## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 1427-1462.

### Maximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time series

#### Abstract

The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution to a sample of block maxima extracted from an observed stretch of a time series. The method is usually validated under two simplifying assumptions: the block maxima should be distributed exactly according to an extreme-value distribution and the sample of block maxima should be independent. Both assumptions are only approximately true. The present paper validates that the simplifying assumptions can in fact be safely made.

For general triangular arrays of block maxima attracted to the Fréchet distribution, consistency and asymptotic normality is established for the maximum likelihood estimator of the parameters of the limiting Fréchet distribution. The results are specialized to the common setting of block maxima extracted from a strictly stationary time series. The case where the underlying random variables are independent and identically distributed is further worked out in detail. The results are illustrated by theoretical examples and Monte Carlo simulations.

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 1427-1462.

**Dates**

Received: March 2016

Revised: September 2016

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980900

**Digital Object Identifier**

doi:10.3150/16-BEJ903

**Mathematical Reviews number (MathSciNet)**

MR3706798

**Zentralblatt MATH identifier**

06778369

**Keywords**

asymptotic normality block maxima method heavy tails maximum likelihood estimation stationary time series triangular arrays

#### Citation

Bücher, Axel; Segers, Johan. Maximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time series. Bernoulli 24 (2018), no. 2, 1427--1462. doi:10.3150/16-BEJ903. https://projecteuclid.org/euclid.bj/1505980900

#### Supplemental materials

- Supplement to “Maximum likelihood estimation for the Fréchet distribution based on block maxima extracted from a time series”. The supplementary material Bücher and Segers [5] contains a lemma on moment convergence of block maxima used in the proof of Theorem 4.2 (in Section C), the proof of Lemma 5.1 (in Section D) and the proofs of auxiliary lemmas from Section B (in Section E). Furthermore, we present additional Monte Carlo simulation results to quantify the finite-sample bias and variance of the maximum likelihood estimator (in Section F).Digital Object Identifier: doi:10.3150/16-BEJ903SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.