq i\leq n=n_{1}\times n_{2}\times\cdots\times n_{N}\right\}$ using for instance a triangular array notation and a lexicographic ordering. For this notation the mixing conditions $C_{M}$ and $C_{R}$ (regularity) are written as.

Condition $C^{\prime}_{M}$ (mixing condition). Let’s $(l_{n})_{n\in\mathbb{N}^{*}}$ be a sequence of integers such that $1\leq l_{n}\leq{n}$; set $\mathcal{B}_{m}^{j}=\sigma(X_{i},m\leq i\leq j)$ be $\sigma$-fields generated by the random variables $(X_{i})_{i}$ with $m\leq i\leq j$. The $\beta$-mixing condition is given by:

$$\beta(l_{n}):=\underset{m\in\mathbb{N}^{*}}{sup}\mathbb{E}\left[\underset{A\in\mathcal{B}_{l_{n}+m+1}^{+\infty}}{sup}|\mathbb{P}(A|\mathcal{B}_{{1}}^{m})-\mathbb{P}(A)|\right]\underset{l_{n}\rightarrow\infty}{\longrightarrow}0.$$

(36)

See [18] for a discussion on the $\beta$-mixing and examples.

Condition $C^{\prime}_{R}$ (regularity). There is $\epsilon>0$, a function $r:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$, and $({l}_{n})$ defined above is such that $l_{n}=\text{o}(n/k_{n})$; and when $n\rightarrow\infty$

• (a${}^{\prime}$) $\frac{\beta(l_{n})}{l_{n}}n+l_{n}\frac{\text{log}^{2}k_{n}}{\sqrt{k_{n}}}\rightarrow 0$;

• (b${}^{\prime}$) $\frac{n}{l_{n}k_{n}}\text{Cov}\left({{\underset{i=1}{\overset{l_{n}}{\sum}}}}{1}_{\{X_{i}>F^{\leftarrow}(1-k_{n}x/n)\}},{{\underset{i=1}{\overset{l_{n}}{\sum}}}}{1}_{\{X_{i}>F^{\leftarrow}(1-k_{n}y/n)\}}\right)\rightarrow r(x,y),$ $\forall\ 0\leq x,y\leq 1+\epsilon$;

• (c${}^{\prime}$) there exists a constant $C$ such that :

  $$\frac{n}{l_{n}k_{n}}E\left[\left({{\underset{i=1}{\overset{l_{n}}{\sum}}}}{1}_{\{F^{\leftarrow}(1-k_{n}y/n)<X_{i}\leq F^{\leftarrow}(1-k_{n}x/n)\}}\right)^{4}\right]\leq C(y-x)\quad\forall\ 0\leq x<y\leq 1+\epsilon.$$

A. Proof of Theorem II.1

To establish the proof of the theorem, we need the following proposition.

Proposition VI.1. Let $\left\{X_{\mathbf{i}},\mathbf{i}\in{\mathbb{Z}}^{N}\right\}$ be a $\beta$ -mixing stationary spatial process with a distribution function $F$ ; verifying $C_{A}$ and $C_{R}$ and $K$ a function verifying $C_{K}$ . Let $(k_{n})$ be an intermediate sequence such that $\sqrt{k_{n}}\mathcal{A}(b(n/k_{n}))\to\lambda$ , as $n\to\infty$ . For all $\epsilon>0$ , by Skorohod construction, there exist a function $\tilde{\mathcal{A}}\sim\mathcal{A}$ and a Gaussian centred process $(W(t))_{t\in[0,1]}$ with covariance function $r$ such that, as $n\rightarrow\infty$ ,

$$\underset{t\in(0,1]}{\sup}t^{1/2+\epsilon}\left|\sqrt{k_{n}}\left(\log\left(\frac{Q_{n}(t)}{\text{U}(b(n/k_{n}))}\right)+\frac{\gamma}{\int_{0}^{1}K(s)ds}\log t\right)-\gamma t^{-1}W(t)\right.$$

$${}-\left.\sqrt{k_{n}}\tilde{\mathcal{A}}(b(n/k_{n}))\dfrac{t^{-\rho}-1}{\rho}\right|\overset{\text{a.s}}{\longrightarrow}0.$$

(37)

Proof of Proposition VI.1. Suppose the relationship (12) (from the condition $C_{A}$) hold. By applying Theorem B$.2.18$ in [15], we get:

