Ordinal patterns in clusters of subsequent extremes of regularly varying time series

Abstract

In this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns giving the relative positions of the data points within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account, the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show their asymptotic normality under appropriate mixing conditions. The performance of the estimators is demonstrated in a simulated example and a real data application to discharge data of the river Rhine.

Appendix: Proofs

Proof of Proposition 1

We first note that, by Eq. (11), there exists some C > 0 such that α_h ≤ Cn^−δ for all $n \in \mathbb {N}$. By a straightforward computation, for any intermediate sequence $(r_{n})_{n \in \mathbb {N}}$ with $r_{n} \to \infty$ and r_n/n → 0, we then obtain

$$ \begin{array}{@{}rcl@{}} && \text{ Var }\left( \frac{\widehat{P}_{n,u_{n}}(A)}{\mathbb{P}(X_{0}>u_{n})}\right) \\ &=& \frac{1}{n^{2} \mathbb{P}(X_{0}>u_{n})^{2}} \sum\limits_{k=1}^{n-t} \sum\limits_{l=1}^{n-t} \text{ Cov }\left( \mathbf{1}_{\{(X_{i})_{i=k-1}^{k+t} \in u_{n}A\}}, \mathbf{1}_{\{(X_{i})_{i=l-1}^{l+t} \in u_{n}A\}}\right) \\ &\leq& \frac{2}{n \mathbb{P}(X_{0}>u_{n})^{2}} {\sum}_{h=0}^{n-t-1} \left| \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A, (X_{i})_{i=h-1}^{h+t} \in u_{n} A) \right.\\ & &\left.- \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A) \mathbb{P}((X_{i})_{i=h-1}^{t} \in u_{n} A) \right| \\ &\leq& \frac{2}{n \mathbb{P}(X_{0} > u_{n})} \sum\limits_{h=0}^{r_{n}} \left| \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A, (X_{i})_{i=h-1}^{h+t} \in u_{n} A \mid X_{0} > u_{n}) \right.\\ & &\left.- \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A \mid X_{0} > u_{n})\mathbb{P}((X_{i})_{i=h-1}^{t} \in u_{n} A) \right| \\ & &+ \frac{2}{n \mathbb{P}(X_{0} > u_{n})^{2}} \sum\limits_{h=r_{n}+1}^{\infty} \alpha_{h-t-2} \\ &\leq& \frac{2 (r_{n} +1)}{n \mathbb{P}(X_{0} > u_{n})} + \frac{2 C }{ n \mathbb{P}(X_{0} > u_{n})^{2}} \sum\limits_{h=r_{n}-t-1}^{\infty} h^{-\delta}\\ &\sim& \frac{2 (r_{n} +1)}{n \mathbb{P}(X_{0} > u_{n})} + \frac{2 C (r_{n} - t - 1)^{1-\delta}}{(\delta-1) n \mathbb{P}(X_{0} > u_{n})^{2}}. \end{array} $$

Setting $r_{n} = \mathbb {P}(X_{0} > u_{n})^{-1/\delta }$, the right-hand side is asymptotically equal to

$$ \frac{2 (1 + C/(\delta-1)) + o(1)}{n \mathbb{P}(X_{0} > u_{n})^{1+1/\delta}} = \frac{2 (1 + C/(\delta-1)) + o(1)}{[n^{\delta/(1+\delta)} \mathbb{P}(X_{0} > u_{n})]^{1+1/\delta}} \to 0. $$

Thus, by Chebychev’s inequality, this implies that

$$ \frac{\widehat{P}_{u_{n},n}(A) - \mathbb{E}(\widehat{P}_{u_{n},n}(A))}{\mathbb{P}(X_{0} > u_{n})} \to_{p} 0.$$

Since, by regular variation,

$$\frac{\mathbb{E}(\widehat{P}_{u_{n},n}(A))}{\mathbb{P}(X_{0} > u_{n})} = \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A \mid X_{0} > u_{n}) \to \mu_{\{-1,\ldots,t\}}(A), \qquad n \to \infty,$$

we obtain that $\widehat {P}_{u_{n},n}(A)/\mathbb {P}(X_{0} > u_{n}) \to _{p} \mu _{\{-1,\ldots ,t\}}(A)$. The same result can be obtained if we replace A by A₀. An application of the continuous mapping theorem for convergence in probability completes the proof. □

