Improved interexceedance-times-based estimator of the extremal index using truncated distribution

Abstract

The extremal index is an important parameter in the characterization of extreme values of a stationary sequence. This paper presents a novel approach to estimation of the extremal index based on truncation of interexceedance times. The truncated estimator based on the maximum likelihood method is derived together with its first-order bias. The estimator is further improved using penultimate approximation to the limiting mixture distribution. In order to assess the performance of the proposed estimator, a simulation study is carried out for various stationary processes satisfying the local dependence condition $D^{(k)}(u_n)$. An application to daily maximum temperatures at Uccle, Belgium, is also presented.

Appendix

1.1 Derivation of the bias corrections

1.1.1 Bias correction of the estimator $\widehat{\theta }$ under the limit distribution

Let $T_1, \dots , T_{N-1}$ be a sequence of interexceedance times obtained from an underlying series $X_1, \dots , X_n$ for a high enough threshold u. Moreover, let us assume that the underlying series satisfies the $D^{(D+1)}(u_n)$ condition for certain non-negative integer D. On that account, times $T_1, \dots , T_{N-1}$ exceeding D can be treated as approximately independent inter-cluster times (Ferro and Segers 2003).

Using the delta method (Casella and Berger 2002), we may establish the asymptotic approximation for the estimator (7), i.e.

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}\widehat{\theta } \approx \frac{{{\,\mathrm{\mathrm {E}}\,}}U}{{{\,\mathrm{\mathrm {E}}\,}}V} - \frac{{{\,\mathrm{\mathrm {cov}}\,}}(U,V)}{({{\,\mathrm{\mathrm {E}}\,}}V)^2} + \frac{{{\,\mathrm{\mathrm {E}}\,}}U}{({{\,\mathrm{\mathrm {E}}\,}}V)^3} {{\,\mathrm{\mathrm {var}}\,}}V, \end{aligned}$$

(13)

where $U = \sum _{i=1}^{N-1} \varvec{1}_{[T_{i}> D]}$ and $V = \overline{F}(u) \sum _{i=1}^{N-1} (T_i-D) \varvec{1}_{[T_{i}> D]}$. Let us assume the variable $\overline{F}(u) T_i$ follows the limiting distribution $F_{\theta }$ from (2), i.e. $P(\overline{F}(u) T_i \le x) = 1-\theta \exp (-\theta x)$ for $x \ge 0$. For the sake of simplicity, we put $d = \overline{F}(u) D$ and obtain

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}\left( \varvec{1}_{[T_i>D]} \right) = P(T_i>D) = 1-P\left( \overline{F}(u) T_i \le d\right) = \theta \mathrm e^{-\theta d}. \end{aligned}$$

(14)

Notice that for $x>0$ it holds

$$\begin{aligned} P\left( \overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]}> x\right) = P\left( \overline{F}(u) (T_i-D)> x \mid T_{i}> D\right) \cdot P(T_i>D). \end{aligned}$$

(15)

Let us take $P(T_i>D) = \theta \mathrm e^{-\theta d}$ from (14), and suppose $P\left( \overline{F}(u) (T_i-D)> x \mid T_{i}> D\right) = \mathrm e^{-\theta x}$ from (6). Since $\overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]}$ is a non-negative random variable, by applying (15) we obtain

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}(\overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]})&= \int _0^{\infty } P\left( \overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]}> x\right) \,\mathrm {d}x \\&= \theta \mathrm e^{-\theta d} \int _0^{\infty } e^{-\theta x} \,\mathrm {d}x = \mathrm e^{-\theta d},\\ {{\,\mathrm{\mathrm {E}}\,}}(\overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]})^2&= 2\int _0^{\infty } x P\left( \overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]} > x\right) \,\mathrm {d}x\\&= 2 \theta \mathrm e^{-\theta d} \int _0^{\infty } x \mathrm e^{-\theta x} \,\mathrm {d}x = \frac{2}{\theta } e^{-\theta d}. \end{aligned}$$