$\forall\ \epsilon$, $\delta>0$ $\exists\ u_{0}=u_{0}(\epsilon,\delta)$ such that $\forall\ ux\geq u_{0}$;

$$\left|\frac{\log(\text{U}(ux)/\text{U}(u))-{\gamma}\log(x)}{\tilde{\mathcal{A}}(u)}-\frac{x^{\rho}-1}{\rho}\right|\leq\epsilon x^{\rho}\max(x^{\delta},x^{-\delta}).$$

(38)

Set $X_{i}=U(Y_{i})$ where $Y_{i}$ follows a standard Pareto distribution. Then $(Y_{\mathbf{i}})_{\mathbf{i}\in\mathbb{Z}^{N}}$ is stationary and $\beta$-mixing satisfying the regular variation ($C_{R}$). Then, since $Q_{n}(t)=U(Y_{n-\lfloor k_{n}t\rfloor,n})$ and according to Theorem $2.1$ in Drees $(2003)$ and under Skorohod construction, there exists a centred Gaussian process $(W(t))_{t\in[0,1]}$ with covariance function $r$ such that for all $\epsilon>0$

$$\underset{t\in(0,1]}{\sup}t^{1/2+\epsilon}\left|\sqrt{k_{n}}\left(t\frac{Y_{n-\lfloor k_{n}t\rfloor,n}}{b(n/k_{n})}-1\right)-t^{-1}W(t)\right|\rightarrow 0,\ \text{a.s.}$$

(39)

as $n\rightarrow\infty$. The inequality $(38)$ gives, for all $n>n_{0}(\epsilon,\delta)$:

$$\Bigg{|}\log(Q_{n}(t))-\log\left(\text{U}(b(n/k_{n}))\right)-{\gamma}\text{log}\left(\frac{1}{b(n/k_{n})}Y_{n-\lfloor k_{n}t\rfloor,n}\right)$$

$${}-{\tilde{\mathcal{A}}(b(n/k_{n}))}\frac{\left(\frac{1}{b(n/k_{n})}Y_{n-\lfloor k_{n}t\rfloor,n}\right)^{\rho}-1}{\rho}\Bigg{|}$$

$${}\leq\epsilon\left|{\tilde{\mathcal{A}}(b(n/k_{n}))}\right|\left(\frac{1}{b(n/k_{n})}Y_{n-\lfloor k_{n}t\rfloor,n}\right)^{\rho+\delta}.$$

So,

$$t^{1/2+\epsilon}\left|\sqrt{k_{n}}\left(\text{log}\left(\frac{Q_{n}(t)}{\text{U}(b(n/k_{n}))}\right)+\frac{\gamma}{\int_{0}^{1}K(s)ds}\text{log}(t)\right)-\gamma t^{-1}W(t)\right.$$

$${}-\sqrt{k_{n}}{\tilde{\mathcal{A}}(b(n/k_{n}))}\frac{t^{-\rho}-1}{\rho}+\sqrt{k_{n}}{\tilde{\mathcal{A}}(b(n/k_{n}))}\frac{1}{\rho}\left(t^{-\rho}-\left(\frac{1}{b(n/k_{n})}Y_{n-\lfloor k_{n}t\rfloor,n}\right)^{\rho}\right)$$

$${}-\left.\gamma\left\{\sqrt{k_{n}}\left(\log\left(\frac{1}{b(n/k_{n})}Y_{n-\lfloor k_{n}t\rfloor,n}\right)+\frac{1}{\int_{0}^{1}K(s)ds}\log(t)\right)-t^{-1}W(t)\right\}\right|$$

$${}\leq\epsilon\sqrt{k_{n}}\left|{\tilde{\mathcal{A}}(b(n/k_{n}))}\right|t^{1/2+\epsilon}\left(\frac{1}{b(n/k_{n})}\right).$$

Since $\frac{1}{b(n/k_{n})}Y_{n-\lfloor k_{n}t\rfloor,n}\geq 1$, by choosing $\delta\in(0,-\rho)$ the right term tends to $0$ when $\epsilon\rightarrow 0$. Thus under the convergence $(39)$ the proof of the Proposition VI.1 is obtained as in [16].