Proof of Theorem 1

We prove the equivalent statement that all linear combinations of the form

$$ \sum\limits_{j=0}^{N} a_{j} \sqrt{n \mathbb{P}(X_{0} > u_{n})} \left( \frac{\widehat{P}_{n,u_{n}}(A_{j})}{\mathbb{P}(X_{0} > u_{u})} - \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j} \mid X_{0} > u_{n})\right), $$

$a_{0},\ldots ,a_{N} \in \mathbb {R}$, converge in distribution to a centered normal distribution with the corresponding variance $\sum \nolimits _{j=0}^{N} \sum \nolimits _{l=0}^{N} a_{j} a_{l} \sigma _{jl}$. To this end, define

$$ Z_{n,k} = \frac 1 {\sqrt{n \mathbb{P}(X_{0}>u_{n})}} {\sum}_{j=0}^{N} a_{j} \left( \mathbf{1}_{\{(X_{i})_{i=k-1}^{k+t} \in u_{n} A_{j}\}} - \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j}) \right).$$

We note that, for each $n \in \mathbb {N}$, the random variable ${\sum }_{k=1}^{l} Z_{n,k}$ is centered and that its variance converges

$$ \begin{array}{@{}rcl@{}} && \text{ Var }\left( \sum\nolimits_{k=1}^{n} Z_{n,k}\right) \\ &=& \sum\limits_{j=0}^{N} \sum\limits_{l=0}^{N} a_{j} a_{l} \sum\limits_{h=-(n-1)}^{n-1} \frac{n-|h|}{n} \cdot \frac{\text{ Cov }(\mathbf{1}_{\{(X_{i})_{i=-1}^{t} \in u_{n} A_{j}\}},\mathbf{1}_{\{(X_{i})_{i=h-1}^{h+t} \in u_{n} A_{l}\}})}{\mathbb{P}(X_{0} > u_{n})}\\ &\to& \sum\nolimits_{j=0}^{N} \sum\nolimits_{l=0}^{N} a_{j} a_{l} \sigma_{jl} \end{array} $$

to the desired quantity, which can be verified analogously to the proof of Lemma 1, i.e. the proof of Thm. 3.1 in Davis and Mikosch (2009).

It remains to show that the asymptotic distribution is normal. To this end, we verify that the triangular scheme {Z_n,k}_k= 1,…,n, $n \in \mathbb {N}$, satisfies the conditions of Thm. 4.4 in Rio (2017):

• At first, all the variables Z_n,k are required to be centered and have finite variance which holds true as they are bounded.

• Secondly, we need to verify that

  $$ \begin{array}{@{}rcl@{}} \limsup_{n \to \infty} \max_{l=1,\ldots,n} \text{ Var }\left( \sum\nolimits_{k=1}^{l} Z_{n,k}\right) < \infty, \end{array} $$

  (24)

  which again can be shown analogously to the proof of Lemma 1.

• The third and last condition to be verified is

  $$ \lim_{x \to \infty} n {{\int}_{0}^{1}} \alpha^{-1}(x) Q_{n,a}^{2}(x) \min\{ \alpha^{-1}(x) Q_{n,a}(x), 1\} \mathrm{d} x = 0 $$

  (25)

  where α^− 1 and Q_n,a denote the inverse functions of h↦α_h and

  $$u \mapsto \mathbb{P}(|Z_{n,k}| > u), $$

  respectively. Noting that Q_n,a can be bounded via the relation

  $$ Q_{n,a}(x) \leq \sum\nolimits_{j=1}^{N} |a_{j}| Q_{n,e_{j}}\left( \frac{x}{N}\right), \quad x \in [0,1], $$

  where e_j is the j th standard basis vector in $\mathbb {R}^{N}$, j = 1,…,N, it suffices to verify

  $$ \lim_{x \to \infty} n {{\int}_{0}^{1}} \alpha^{-1}(x) Q_{n,e_{j}}^{2}\left( \frac{x}{N}\right) \min\left\{ \alpha^{-1}(x) Q_{n,e_{j}}\left( \frac{x}{N}\right), 1\right\} \mathrm{d} x = 0 $$