Accounting for the independence of the large interexceedance times, we have

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}U&= (N-1) {{\,\mathrm{\mathrm {E}}\,}}\left( \varvec{1}_{[T_i>D]} \right) = (N-1)\theta \mathrm e^{-\theta d},\\ {{\,\mathrm{\mathrm {E}}\,}}V&= (N-1) {{\,\mathrm{\mathrm {E}}\,}}(\overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]}) = (N-1)\mathrm e^{-\theta d},\\ {{\,\mathrm{\mathrm {var}}\,}}V&= (N-1) \left[ {{\,\mathrm{\mathrm {E}}\,}}(\overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]})^2 - \left( {{\,\mathrm{\mathrm {E}}\,}}(\overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]}) \right) ^2 \right] \\&= (N-1) e^{-\theta d} \left( \frac{2}{\theta } - e^{-\theta d} \right) ,\\ {{\,\mathrm{\mathrm {cov}}\,}}(U,V)&= (N-1) \left[ {{\,\mathrm{\mathrm {E}}\,}}(\overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]}) - {{\,\mathrm{\mathrm {E}}\,}}(\varvec{1}_{[T_{i}> D]}) {{\,\mathrm{\mathrm {E}}\,}}(\overline{F}(u) (T_i-D) \varvec{1}_{[T_{i}> D]}) \right] \\&= (N-1) \mathrm e^{-\theta d}(1-\theta \mathrm e^{-\theta d}), \end{aligned}$$

where ${{\,\mathrm{\mathrm {cov}}\,}}(\varvec{1}_{[T_i>D]}, \overline{F}(u) (T_j-D) \varvec{1}_{[T_j > D]}) = 0$ for $i \ne j$ by the assumption. We put the foregoing into (13) and obtain

$${{\,\mathrm{\mathrm {E}}\,}}\widehat{\theta } \approx \theta + \frac{1}{N-1}\mathrm e^{\theta d} = \theta + \frac{1}{N-1} + \frac{\theta d}{N-1} + O(\theta ^2).$$

1.1.2 Bias correction under the penultimate approximation

Let us assume the distribution of each time $T_i$ is given by $P(T_i > n) = \theta p^{n \theta }$, $n=1, 2, \dots$, where $\theta \in (0,1]$ and $p \in (0,1)$. Further on, it will be more convenient to work with the probability mass function $h(n) = P(T_i> n-1) - P(T_i > n)$ of $T_i$. This takes the form of

$$\begin{aligned} h(n) = \left\{ \begin{array}{ll} 1-\theta p^{\theta }, &{} n=1,\\ \theta p^{n \theta } \frac{1-p^{\theta }}{p^{\theta }} &{}, n=2, 3, \dots . \end{array} \right. \end{aligned}$$

(16)

Let us again assume the times $T_1, \dots , T_{N-1}$ exceeding D can be treated as approximately independent. Let us denote $U = \sum _{i=1}^{N-1} \varvec{1}_{[T_i>D]}$ and $V = (1-p) \sum _{i=1}^{N-1} (T_i-D) \varvec{1}_{[T_i>D]}$. From (13) we may derive the first-order bias of the estimator $\widehat{\theta } = U/V$ under the penultimate distribution (16). Since

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}\left( \varvec{1}_{[T_i>D]}\right)&= P(T_i>D) = \theta p^{D \theta },\\ {{\,\mathrm{\mathrm {E}}\,}}\left( T_i \varvec{1}_{[T_i>D]}\right)&= \sum _{n=1}^{\infty } n \varvec{1}_{[n>D]} h(n) = \sum _{n=D+1}^{\infty } n \theta p^{n \theta } \frac{1-p^{\theta }}{p^{\theta }} = \frac{\theta p^{D\theta }}{1-p^{\theta }}\left[ 1+D(1-p^{\theta })\right] ,\\ {{\,\mathrm{\mathrm {E}}\,}}\left( T_i^2 \varvec{1}_{[T_i>D]}\right)&= \sum _{n=1}^{\infty } n^2 \varvec{1}_{[n>D]} h(n) = \sum _{n=D+1}^{\infty } n^2 \theta p^{n \theta } \frac{1-p^{\theta }}{p^{\theta }} \\&= \frac{\theta p^{D \theta }}{(1-p^{\theta })^2} \left( p^{\theta } + \left[ 1+D(1-p^{\theta }) \right] ^2 \right) ,\\ \end{aligned}$$

and ${{\,\mathrm{\mathrm {cov}}\,}}\left( \varvec{1}_{[T_i>D]}, T_i \varvec{1}_{[T_i>D]} \right) = {{\,\mathrm{\mathrm {E}}\,}}\left( T_i \varvec{1}_{[T_i>D]} \right) - {{\,\mathrm{\mathrm {E}}\,}}\left( \varvec{1}_{[T_i>D]} \right) {{\,\mathrm{\mathrm {E}}\,}}\left( T_i \varvec{1}_{[T_i>D]} \right) ,$ straightforward computations lead us to