Proof of Theorem II.1.: From Proposition VI.1, we deduce that:

$$\sqrt{k_{n}}\left\{\hat{\gamma}_{{k}_{n}}(K)+\frac{\gamma}{\int_{0}^{1}K(s)ds}\int\limits_{0}^{1}\log(t)d(tK(t))\right\}=\gamma\int\limits_{0}^{1}(t^{-1}W(t)-W(1))d(tK(t))$$

$${}+\sqrt{k_{n}}\tilde{\mathcal{A}}(b(n/k_{n}))\int\limits_{0}^{1}\frac{t^{-\rho}-1}{\rho}d(tK(t))+\text{o}(1)\int\limits_{0}^{1}t^{-1/2-\epsilon}d(tK(t)).$$

Using an integration by part, we can write:

$$\int\limits_{0}^{1}\text{log}(t)d(tK(t))=-\int\limits_{0}^{1}K(s)ds\quad\text{and}\quad\int\limits_{0}^{1}\frac{t^{-\rho}-1}{\rho}d(tK(t))=\int\limits_{0}^{1}t^{-\rho}K(t)dt.$$

Hence

$$\sqrt{k_{n}}\left\{\hat{\gamma}_{{k}_{n}}(K)-\gamma-\tilde{\mathcal{A}}(b(n/k_{n}))\int\limits_{0}^{1}t^{-\rho}K(t)dt\right\}=\gamma\int\limits_{0}^{1}(t^{-1}W(t)-W(1))d(tK(t))$$

$${}+\text{o}(1)\int\limits_{0}^{1}t^{-1/2-\epsilon}d(tK(t)).$$

By taking $0<\epsilon<1/2-\tau$, we get the convergence of $\int_{0}^{1}t^{-1/2-\epsilon}d(tK(t))$. Thus, this ends the proof.

B. Proof of Corollary II.1

Indeed, the term $\mathcal{A}(b(n/k_{n}))\int_{0}^{1}t^{-\rho}K(t)dt$ outcome from (13) is the bias of the estimator; and as $\sqrt{k_{n}}\mathcal{A}(b(n/k_{n}))\longrightarrow\lambda$ then we get the asymptotic bias $\lambda\mathcal{AB}(K)=\lambda\int_{0}^{1}t^{-\rho}K(t)dt$.

The variance $\mathcal{AV}(K)$ is obtained from the Gaussian centered process $(W(t))_{t\in[0.1]}$ covariance function $r$; that is

$$\displaystyle\mathcal{AV}(K)=\gamma^{2}\text{E}\left[\left(\int\limits_{0}^{1}\left[t^{-1}W(t)-W(1)\right]d(tK(t))\right)^{2}\right].$$

The proofs of Corollaries II.2 and II.3 are straightforward and follow the same lines as the Corollary II.1.

C. Proof of Theorem II.2

Let

$$K_{S^{*}}(t)=\frac{\rho^{2}}{1-2\rho}-\frac{\rho^{2}}{1-\rho}t^{-\rho}:=\alpha K_{1,\rho}(t)+\beta K_{2,\rho}(t)\quad\text{and}$$

$$K_{\hat{S}^{*}}(t)=\frac{\hat{\rho}_{k_{n,\rho}}^{2}}{1-2\hat{\rho}_{k_{n,\rho}}}-\frac{\hat{\rho}_{k_{n,\rho}}^{2}}{1-\hat{\rho}_{k_{n,\rho}}}t^{-\hat{\rho}_{k_{n,\rho}}}:=\hat{\alpha}K_{1,\hat{\rho}_{k_{n,\rho}}}(t)+\hat{\beta}K_{2,\hat{\rho}_{k_{n,\rho}}}(t),$$

where $\hat{\alpha}$ and $\hat{\beta}$ are consistent estimators of $\alpha$ and $\beta$, respectively.

Let us first, give the following decomposition

$$\sqrt{k_{n}}\left(\hat{\gamma}_{k_{n}}(K_{\hat{S}^{*}})-\gamma\right)=\sqrt{k_{n}}\left(\hat{\gamma}_{k_{n}}(K_{{S}^{*}})-\gamma\right)+\sqrt{k_{n}}\left(\hat{\gamma}_{k_{n}}(K_{\hat{S}^{*}})-\hat{\gamma}_{k_{n}}(K_{{S}^{*}})\right).$$

(40)