  for j = 1,…,N. By Eq. (14), we have α^− 1(x) ≤ (x/C)^− 1/δ for some C > 0 and

  $$ \begin{array}{@{}rcl@{}} Q_{n,e_{j}}\left( \frac{x}{N}\right) = \begin{cases} \frac{1 - \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})}{\sqrt{n \mathbb{P}(X_{0} > u_{n})}}, & 0 \leq x < N \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j}), \\ \frac{\mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})}{\sqrt{n \mathbb{P}(X_{0} > u_{n})}}, & N \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j}) \leq x \leq 1, \end{cases} \end{array} $$

  for sufficiently large n. Thus, we have

  $$ \begin{array}{@{}rcl@{}} && n {{\int}_{0}^{1}} \alpha^{-1}(x) Q_{n,e_{j}}^{2}\left( \frac{x}{N}\right) \min\left\{ \alpha^{-1}(x) Q_{n,e_{j}}\left( \frac{x}{N}\right), 1\right\} \mathrm{d} x \\ &\leq& n {\int}_{0}^{N \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})} (x/C)^{-1/\delta} (n \mathbb{P}(X_{0}>u_{n}))^{-1} \\ && \min\{1,(x/C)^{-1/\delta} (n \mathbb{P}(X_{0}>u_{n})^{-1/2}\} \mathrm{d} x \\ && + n {\int}_{N \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})}^{1} (x/C)^{-2/\delta} \left( \frac{\mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})}{\sqrt{n \mathbb{P}(X_{0} > u_{n})}}\right)^{3} \mathrm{d} x \\ &=:& I_{1} + I_{2}. \end{array} $$

  For the assessment of the integral term I₂, we employ the upper bound

  $$ \left( \frac x C\right)^{-2/\delta} \leq (C/N)^{2/\delta} \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})^{-2/\delta} $$

  and obtain

  $$ \begin{array}{@{}rcl@{}} I_{2} \leq{}& n {{\int}_{0}^{1}} \left( \frac{C}{N}\right)^{\frac{2}{\delta}} \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})^{-\frac{2}{\delta}} \left( \frac{\mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})}{\sqrt{n \mathbb{P}(X_{0} > u_{n})}}\right)^{3} \mathrm{d} x \\ \leq{}& \left( \frac{C}{N}\right)^{\frac{2}{\delta}} \frac 1 {\sqrt{n}} \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})^{\frac{3\delta-4}{2\delta}} \left( \frac{\mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})}{\mathbb{P}(X_{0} > u_{n})}\right)^{3/2} \longrightarrow 0 \end{array} $$

  using that $\mathbb {P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j}) / \mathbb {P}(X_{0} > u_{n}) \to \mu _{\{-1,\ldots ,t\}}(A_{j}) < \infty$ and $\mathbb {P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j}) \to 0$ as $n \to \infty$.

  For the assessment of the integral I₁, we distinguish between the two cases δ = 2 and δ > 2. In the case δ = 2, we have

  $$ \begin{array}{@{}rcl@{}} I_{1} &\leq& n {\int}_{0}^{N [ n \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j}) ]^{-1}} C^{1/2} x^{-1/2} (n \mathbb{P}(X_{0} > u_{n}))^{-1} \mathrm{d} x \\ && + n {\int}_{N [ n \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j}) ]^{- 1}}^{N \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})} C x^{-1} \sqrt{n \mathbb{P}(X_{0} > u_{n})}^{-3} \mathrm{d}x \\ &=& \sqrt{2CN} [ n \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j}) ]^{-1/2} \mathbb{P}(X_{0} > u_{n})^{-1} \\ && + C n^{-1/2} \mathbb{P}(X_{0} > u_{n})^{-3/2} \log(\mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})) \\ && + C n^{-1/2} \mathbb{P}(X_{0} > u_{n})^{-3/2} \log(n \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j}) ) \\ &=& \sqrt{2CN} n^{-1/2} \mathbb{P}(X_{0} >u_{n})^{-3/2} \cdot \sqrt{\frac{\mathbb{P}(X_{0} > u_{n})}{\mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})}} \\ && + C n^{-1/2} \mathbb{P}(X_{0} > u_{n})^{-3/2} \cdot \left\{ - \log(\mathbb{P}(X_{0} > u_{n})) + \log(n \mathbb{P}(X_{0} > u_{n})^{3} ) \right.\\ &&\left. + 2 \log\left( \frac{\mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})}{\mathbb{P}(X_{0} > u_{n})}\right) \right\} \end{array} $$

