Weak convergence of a pseudo maximum likelihood estimator for the extremal index

Cluster indices describe extremal behaviour of stationary time series. We consider their sliding blocks estimators. Using a modern theory of multivariate, regularly varying time series, we obtain central limit theorems under conditions that can be easily verified for a large class of models. In particular, we show that in the Peaks-Over-Threshold framework, sliding and disjoint blocks estimators have the same limiting variance.

Section: Existing Resultsmentioning

confidence: 99%

Estimation of cluster functionals for regularly varying time series: sliding blocks estimators

Cissokho

Kulik

2021

“…Moreover, the sample of sliding block maxima carries more information than the sample of disjoint block maxima, which suggests the possibility of more accurate inference. Robert et al (2009), Northrop (2015) and Berghaus and Bücher (2016) applied this idea to the estimation of the extremal index, a summary measure for the strength of serial dependence between extremes. They found that estimators based on sliding blocks were indeed more efficient than their counterparts based on disjoint blocks.…”

Section: Introductionmentioning

confidence: 99%

Inference for heavy tailed stationary time series based on sliding blocks

Bücher¹,

Segers²

2018

Self Cite

The block maxima method in extreme value theory consists of fitting an extreme value distribution to a sample of block maxima extracted from a time series. Traditionally, the maxima are taken over disjoint blocks of observations. Alternatively, the blocks can be chosen to slide through the observation period, yielding a larger number of overlapping blocks. Inference based on sliding blocks is found to be more efficient than inference based on disjoint blocks. The asymptotic variance of the maximum likelihood estimator of the Fréchet shape parameter is reduced by more than 18%. Interestingly, the amount of the efficiency gain is the same whatever the serial dependence of the underlying time series: as for disjoint blocks, the asymptotic distribution depends on the serial dependence only through the sequence of scaling constants. The findings are illustrated by simulation experiments and are applied to the estimation of high return levels of the daily log-returns of the Standard & Poor's 500 stock market index.

“…to, e.g., denote the disjoint blocks CFG-estimator based on theŶ ni and the sliding blocks madogram-estimator based on theẐ ni , respectively. Note that the four estimators of the formθ yn m,R,1 ,θ zn m,R,1 , m ∈ {db, sb}, are the (pseudo) maximum likelihood (PML) estimators considered in [3].…”

Section: Definition Of the Estimatorsmentioning

confidence: 99%

“…The proposed estimators typically depend on two or, arguably preferable, one parameter to be chosen by the statistician. The present paper is on a class of method of moments estimators (based on the blocks method), which improves upon a recent estimator proposed by Paul Northrop in [22] and analyzed theoretically in [3]. Some notations and assumptions are necessary for the motivation of the new class of estimators.…”

Section: Introductionmentioning

confidence: 99%

“…The sequence is assumed to have an extremal index θ ∈ (0, 1], i.e., for any τ > 0, there exists a sequence where F ← (z) = inf{y ∈ R : F (y) ≥ z} denotes the generalized inverse of F evaluated at z ∈ R. In other words, for large block length b, Y 1:b approximately follows an exponential distribution with parameter θ, denoted by Exp(θ) throughout. This inspired [22] and [3] to estimate θ by the maximum likelihood estimator for the exponential distribution; see Section 2 below for details on how to arrive at an observable (rank-based) approximate sample from the Exp(θ)-distribution based on an observed stretch of length n from the time series (X s ) s∈N .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Method of moments estimators for the extremal index of a stationary time series

Bücher¹,

Jennessen²

2020

Self Cite

The extremal index θ, a number in the interval [0, 1], is known to be a measure of primal importance for analyzing the extremes of a stationary time series. New rank-based estimators for θ are proposed which rely on the construction of approximate samples from the exponential distribution with parameter θ that is then to be fitted via the method of moments. The new estimators are analyzed both theoretically as well as empirically through a large-scale simulation study. In specific scenarios, in particular for time series models with θ ≈ 1, they are found to be superior to recent competitors from the literature.