According to Corollary II.2, the first term on the right converges to Gaussian distribution. So it remains to prove that the second term tends to $0$ in probability. The proof follows the same lines as those of Theorem $2$ in [10]. The difference in our approach is in managing the assumptions of the $C_{R}$ and $C_{K}$ conditions since we assumed that $K$ is not necessarily a kernel function. We have:

$$\sqrt{k_{n}}\left(\hat{\gamma}_{k_{n}}(K_{\hat{S}^{*}})-\hat{\gamma}_{k_{n}}(K_{{S}^{*}})\right)=\sqrt{k_{n}}\left\{\int\limits_{0}^{1}\text{log}\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{\hat{S}^{*}}(t))-\int\limits_{0}^{1}\text{log}\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{{S}^{*}}(t))\right\}$$

$${}=\sqrt{k_{n}}\left\{\hat{\alpha}\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}dt-\alpha\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}dt\right.$$

$${}+\left.\hat{\beta}\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{2,\hat{\rho}_{k_{n,\rho}}}(t))-\beta\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{2,\rho}(t))\right\}$$

$${}=(\hat{\alpha}-\alpha)\sqrt{k_{n}}\left\{\int\limits_{0}^{1}\text{log}\frac{Q_{n}(t)}{Q_{n}(1)}dt-\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{2,\rho}(t))\right\}$$

$${}+(\hat{\alpha}-\alpha+\hat{\beta}-\beta)\sqrt{k_{n}}\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{2,\rho}(t))$$

$${}+\hat{\beta}\sqrt{k_{n}}\left\{\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{2,\hat{\rho}_{k_{n,\rho}}}(t))-\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{2,\rho}(t))\right\}$$

$${}=:T_{1}+T_{2}+T_{3}.$$

(41)

Let us evaluate the three terms.

Using Corollary II.1 and the fact that $\hat{\alpha}$ and $\hat{\beta}$ are consistent estimators of $\alpha$ and $\beta$, respectively, we have $T_{1}=\text{o}_{\mathbb{P}}(1)$ and $T_{2}=\text{o}_{\mathbb{P}}(1)$.

It remains to deal with Term $T_{3}$ .

Noting that $\log\frac{Q_{n}(t)}{Q_{n}(1)}=\log\frac{Q_{n}(t)}{\text{U}(b(n/k_{n}))}-\log\frac{Q_{n}(1)}{\text{U}(b(n/k_{n}))}$, the Proposition VI.1 gives for all $\epsilon\in(0,1/2)$

$$\log\frac{Q_{n}(t)}{Q_{n}(1)}=\frac{\gamma}{\int_{0}^{1}K(s)ds}(-\log(t))+\frac{\gamma}{\sqrt{k_{n}}}[t^{-1}W(t)-W(1)]$$

$${}+\tilde{\mathcal{A}}(b(n/k_{n}))\frac{t^{-\rho}-1}{\rho}+\frac{\text{o}(1)}{\sqrt{k_{n}}}t^{-\epsilon-1/2}.$$

Thus we get:

$$\sqrt{k_{n}}\left\{\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{2,\hat{\rho}_{k_{n,\rho}}}(t))-\int\limits_{0}^{1}\log\frac{Q_{n}(t)}{Q_{n}(1)}d(tK_{2,\rho}(t))\right\}$$

$${}=\gamma\sqrt{k_{n}}\left\{\frac{1}{\int_{0}^{1}K_{2,\hat{\rho}_{k_{n,\rho}}}(t)dt}\int\limits_{0}^{1}(-\log(t))d(tK_{2,\hat{\rho}_{k_{n,\rho}}}(t))\right.$$

$${}-\left.\frac{1}{\int_{0}^{1}K_{2,{\rho}}(t)dt}\int\limits_{0}^{1}(-\log(t))d(tK_{2,\rho}(t))\right\}$$

$${}+\gamma\left\{\int\limits_{0}^{1}[t^{-1}W(t)-W(1)]d(tK_{2,\hat{\rho}_{k_{n,\rho}}}(t))-\int\limits_{0}^{1}[t^{-1}W(t)-W(1)]d(tK_{2,\rho}(t))\right\}$$

$${}+\sqrt{k_{n}}\tilde{\mathcal{A}}(b(n/k_{n}))\left\{\int\limits_{0}^{1}\frac{t^{-\rho}-1}{\rho}d(tK_{2,\hat{\rho}_{k_{n,\rho}}}(t))-\int\limits_{0}^{1}\frac{t^{-\rho}-1}{\rho}d(tK_{2,\rho}(t))\right\}$$

$${}+\text{o}(1)\left\{\int\limits_{0}^{1}t^{-\frac{1}{2}-\epsilon}d(tK_{2,\hat{\rho}_{k_{n,\rho}}}(t))-\int\limits_{0}^{1}t^{-\frac{1}{2}-\epsilon}d(tK_{2,\rho}(t))\right\}$$

$${}=:A+B+C+D.$$

The term $A$ converges by using an integration by part.