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}U&= (N-1){{\,\mathrm{\mathrm {E}}\,}}\left( \varvec{1}_{[T_i>D]} \right) = (N-1)\theta p^{D \theta },\\ {{\,\mathrm{\mathrm {E}}\,}}V&= (1-p)(N-1) \left[ {{\,\mathrm{\mathrm {E}}\,}}\left( T_i \varvec{1}_{[T_i>D]}\right) - D {{\,\mathrm{\mathrm {E}}\,}}\left( \varvec{1}_{[T_i>D]}\right) \right] = (N-1) \theta p^{D \theta } \frac{1-p}{1-p^{\theta }}, \end{aligned}$$

Since ${{\,\mathrm{\mathrm {cov}}\,}}(\varvec{1}_{[T_i>D]}, T_j \varvec{1}_{[T_j>D]}) = {{\,\mathrm{\mathrm {cov}}\,}}(\varvec{1}_{[T_i>D]}, \varvec{1}_{[T_j>D]}) = {{\,\mathrm{\mathrm {cov}}\,}}(T_i \varvec{1}_{[T_i>D]}, T_j \varvec{1}_{[T_j>D]}) = 0$ for $i \ne j$ by the assumption, we obtain

$$\begin{aligned} {{\,\mathrm{\mathrm {cov}}\,}}(U,V)&= (1-p) {{\,\mathrm{\mathrm {cov}}\,}}\left( \sum _{i=1}^{N-1} \varvec{1}_{[T_i>D]}, \sum _{i=1}^{N-1} (T_i-D)\varvec{1}_{[T_i>D]} \right) \\&= (1-p) \left[ \sum _{i=1}^{N-1} {{\,\mathrm{\mathrm {cov}}\,}}\left( \varvec{1}_{[T_i>D]}, T_i \varvec{1}_{[T_i>D]} \right) - D \sum _{i=1}^{N-1} {{\,\mathrm{\mathrm {var}}\,}}\left( \varvec{1}_{[T_i>D]} \right) \right] ,\\ {{\,\mathrm{\mathrm {var}}\,}}V&= (1-p)^2 (N-1) {{\,\mathrm{\mathrm {var}}\,}}\left[ (T_i-D) \varvec{1}_{[T_i>D]} \right] = (1-p)^2 (N-1) \left[ {{\,\mathrm{\mathrm {var}}\,}}(T_i \varvec{1}_{[T_i>D]}) \right. \\&\left. \quad\;+ D^2 {{\,\mathrm{\mathrm {var}}\,}}(\varvec{1}_{[T_i>D]}) - 2D {{\,\mathrm{\mathrm {cov}}\,}}(T_i \varvec{1}_{[T_i>D]}, \varvec{1}_{[T_i>D]}) \right] \end{aligned}$$

Since $\varvec{1}_{[T_i>D]}$ is the Bernoulli random variable equal to 1 with probability $\theta p^{D \theta }$, we have ${{\,\mathrm{\mathrm {var}}\,}}(\varvec{1}_{[T_i>D]}) = \theta p^{D \theta } (1- \theta p^{D \theta })$. It yields

$$\begin{aligned} {{\,\mathrm{\mathrm {cov}}\,}}(U,V)&= (N-1) \theta p^{D \theta } (1-\theta p^{D \theta }) \frac{1-p}{1-p^{\theta }},\\ {{\,\mathrm{\mathrm {var}}\,}}V&= (N-1)\frac{ p^{D \theta }}{\left( 1-p^{\theta }\right) ^2}\left[ \theta (1+p^{\theta })-\theta ^2p^{D\theta }\right] (1-p)^2. \end{aligned}$$

Hence, if $T_i$ is described by the penultimate distribution (16), from (13) we obtain

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}\widehat{\theta } \approx \frac{1-p^{\theta }}{1-p} + \frac{1}{N-1}\frac{(1-p^{\theta })p^{\theta }}{(1-p)\theta p^{D \theta }}. \end{aligned}$$

(17)

If we replace $\overline{F}(u)$ by $1-p$ in (9) and substitute $\widehat{\theta }$ by (17), then

