Threshold selection for extremal index estimation

ABSTRACT

We propose a new threshold selection method for nonparametric estimation of the extremal index of stochastic processes. The discrepancy method was proposed as a data-driven smoothing tool for estimation of a probability density function. Now it is modified to select a threshold parameter of an extremal index estimator. A modification of the discrepancy statistic based on the Cramér–von Mises–Smirnov statistic ${\omega }^2$ is calculated by k largest order statistics instead of an entire sample. Its asymptotic distribution as $k\to \mathrm{\infty }$ is proved to coincide with the ${\omega }^2$-distribution. Its quantiles are used as discrepancy values. The convergence rate of an extremal index estimate coupled with the discrepancy method is derived. The discrepancy method is used as an automatic threshold selection for the intervals and K-gaps estimators. It may be applied to other estimators of the extremal index. The performance of our method is evaluated by simulated and real data examples.

KEYWORDS:

• Cramér–von Mises–Smirnov statistic
• discrepancy method
• extremal index
• nonparametric estimation
• threshold selection

AMS SUBJECT CLASSIFICATION::

• 62G32

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The connection between (1(1) ${\omega }_n^2=n{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{(F_n(x)-F(x))}^2\mathrm{d}F(x)$(1) ) and (2(2) ${\hat{\omega }}_n^2(h)=\sum _{i=1}^n{({\hat{F}}_h(X_{i,n})-\frac{i-0.5}{n})}^2+\frac{1}{12n}$(2) ) can be found in Markovich (2007, p. 81).

2 Theoretically, events $\{T_i=1\}$ are allowed. In practice, such cases related to single inter-arrival times between consecutive exceedances are meaningless.

3 The modification $({\hat{\omega }}_n^2-0.4/n+0.6/n^2)(1+1/n)$ of classical statistic (2(2) ${\hat{\omega }}_n^2(h)=\sum _{i=1}^n{({\hat{F}}_h(X_{i,n})-\frac{i-0.5}{n})}^2+\frac{1}{12n}$(2) ) eliminates the dependence of the percentage points of the C–M–S statistic on the sample size (Stephens 1974). For n>40 it changes the statistic on less than one percent. One can use the modification with regard to ${\tilde{\omega }}_L^2(\hat{\theta })$ for finite L due to the closeness of its distribution to the limit distribution of the C–M–S statistic by Theorem 3.2.

Additional information

Funding

The work of N.M. Markovich in Sections 1, 2, 4 and 5 was supported by the Russian Science Foundation [grant number 22-21-00177]. The work of I. V. Rodionov in Section 3 and proofs in Markovich and Rodionov (2022) was performed at the Institute for Information Transmission Problems (Kharkevich Institute) of the Russian Academy of Sciences with the support of the Russian Science Foundation (grant No. 21-71-00035).