The term $B$ is

$$\displaystyle B=\gamma\left\{\int\limits_{0}^{1}[t^{-1}W(t)-W(1)]\left(K_{2,\hat{\rho}_{k_{n,\rho}}}(t)-K_{2,\rho}(t)\right)dt\right.$$

$${}+\left.\int\limits_{0}^{1}[t^{-1}W(t)-W(1)]t\left(K^{{}^{\prime}}_{2,\hat{\rho}_{k_{n,\rho}}}(t)-K^{{}^{\prime}}_{2,\rho}(t)\right)dt\right\}.$$

Let us consider $\epsilon\in(0,1)$ and $\tilde{\rho}$ a random value between $\rho$ and $\hat{\rho}_{k_{n\rho}}$. We have

$$\left|\int\limits_{0}^{1}[t^{-1}W(t)-W(1)]\left(K_{2,\hat{\rho}_{k_{n,\rho}}}(t)-K_{2,\rho}(t)\right)dt\right|$$

$${}\leq\int\limits_{0}^{1}\left|t^{-1}W(t)-W(1)\right|\left|K_{2,\hat{\rho}_{k_{n,\rho}}}(t)-K_{2,\rho}(t)\right|dt$$

$${}\leq(1-\hat{\rho}_{k_{n,\rho}})\int\limits_{0}^{1}\left|t^{-1}W(t)-W(1)\right|\left|t^{-\hat{\rho}_{k_{n,\rho}}}-t^{-\rho}\right|dt$$

$${}+\left|\hat{\rho}_{k_{n,\rho}}-\rho\right|\int\limits_{0}^{1}\left|t^{-1}W(t)-W(1)\right|t^{-\rho}dt$$

$${}\leq(1-\hat{\rho}_{k_{n,\rho}})\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|t^{-1}W(t)-W(1)\right|\underset{t\in(0,1]}{\sup}t^{\frac{1}{4}}\left|t^{-\hat{\rho}_{k_{n,\rho}}}-t^{-\rho}\right|\int\limits_{0}^{1}t^{-\dfrac{3}{4}-\epsilon}dt$$

$${}+\left|\hat{\rho}_{k_{n,\rho}}-\rho\right|\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|t^{-1}W(t)-W(1)\right|\underset{t\in(0,1]}{\sup}t^{\frac{1}{4}-\rho}\int\limits_{0}^{1}t^{-\dfrac{3}{4}-\epsilon}dt$$

$${}\overset{*}{\leq}\frac{4}{1-4\epsilon}\left|\hat{\rho}_{k_{n,\rho}}-\rho\right|(1-\hat{\rho}_{k_{n,\rho}})\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|t^{-1}W(t)-W(1)\right|\underset{t\in(0,1]}{\sup}\left(-\text{log}t\right)t^{\frac{3}{4}-\tilde{\rho}}$$

$${}+\frac{4}{1-4\epsilon}\left|\hat{\rho}_{k_{n,\rho}}-\rho\right|\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|t^{-1}W(t)-W(1)\right|=\text{o}_{\mathbb{P}}(1).$$

The inequality $\overset{*}{\leq}$ is justified by:

set $h(t)=t^{\frac{1}{4}-\hat{\rho}_{k_{n,\rho}}}-t^{\frac{1}{4}-\rho}=t^{\frac{1}{4}-\tilde{\rho}}(t^{\tilde{\rho}-\hat{\rho}_{k_{n,\rho}}}-t^{\tilde{\rho}-\rho})$. A Taylor expansion of the term $(t^{\tilde{\rho}-\hat{\rho}_{k_{n,\rho}}}-t^{\tilde{\rho}-\rho})$ gives

$$\displaystyle h(t)\simeq t^{\frac{1}{4}-\tilde{\rho}}(\rho-\hat{\rho}_{k_{n,\rho}})\text{log}(t),$$

and we get

$$\displaystyle\underset{t\in(0,1]}{\sup}\left|h(t)\right|\leq\left|\hat{\rho}_{k_{n,\rho}}-\rho\right|\underset{t\in(0,1]}{\text{sup}}t^{\frac{1}{4}-\tilde{\rho}}\left(-\text{log}(t)\right).$$