  which vanishes as $n \to \infty$ because of Eq. (16) and $\lim _{x \to \infty } x^{-1/2} \log (x) = 0$. For δ > 2, we obtain

  $$ \begin{array}{@{}rcl@{}} I_{1} &\leq& n {\int}_{0}^{N \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})} C^{2/\delta} x^{-2/\delta} \sqrt{n \mathbb{P}(X_{0} > u_{n})}^{-3} \mathrm{d}x \\ &=& \frac{\delta}{\delta-2} N^{1-\frac{2}{\delta}} C^{\frac{2}{\delta}} \frac{1}{\sqrt{n}} \mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})^{1 - \frac{2}{\delta}} \mathbb{P}(X_{0} > u_{n})^{-3/2} \\ &=& \frac{\delta}{\delta-2} N^{1-\frac{2}{\delta}} C^{\frac{2}{\delta}} \frac{1}{\sqrt{n}} \mathbb{P}(X_{0} > u_{n})^{-\frac 12 \left( \frac{4+\delta}{\delta}\right)} \left( \frac{\mathbb{P}((X_{i})_{i=-1}^{t} \in u_{n} A_{j})}{\mathbb{P}(X_{0} > u_{n})}\right)^{1 - \frac{2}{\delta}} \end{array} $$

  which vanishes as $n \to \infty$ because of Eq. (15).

□

Proof of Proposition 2

We verify the validity of both Eqs. (12) and (13). By Eq. (21), the series considered in Eq. (12) can the bounded by

$$ \begin{array}{@{}rcl@{}} \frac{1}{\mathbb{P}(X_{0} > u_{n})} \sum\nolimits_{h=r_{n}}^{\infty} \alpha_{h} &\leq& \frac{2}{\mathbb{P}(X_{0} > u_{n})} \sum\nolimits_{h=r_{n}}^{\infty} \sum\nolimits_{s=0}^{\infty} (s+1)\left( 2 - \theta(s + h)\right)\\ &=& \frac{2}{\mathbb{P}(X_{0} > u_{n})} \sum\nolimits_{t=0}^{\infty} \sum\nolimits_{h^{\prime}=0}^{t} (t - h^{\prime}+1)\left( 2 - \theta(t + r_{n})\right) \\ &=& \frac{1}{\mathbb{P}(X_{0} > u_{n})} \sum\nolimits_{t=0}^{\infty} (t+1)(t+2) \left( 2 - \theta(t+r_{n})\right), \end{array} $$

where we set $h^{\prime } = h - r_{n}$ and $t = s + h^{\prime }$.

The asymptotic relation $\mathbb {P}(X_{0} > u_{n}) = 1 - \exp (-u_{n}^{-\alpha }) \sim u_{n}^{-\alpha }$ then yields that Eq. (22) is a sufficient condition for Eq. (12).

In order to simplify condition (13), we first note that

$$ \mathbb{P}(X_{h} > u_{n} \mid X_{0} > u_{n}) = \frac{2(1-\exp(-u_{n}^{-\alpha})) - (1-\exp(-\theta(h)u_{n}^{-\alpha}))}{1 - \exp(-u_{n}^{-\alpha})}. $$

Using the inequality $x - x^{2}/2 \leq 1 - \exp (-x) \leq x$ for all x ≥ 0, we obtain the bounds

$$ \begin{array}{@{}rcl@{}} & 2 - \theta(h) - \frac{u_{n}^{-\alpha}}2 \leq{} \mathbb{P}(X_{h} > u_{n} \mid X_{0} > u_{n}) \\ \leq{}& \frac{2 - \theta(h) + \frac{\theta(h)^{2} u_{n}^{-\alpha}}{2}}{1 - \frac {u_{n}^{-\alpha}} 2 } \sim 2 - \theta(h) + \frac{\theta(h)^{2}}{2} u_{n}^{-\alpha} \leq 2 - \theta(h) + 2 u_{n}^{-\alpha}. \end{array} $$

(26)