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}\widehat{\theta }^\mathrm{BC} \approx \frac{1}{N-1+(1-p)D} \left[ (N-1)\left( \frac{1-p^{\theta }}{1-p} + \frac{1}{N-1}\frac{(1-p^{\theta })p^{\theta }}{(1-p)\theta p^{D \theta }} \right) - 1 \right] \\ = \theta + \frac{1-p}{2 (N-1)} \left[ 1 + \theta (N-4) - \theta ^2 (N-1) \right] + O((1-p)^2), \end{aligned}$$

where the last equality follows from the Taylor expansion as $p \rightarrow 1$.

1.2 Proof of Theorem 2

The bias-corrected estimator $\widehat{\theta }^\mathrm{BC}$ 

Let us denote $\overline{F}(u_n)$ by $p_n$. First, let us deal with the bias-corrected estimator $\widehat{\theta }^\mathrm{BC}$ that is of the form

$$\widehat{\theta }^\mathrm{BC} = \frac{N-1}{N-1+p_n D} \cdot \frac{\sum _{i=1}^{N-1} \varvec{1}_{[T_i>D]}}{p_n \sum _{i=1}^{N-1} (T_i-D) \varvec{1}_{[T_i>D]}} - \frac{1}{N-1+p_n D},$$

where $N = \sum _{i=1}^n \varvec{1}_{[X_i > u_n]}$. From Lemma B.3 in Ferro and Segers (2002) we already know $N/(np_n) \xrightarrow {p}1$, i.e. $N = np_n (1+o_p(1))$. In particular, $N \xrightarrow {p}\infty$ for $n \rightarrow \infty$. Since $p_n = o(1)$, it is obvious that $(N-1)/(N-1+p_n D) \xrightarrow {p}1$, and $1/(N-1+p_n D) \xrightarrow {p}0$.

The term $\sum _{i=1}^{N-1} \varvec{1}_{[T_i>D]}$

Let us define

$$Y_n = \sum _{i=1}^{n-D} \varvec{1}_{[X_i>u_n, M_{i,i+D} \le u_n]}.$$

It should be pointed out that $Y_n = \sum _{i=1}^{N-1} \varvec{1}_{[T_i>D]} + \varvec{1}_{[j_N \le n-D]}$, where $j_N$ denotes the largest exceedance time of $u_n$ (i.e. the last exceedance) within the series $X_1, \dots , X_n$. As shown in Ferro and Segers (2002), it holds $n-j_N = o_p(n)$. This particularly means that $n-j_N \xrightarrow {p}\infty$; therefore $\varvec{1}_{[j_N \le n-D]} = \varvec{1}_{[n-j_N \ge D]} \xrightarrow {p}1$.

Based on the stationarity and the $D^{(D+1)}(u_n)$ condition, it is implied that

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}Y_n&= (n-D) P\left( X_1>u_n\right) P\left( M_{1,1+D} \le u_n \mid X_1>u_n\right) = (n-D) p_n \theta (1 + o(1)). \end{aligned}$$

Let us put $I_i = \varvec{1}_{[X_i>u_n, M_{i,i+D} \le u_n]}$ for integer $1 \le i \le n-D$. Notice the m-dependence implies ${{\,\mathrm{\mathrm {cov}}\,}}(I_i, I_j) = 0$ for all $j > i+D+m$, and the stationarity implies that ${{\,\mathrm{\mathrm {E}}\,}}I_i = {{\,\mathrm{\mathrm {E}}\,}}I_1$. We obtain

$$\begin{aligned} {{\,\mathrm{\mathrm {var}}\,}}Y_n&= \sum _{i=1}^{n-D} \sum _{j=1}^{n-D} {{\,\mathrm{\mathrm {cov}}\,}}\left( I_i, I_j \right) \le 2 \sum _{i=1}^{n-D} \sum _{j=i}^{n-D} {{\,\mathrm{\mathrm {cov}}\,}}\left( I_i, I_j \right) = 2 \sum _{i=1}^{n-2D-m} \ \sum _{j=i}^{i+D+m} {{\,\mathrm{\mathrm {cov}}\,}}\left( I_i, I_j \right) \\&= 2 \sum _{i=1}^{n-2D-m} \ \sum _{j=i}^{i+D+m} \left[ {{\,\mathrm{\mathrm {E}}\,}}(I_i I_j) - {{\,\mathrm{\mathrm {E}}\,}}I_1 \cdot {{\,\mathrm{\mathrm {E}}\,}}I_1 \right] \end{aligned}$$

where ${{\,\mathrm{\mathrm {E}}\,}}I_1 = P\left( X_1> u_n\right) P\left( M_{1,D+1} \le u_n \mid X_1 > u_n\right) = p_n \theta (1+o(1))$ based on the $D^{(D+1)}(u_n)$ condition, and