In the same way we have:

$$\left|\int\limits_{0}^{1}[t^{-1}W(t)-W(1)]t\left(K^{\prime}_{2,\hat{\rho}_{k_{n,\rho}}}(t)-K^{\prime}_{2,\rho}(t)\right)dt\right|$$

$${}\leq\int\limits_{0}^{1}\left|t^{-1}W(t)-W(1)\right|t\left|K^{\prime}_{2,\hat{\rho}_{k_{n,\rho}}}(t)-K^{\prime}_{2,\rho}(t)\right|dt$$

$${}\leq-\hat{\rho}_{k_{n,\rho}}(1-\hat{\rho}_{k_{n,\rho}})\int\limits_{0}^{1}\left|t^{-1}W(t)-W(1)\right|\left|t^{-\hat{\rho}_{k_{n,\rho}}}-t^{-\rho}\right|dt$$

$${}-\left(\hat{\rho}_{k_{n,\rho}}+\rho\right)\left|\hat{\rho}_{k_{n,\rho}}-\rho\right|^{2}\int\limits_{0}^{1}\left|t^{-1}W(t)-W(1)\right|t^{-\rho}dt$$

$${}\leq-\hat{\rho}_{k_{n,\rho}}(1-\hat{\rho}_{k_{n,\rho}})\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|t^{-1}W(t)-W(1)\right|\underset{t\in(0,1]}{\sup}t^{\frac{1}{4}}\left|t^{-\hat{\rho}_{k_{n,\rho}}}-t^{-\rho}\right|\int\limits_{0}^{1}t^{-\dfrac{3}{4}-\epsilon}dt$$

$${}-\left(\hat{\rho}_{k_{n,\rho}}+\rho\right)\left|\hat{\rho}_{k_{n,\rho}}-\rho\right|^{2}\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|t^{-1}W(t)-W(1)\right|\underset{t\in(0,1]}{\sup}t^{\frac{1}{4}-\rho}\int\limits_{0}^{1}t^{-\dfrac{3}{4}-\epsilon}dt$$

$${}\overset{*}{\leq}-\hat{\rho}_{k_{n,\rho}}\frac{4}{1-4\epsilon}\left|\hat{\rho}_{k_{n,\rho}}-\rho\right|(1-\hat{\rho}_{k_{n,\rho}})\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|t^{-1}W(t)-W(1)\right|\underset{t\in(0,1]}{\sup}\left(-\text{log}t\right)t^{\frac{3}{4}-\tilde{\rho}}$$

$${}-\frac{4}{1-4\epsilon}\left(\hat{\rho}_{k_{n,\rho}}+\rho\right)\left|\hat{\rho}_{k_{n,\rho}}-\rho\right|^{2}\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|t^{-1}W(t)-W(1)\right|=\text{o}_{\mathbb{P}}(1).$$

Then we get $B=\text{o}_{\mathbb{P}}(1)$.

By making an integration by part $C$ is

$$C=\sqrt{k_{n}}\tilde{\mathcal{A}}(b(n/k_{n}))\left\{\left[t\frac{t^{-\rho}-1}{\rho}\left(K_{2,\hat{\rho}_{k_{n,\rho}}}(t)-K_{2,\rho}(t)\right)\right]_{0}^{1}\right.$$

$${}+\left.\int\limits_{0}^{1}t^{-\rho}\left(K_{2,\hat{\rho}_{k_{n,\rho}}}(t)-K_{2,\rho}(t)\right)dt\right\},$$

and as $\sqrt{k_{n}}\tilde{\mathcal{A}}(b(n/k_{n}))\rightarrow\lambda$ and $\int_{0}^{1}t^{-\rho}\left(K_{2,\hat{\rho}_{k_{n,\rho}}}(t)-K_{2,\rho}(t)\right)dt$ converges so we have $C=\text{o}_{\mathbb{P}}(1)$.

Similarly, an integration by part allows us to conclude that $D=\text{o}_{\mathbb{P}}(1)$.

In short, we have $T_{3}=\text{o}_{\mathbb{P}}(\hat{\beta})$ and since $\hat{\beta}<1$ we have

$$\sqrt{k_{n}}\left(\hat{\gamma}_{k_{n}}(K_{\hat{S}^{*}})-\hat{\gamma}_{k_{n}}(K_{{S}^{*}})\right)=\text{o}_{\mathbb{P}}(1).$$

This ends the proof of Theorem II.2.

D. Proof of Theorem III.1

To do this, we only need to show the asymptotic normality of $\frac{\sqrt{k_{n}}}{\text{log}\frac{1}{pb(n/k_{n})}}\text{log}\frac{\hat{x}_{p,\xi}}{{x}_{p}}$.