As the sequences $r_{n}, u_{n} \to \infty$ are chosen such that $r_{n} \mathbb {P}(X_{0} > u_{n}) \to 0$ as $n \to \infty$, we obtain that, for every $k \in \mathbb {N}$,

$$ \sum\nolimits_{h=k}^{r_{n}} u_{n}^{-\alpha} \leq \frac{r_{n}}{u_{n}^{\alpha}} = \frac{r_{n} \mathbb{P}(X_{0} > u_{n})}{u_{n}^{\alpha} \mathbb{P}(X_{0} > u_{n})} \to 0 \quad (n \to \infty),$$

and consequently, from the inequalities above, Eq. (13) holds if and only if Eq. (23) holds. □

Proof of Corollary 2

We first note that, for the Brown–Resnick model,

$$\theta(h) = 2 {\Phi}\left( \sqrt{\gamma(h)/2}\right), \qquad h \in \mathbb{Z},$$

cf. Kabluchko et al. (2009). In particular, using $1 - {\Phi }(x) \sim x^{-1} \varphi (x)$ as $x \to \infty$, we obtain for large h that

$$ 2 - \theta(h) \sim \frac{2}{\sqrt{\pi \gamma(h)}} \exp\left( -\frac{\gamma(h)}{4}\right). $$

Thus, similarly to Davis et al. (2013), we can employ (21) to bound the α-mixing coefficients by

$$ \begin{array}{@{}rcl@{}} \alpha_{h} &\leq& \frac{4}{\sqrt{\pi C}} \sum\nolimits_{s=0}^{\infty} \frac{s+1}{(s+h)^{\varepsilon/2}} \exp\left( -\frac 1 4 C \cdot (s+h)^{\varepsilon}\right)\\ &\leq& \frac{4}{\sqrt{\pi C}} \sum\nolimits_{s=0}^{\infty} \frac{s+1}{s^{\varepsilon/2}} \exp\left( -\frac 1 8 C s^{\varepsilon} - \frac 1 8 C h^{\varepsilon}\right) \leq C_{\alpha} \exp\left( - \frac 1 8 C h^{\varepsilon}\right) \end{array} $$

for some appropriate constant C_α > 0, that is, the α-mixing coefficients α_h decay at an exponential rate. As Eq. (14) holds for every δ > 0, Eq. (15) simplifies to $\lim _{n \to \infty } n^{1+\varepsilon ^{*}} \mathbb {P}(X_{0} > u_{n}) = \lim _{n \to \infty } n^{1-\varepsilon ^{*}}/u_{n} = \infty$ for some ε^∗ > 0, which is true for $u_{n} \sim n^{\beta _{1}}$. Thus, the assumptions of Theory 1 reduce to Condition (M).

Choosing $u_{n} \sim n^{\beta _{1}}$ and $r_{n} \sim n^{\beta _{2}}$ (Buhl and Klüppelberg 2018, see also; Buhl et al. 2019), we have $u_{n} \to \infty$, $r_{n} \to \infty$, $n \mathbb {P}(X_{0} > u_{n}) \sim n^{1 - \beta _{1}} \to \infty$ and $r_{n} \mathbb {P}(X_{0} > u_{n}) \sim n^{\beta _{2} - \beta _{1}} \to 0$ as $n \to \infty$. Furthermore, similarly to the assessment above,

$$ \begin{array}{@{}rcl@{}} & u_{n} \sum\nolimits_{h=1}^{\infty} h^{2} \left( 2 - \theta(h+r_{n})\right) \\ \leq{}& u_{n} \frac 4 {\sqrt{\pi C}} \exp\left( - \frac 1 8 C r_{n}^{\varepsilon} \right) \cdot \sum\nolimits_{h=1}^{\infty} h^{2-\varepsilon/2} \exp\left( - \frac 1 8 C h^{\varepsilon} \right) \stackrel{n \to \infty}{\longrightarrow} 0, \end{array} $$

i.e. Eq. (22) holds. We also obtain

$$ {\sum}_{h=k}^{r_{n}} \left( 2 - \theta(h)\right) \leq \exp\left( - \frac 1 8 C (k-1)^{\varepsilon}\right) {\sum}_{h=1}^{\infty} \frac{4}{\sqrt{\pi C h^{\varepsilon}}} \exp\left( - \frac 1 8 C h^{\varepsilon}\right) \stackrel{k \to \infty}{\longrightarrow} 0 $$

which implies Eq. (23). Consequently, the assertion of the corollary follows from Proposition 2. □