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}(I_i I_j)&= P\left( X_i> u_n, M_{i,i+D} \le u_n, X_j> u_n, M_{j,j+D} \le u_n\right) \\&= P\left( X_j> u_n, M_{j,j+D} \le u_n \mid X_i> u_n, M_{i,i+D} \le u_n\right) P\left( M_{i,i+D} \le u_n \mid X_i> u_n\right) \\&\quad\times P(X_i > u_n) \le p_n \theta (1 + o(1)), \end{aligned}$$

for $i+D+1 \le j \le i+D+m$. For $j \le i+D$ the events $I_i$ and $I_j$ conflict; therefore ${{\,\mathrm{\mathrm {E}}\,}}(I_i I_j) = 0$. Based on this, ${{\,\mathrm{\mathrm {var}}\,}}Y_n \le 2 (n-2D-m) m \left[ p_n \theta - p_n^2 \theta ^2 +o(1) \right] \le 2 n m p_n \theta (1+o(1))$.

Putting ${{\,\mathrm{\mathrm {E}}\,}}Y_n$ and ${{\,\mathrm{\mathrm {var}}\,}}Y_n$ together, the expectation ${{\,\mathrm{\mathrm {E}}\,}}[ \left( Y_n \{n p_n \theta (1+o(1))\}^{-1} - 1 \right) ^2 ]$ is bounded by some positive constant times $2 m (n p_n \theta )^{-1}$. Since $2 m (n p_n \theta )^{-1} \rightarrow 0$, this gives $Y_n = n p_n \theta (1+o_p(1))$.

The term $\sum _{i=1}^{N-1} (T_i-D) \varvec{1}_{[T_i>D]}$ 

Let us define

$$Z_n = \sum _{i=1}^{n-D} \Big [ \varvec{1}_{[X_i>u_n, M_{i,i+D} \le u_n]} \cdot \sum _{j=1}^{n-i} \varvec{1}_{[M_{i,i+j} \le u_n]} \Big ].$$

It can be seen that $Z_n = \sum _{i=1}^{N-1} (T_i-1) \varvec{1}_{[T_i>D]} + (n-j_N) \varvec{1}_{[j_N \le n-D]}$, and

$$\begin{aligned} \sum _{i=1}^{N-1} (T_i-D) \varvec{1}_{[T_i>D]}&= Z_n - (n-j_N) \varvec{1}_{[j_N \le n-D]} - (D-1) \sum _{i=1}^{N-1} \varvec{1}_{[T_i>D]}\\&= Z_n + o_p(n) - (D-1) (Y_n + O_p(1)). \end{aligned}$$

Moreover, we can take advantage of the following expression

$$Z_n = \sum _{i=1}^{n-D} \Big [ \varvec{1}_{[X_i>u_n, M_{i,i+D} \le u_n]} \Big ( D + \sum _{j=1}^{n-D-i} \varvec{1}_{[M_{i,i+D+j} \le u_n]} \Big ) \Big ] = DY_n + V_n,$$

where

$$V_n = \sum _{i=1}^{n-D} \sum _{j=1}^{n-D-i} \varvec{1}_{[X_i>u_n, M_{i,i+D+j} \le u_n]}.$$

Therefore $\sum _{i=1}^{N-1} (T_i-D) \varvec{1}_{[T_i>D]} = V_n + Y_n + o_p(n)$.

To establish consistency of the estimator $\widehat{\theta }^\mathrm{BC}$, we only need to show $n^{-1}V_n \xrightarrow {p}1$. Thus, we need to prove (i) $n^{-1} {{\,\mathrm{\mathrm {E}}\,}}V_n \rightarrow 1$ and (ii) $n^{-2} {{\,\mathrm{\mathrm {E}}\,}}(V_n^2) \rightarrow 1$.