We have the decomposition below:

$$\frac{\sqrt{k_{n}}}{\text{log}\frac{1}{pb(n/k_{n})}}\text{log}\frac{\hat{x}_{p,\xi}}{{x}_{p}}=\frac{\sqrt{k_{n}}}{\text{log}\frac{1}{pb(n/k_{n})}}\left\{\log X_{n-\lfloor tk_{n}\rfloor,n}+\hat{\gamma}_{k_{n}}(K_{\hat{S}^{*}})\log\frac{1}{pb(n/k_{n})}-\log x_{p}\right.$$

$${}-\left.\frac{(1-\xi)(1-2\xi)}{\xi^{2}}[\hat{\gamma}_{k_{n}}(K_{1})-\hat{\gamma}_{k_{n}}(K_{2,\xi})]\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{\xi}-1}{\xi}\right\}$$

$${}=\sqrt{k_{n}}\left(\hat{\gamma}_{k_{n}}(K_{\hat{S}^{*}})-\gamma\right)+\frac{\sqrt{k_{n}}}{\log\frac{1}{pb(n/k_{n})}}\log\frac{Q_{n}(t)}{\text{U}(b(n/k_{n}))}$$

$${}-\frac{\sqrt{k_{n}}}{\log\frac{1}{pb(n/k_{n})}}\left\{\log\frac{\text{U}(\frac{1}{p})}{\text{U}(b(n/k_{n}))}-\gamma\log\frac{1}{pb(n/k_{n})}\right\}$$

$${}-\frac{(1-\xi)(1-2\xi)}{\xi^{2}}\frac{\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{1})-\hat{\gamma}_{k_{n}}(K_{2,\xi})]}{\log\frac{1}{pb(n/k_{n})}}\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{\xi}-1}{\xi}$$

$${}=\sqrt{k_{n}}\left(\hat{\gamma}_{k_{n}}(K_{\hat{S}^{*}})-\gamma\right)+\frac{\sqrt{k_{n}}}{\log\frac{1}{pb(n/k_{n})}}\log\frac{Q_{n}(t)}{\text{U}(b(n/k_{n}))}$$

$${}-\frac{\sqrt{k_{n}}}{\text{log}\frac{1}{pb(n/k_{n})}}\tilde{\mathcal{A}}(b(n/k_{n}))\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{\rho}-1}{\rho}-\frac{\sqrt{k_{n}}}{\text{log}\frac{1}{pb(n/k_{n})}}\tilde{\mathcal{A}}(b(n/k_{n}))$$

$${}\times\left\{\frac{\text{log U}(\frac{1}{p})-\log\text{U}(b(n/k_{n}))-\gamma\log\frac{1}{pb(n/k_{n})}}{\tilde{\mathcal{A}}(b(n/k_{n}))}-\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{\rho}-1}{\rho}\right\}$$

$${}-\frac{(1-\xi)(1-2\xi)}{\xi^{2}}\frac{\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{1})-\hat{\gamma}_{k_{n}}(K_{2,\xi})]}{\log\frac{1}{pb(n/k_{n})}}\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{\xi}-1}{\xi}$$

$${}=:T_{4}+T_{5}-T_{6}-T_{7}-T_{8}.$$

Let us now look at the $5$ terms.

Theorem II.2 ensures the asymptotic normality of the term $T_{4}$

$$\displaystyle T_{4}\overset{d}{\longrightarrow}\mathcal{N}\left(0,\mathcal{AV}(K_{{S}^{*}})\right).$$

Using Proposition VI.1 (for $t=1$ and the fact that $\log(x)\sim x-1$ when $x\rightarrow 1$), we can show

$$T_{5}\overset{\mathbb{P}}{\longrightarrow}0.$$

Indeed,

$$\displaystyle\underset{t\in(0,1]}{\sup}t^{1/2+\epsilon}\left|\sqrt{k_{n}}\log\left(\frac{Q_{n}(t)}{\text{U}(b(n/k_{n}))}\right)-\gamma W(1)\right|$$

$${}\leq\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|\sqrt{k_{n}}\left(\log\left(\frac{Q_{n}(t)}{\text{U}\left(b\left(\frac{n}{k_{n}}\right)\right)}\right)+\frac{\gamma\log(t)}{\int\limits_{0}^{1}K(s)ds}\right)\right.$$

$${}-\left.\gamma t^{-1}W(t)-\sqrt{k_{n}}\tilde{\mathcal{A}}\left(b\left(\frac{n}{k_{n}}\right)\right)\dfrac{t^{-\rho}-1}{\rho}\right|$$

$${}+\underset{t\in(0,1]}{\sup}t^{\frac{1}{2}+\epsilon}\left|\sqrt{k_{n}}\frac{\gamma\text{log}(t)}{\int\limits_{0}^{1}K(s)ds}-\gamma\left(t^{-1}W(t)-W(1)\right)-\sqrt{k_{n}}\tilde{\mathcal{A}}\left(b\left(\frac{n}{k_{n}}\right)\right)\dfrac{t^{-\rho}-1}{\rho}\right|=\text{o}(1).$$

To prove that $T_{6}=\text{o}(1)$, inequation (38) leads to:

$$|T_{7}|\leq\frac{\sqrt{k_{n}}}{\log\frac{1}{pb(n/k_{n})}}|\tilde{\mathcal{A}}(b(n/k_{n}))|$$

$${}\times\left|\frac{\log\text{U}(\frac{1}{p})-\log\text{U}(b(n/k_{n}))-\gamma\log\frac{1}{pb(n/k_{n})}}{\tilde{\mathcal{A}}(b(n/k_{n}))}-\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{\rho}-1}{\rho}\right|$$

$${}\leq\frac{\sqrt{k_{n}}}{\log\frac{1}{pb(n/k_{n})}}|\tilde{\mathcal{A}}(b(n/k_{n}))|\epsilon\left(\frac{1}{pb(n/k_{n})}\right)^{\rho+\delta}=\text{o}(1)$$

for all $0<\delta<-\rho$.

Note that the term $T_{8}$ is a function of $\xi$ which can be a canonical value or an consistent estimator of $\rho$.

• If $\xi=\tilde{\rho}$ then, we have:

  $\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{1})-\hat{\gamma}_{k_{n}}(K_{2,\xi})]=\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{1})-\gamma]-\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{2,\xi})-\gamma]=\text{O}_{\mathbb{P}}(1)$ according to the Corollary II.1. This leads to $T_{8}=\text{o}_{\mathbb{P}}(1)$.

• If $\xi=\hat{\rho}$

  $$T_{8}=\frac{(1-\hat{\rho})(1-2\hat{\rho})}{\hat{\rho}^{2}}\frac{\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{1})-\hat{\gamma}_{k_{n}}(K_{2,\hat{\rho}})]}{\text{log}\frac{1}{pb(n/k_{n})}}$$
  $${}\times\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{{\rho}}-1}{{\rho}}+\frac{(1-\hat{\rho})(1-2\hat{\rho})}{\hat{\rho}^{2}}$$
  $${}\times\frac{\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{1})-\hat{\gamma}_{k_{n}}(K_{2,\hat{\rho}})]}{\log\frac{1}{pb(n/k_{n})}}\left(\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{\hat{\rho}}-1}{\hat{\rho}}-\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{{\rho}}-1}{{\rho}}\right).$$

  However, according to Corollary II.1 and Theorem II.2,

  $$\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{1})-\hat{\gamma}_{k_{n}}(K_{2,\hat{\rho}})]=\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{1})-\gamma]-\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{2,\rho})-\gamma]$$
  $${}-\sqrt{k_{n}}[\hat{\gamma}_{k_{n}}(K_{2,\hat{\rho}})-\hat{\gamma}_{k_{n}}(K_{2,\rho})]=\text{O}_{\mathbb{P}}(1).$$

  The term $T_{8}$ becomes,

  $$T_{8}=\text{o}_{\mathbb{P}}(1)+\text{o}_{\mathbb{P}}(1)\left\{\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{\hat{\rho}}-1}{\hat{\rho}}-\frac{\left(\frac{1}{pb(n/k_{n})}\right)^{{\rho}}-1}{{\rho}}\right\}$$
  $${}=\text{o}_{\mathbb{P}}(1)+\text{o}_{\mathbb{P}}(1)\int\limits_{0}^{\frac{1}{pb(n/k_{n})}}s^{\rho-1}(s^{\hat{\rho}-\rho}-1)ds.$$

  Inspired by [10], we get

  $$\displaystyle\int\limits_{0}^{\frac{1}{pb(n/k_{n})}}s^{\rho-1}(s^{\hat{\rho}-\rho}-1)ds=\text{o}_{\mathbb{P}}(1),$$

which leads to the conclusion that $T_{8}=\text{o}_{\mathbb{P}}(1)$ and therefore we get the proof of Theorem III.1.