By stationarity of the process we obtain

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}V_n&= \sum _{i=1}^{n-D} \sum _{j=1}^{n-D-i} P\left( X_i>u_n, M_{i,i+D+j} \le u_n\right) \\&= p_n \sum _{j=1}^{n-D} (n-D-j) P\left( M_{1,1+D+j} \le u_n \mid X_1>u_n\right) . \end{aligned}$$

We rewrite the sum as an integral and observe

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}V_n&= p_n \int _{1}^{n-D} (n-D-\lceil s \rceil ) P\left( M_{1,1+D+\lceil s \rceil } \le u_n \mid X_1>u_n\right) \,\mathrm {d}s\\&= p_n r_n \int _{r_n^{-1}}^{(n-D)/r_n} (n-D-\lceil r_n s \rceil ) P\left( M_{1,1+D+\lceil r_n s \rceil } \le u_n \mid X_1>u_n\right) \,\mathrm {d}s. \end{aligned}$$

We apply Lemma B.1 and Lemma B.2 from Ferro and Segers (2002) that specify the convergence of the probability of maximum, and by the dominated convergence theorem it follows

$$n^{-1} {{\,\mathrm{\mathrm {E}}\,}}V_n \rightarrow \tau \int _0^{\infty } \theta \mathrm e^{-\theta \tau s} \,\mathrm {d}s = 1.$$

Hereby we have proven the first point concerning $V_n$.

Now we focus on (ii) $n^{-2} {{\,\mathrm{\mathrm {E}}\,}}(V_n^2) \rightarrow 1$. For integers $1 \le i \le n-D$, $1 \le j \le n-D-i$, put $I_{i,j} = \varvec{1}_{[X_i>u_n, M_{i,i+D+j} \le u_n]}$. We may write

$${{\,\mathrm{\mathrm {E}}\,}}(V_n^2) = \sum _{i=1}^{n-D} \ \sum _{j=1}^{n-D-i} \ \sum _{k=1}^{n-D} \ \sum _{l=1}^{n-D-k} {{\,\mathrm{\mathrm {E}}\,}}(I_{i,j} I_{k,l}),$$

where, accounting m-dependence and stationarity, the expected values are as follows

$$\begin{aligned} {{\,\mathrm{\mathrm {E}}\,}}I_{i,j}&= {{\,\mathrm{\mathrm {E}}\,}}I_{1,j},&{{\,\mathrm{\mathrm {E}}\,}}(I_{i,j} I_{k,l})&= 0, \text { for } i < k \le i+D+j,\\ {{\,\mathrm{\mathrm {E}}\,}}(I_{i,j} I_{i,l})&= {{\,\mathrm{\mathrm {E}}\,}}I_{1,\max \{j,l\}},&{{\,\mathrm{\mathrm {E}}\,}}(I_{i,j} I_{k,l})&= {{\,\mathrm{\mathrm {E}}\,}}I_{1,j} \cdot {{\,\mathrm{\mathrm {E}}\,}}I_{1,l}, \text { for } k > i+D+j+m. \end{aligned}$$

We can write ${{\,\mathrm{\mathrm {E}}\,}}(V_n^2) = A_n + B_n + C_n$, where

$$\begin{aligned} A_n&= \sum _{i=1}^{n-D} \ \sum _{j=1}^{n-D-i} \ \sum _{l=1}^{n-D-i} {{\,\mathrm{\mathrm {E}}\,}}I_{1,\max \{ j,l \}},\\ B_n&= 2 \sum _{i=1}^{n-D} \ \sum _{j=1}^{n-D-i} \ \sum _{k=i+D+j+1}^{i+D+j+m} \ \sum _{l=1}^{n-D-k} {{\,\mathrm{\mathrm {E}}\,}}I_{1,\max \{ j,l \}}\\ C_n&= 2 \sum _{i=1}^{n-D} \ \sum _{j=1}^{n-D-i} \ \sum _{k=i+D+j+m+1}^{n-D} \ \sum _{l=1}^{n-D-k} {{\,\mathrm{\mathrm {E}}\,}}I_{1,j} \cdot {{\,\mathrm{\mathrm {E}}\,}}I_{1,l}, \end{aligned}$$

At the same time, ${{\,\mathrm{\mathrm {E}}\,}}I_{1,l} = P\left( X_1>u_n, M_{1,1+D+l} \le u_n\right) \le p_n C \gamma ^{(D+l)/r_n}$ for some $C>0$ and $0< \gamma < 1$, which follows from Lemma B.1 (Ferro and Segers 2002). We obtain

$$\begin{aligned} A_n&\le 2 (n-D) \sum _{j=1}^{n-D} \sum _{l=j}^{n-D} {{\,\mathrm{\mathrm {E}}\,}}I_{1,l} = 2 (n-D) \sum _{l=1}^{n-D} l {{\,\mathrm{\mathrm {E}}\,}}I_{1,l} \le 2(n-D) \sum _{l=1}^{n-D} l p_n C \gamma ^{\frac{D+l}{r_n}},\\ B_n&\le 4m(n-D) \sum _{j=1}^{n-D} \sum _{l=j}^{n-D} {{\,\mathrm{\mathrm {E}}\,}}I_{1,l} = 4m (n-D) \sum _{l=1}^{n-D} l {{\,\mathrm{\mathrm {E}}\,}}I_{1,l} \le 4m(n-D) \sum _{l=1}^{n-D} l p_n C \gamma ^{\frac{D+l}{r_n}}. \end{aligned}$$

Therefore, by the same arguments as in Ferro and Segers (2002), both $A_n$, $B_n$ are $o(n^2)$.

For $C_n$ let us start with changing the summation order and derive that

$$\begin{aligned} C_n&= 2 \sum _{j=1}^{n-D} \sum _{l=1}^{n-D} {{\,\mathrm{\mathrm {E}}\,}}I_{1,j} {{\,\mathrm{\mathrm {E}}\,}}I_{1,l} \sum _{i=1}^{n-D-j} \sum _{k=i+D+j+m+1}^{n-D-l} 1\\&= \sum _{j=1}^{n-D} \sum _{l=1}^{n-D} (n-2D-l-j-m) (n-2D-l-j-m-1)_+ {{\,\mathrm{\mathrm {E}}\,}}I_{1,j} {{\,\mathrm{\mathrm {E}}\,}}I_{1,l}, \end{aligned}$$

where $a_+$ denotes $\max \{ a,0 \}$. Let us rewrite the sum as an integral

$$\begin{aligned} C_n&= \int \limits _{1}^{n-D} \int \limits _{1}^{n-D} (n-2D-\lceil t \rceil -\lceil s \rceil -m) (n-2D-\lceil t \rceil -\lceil s \rceil -m-1)_+ \\&\quad\times p_n^2 P\left( M_{1,1+D+\lceil t \rceil } \le u_n \mid X_1>u_n\right) P\left( M_{1,1+D+\lceil s \rceil } \le u_n \mid X_1>u_n\right) \,\mathrm {d}s \,\mathrm {d}t\\&= p_n^2 r_n^2 \int \limits _{r_n^{-1}}^{\frac{n-D}{r_n}} \int \limits _{r_n^{-1}}^{\frac{n-D}{r_n}} (n-2D-\lceil r_n t \rceil -\lceil r_n s \rceil -m) (n-2D-\lceil r_n t \rceil -\lceil r_n s \rceil -m-1)_+\\&\quad\times P\left( M_{1,1+D+\lceil r_n t \rceil } \le u_n \mid X_1>u_n\right) P\left( M_{1,1+D+\lceil r_n s \rceil } \le u_n \mid X_1>u_n\right) \,\mathrm {d}s \,\mathrm {d}t. \end{aligned}$$

Let us use Lemma B.1 and Lemma B.2 from Ferro and Segers (2002) again, and by the dominated convergence theorem obtain

$$n^{-2} C_n \rightarrow \tau ^2 \int _0^{\infty } \int _0^{\infty } \theta ^2 \mathrm e^{-\theta \tau t} \mathrm e^{-\theta \tau s} \,\mathrm {d}s \,\mathrm {d}t = 1.$$

The estimator $\widetilde{\theta }_\mathrm{T}$ 

From the previous paragraphs we already know $\widehat{\theta }^\mathrm{BC} \xrightarrow {p}\theta$. It remains to show that the estimator $\widetilde{\theta }_\mathrm{T}$ is consistent as well. Let us rewrite the relation (11) to get $\widetilde{\theta }_\mathrm{T} = g_n\left( \widehat{\theta }^\mathrm{BC} \right)$, where

$$g_n(t) = -\frac{p_n}{2(N-1)} + t \left( 1 - \frac{p_n(N-4)}{2(N-1)} \right) + \frac{t^2 p_n}{2}.$$

From the continuous mapping theorem it follows $g_n\left( \widehat{\theta }^\mathrm{BC} \right) \xrightarrow {p}g_n(\theta )$. Since $p_n = o(1)$ and $N = n p_n (1+o_p(1)) = o_p(n)$, it is clear that $g_n(\theta ) \xrightarrow {p}\theta$ for $n \rightarrow \infty$.

$\square$

1.3 Extremal index estimators from the simulation study

Consider a sequence of interexceedance times $T_1, \dots , T_{N-1}$, and let $T_{(1)} \le \dots \le T_{(N-1)}$ be the corresponding order statistics. In Sect. 4, properties of the truncated estimator $\widehat{\theta }_\mathrm{T}$ and the following competitive estimators based on interexceedance times are assessed.

Intervals estimator

The intervals estimator $\widehat{\theta }_{\text {I}}$ of Ferro and Segers (2003) is

$$\begin{aligned} \widehat{\theta }_{\text {I}} = \left\{ \begin{array}{ll} \min \{ 1,\tilde{\theta } \} &{} \quad \mathrm{if}~\max \{T_i:1\le i\le N-1\}\le 2,\\ \min \{ 1,\tilde{\theta }^{*} \} &{} \quad \mathrm{if}~\max \{T_i:1\le i\le N-1\}> 2, \end{array}\right. \end{aligned}$$

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where

$$\begin{aligned} \tilde{\theta } = \frac{2\left( \sum \limits _{i=1}^{N-1}T_i\right) ^2}{(N-1)\sum \limits _{i=1}^{N-1} T_i^2},\qquad \tilde{\theta }^{*} = \frac{2\left[ \sum \limits _{i=1}^{N-1}(T_i-1)\right] ^2}{(N-1)\sum \limits _{i=1}^{N-1} (T_i-1)(T_i-2)}. \end{aligned}$$

K-gap estimator

The K-gap estimator $\widehat{\theta }_{\text {SD}}$ of Süveges and Davison (2010) is based on the sample $S_i^{(K)} = \max \{ T_i-K;0 \}$, $i=1,\dots , N-1$, for some $K \ge 0$. Considering the limit distribution from Süveges and Davison (2010) and under some additional assumptions on the local dependence, the corresponding log-likelihood function is of the form

$$\begin{aligned} \ell _{\text {SD}}\left( \theta \right) = (N-1-N_C) \log (1-\theta ) + 2N_C\log \theta - \theta \sum \limits _{i=1}^{N-1} \overline{F}(u) S_i^{(K)}, \end{aligned}$$

where $N_C = \sum _{i=1}^{N-1} \varvec{1}_{[T_i > K]}$ is the number of interexceedance times exceeding the value of K. The K-gap estimator is obtained by maximizing the log-likelihood function, i.e.

$$\begin{aligned} \widehat{\theta }_{\text {SD}} = {{\,{\text {arg max}}\,}}_{0 \le \theta \le 1} \ell _{\text {SD}}\left( \theta \right) . \end{aligned}$$

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Censored estimator

The estimator $\widehat{\theta }_{\text {C}}$ of Holešovský and Fusek (2020) is based on censoring of the interexceedance times. For a given $D \ge 0$ assume that $T_{(N-N_D-1)} \le D < T_{(N-N_D)}$ for a positive integer $N_D$. The times $T_{(1)}, \dots , T_{(N-N_D-1)}$ are considered censored, while the times exceeding the value of D are considered observed. The corresponding log-likelihood function is

$$\begin{aligned} \ell _{\text {C}}\left( \theta \right)&= (N-1-N_D) \log \left( 1-\theta e^{-\theta \overline{F}(u) D}\right) + \log \frac{(N-1)!}{(N-1-N_D)!}\\&\quad+ 2 N_D \log \theta -\theta \sum _{i=N-N_D}^{N-1} \overline{F}(u) T_{(i)}. \end{aligned}$$

The censored estimator is obtained by maximizing the log-likelihood function, i.e.

$$\begin{aligned} \widehat{\theta }_{\text {C}} = {{\,{\text {arg max}}\,}}_{0 \le \theta \le 1} \ell _{\text {C}}\left( \theta \right) . \end{aligned}$$

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