Weak properties and robustness of t-Hill estimators

Abstract

We describe a novel method of heavy tails estimation based on transformed score (t-score). Based on a new score moment method we derive the t-Hill estimator, which estimates the extreme value index of a distribution function with regularly varying tail. t-Hill estimator is distribution sensitive, thus it differs in e.g. Pareto and log-gamma case. Here, we study both forms of the estimator, i.e. t-Hill and t-lgHill. For both estimators we prove weak consistency in moving average settings as well as the asymptotic normality of t-lgHill estimator in iid setting. In cases of contamination with heavier tails than the tail of original sample, t-Hill outperforms several robust tail estimators, especially in small samples. A simulation study emphasizes the fact that the level of contamination is playing a crucial role. The larger the contamination, the better are the t-score moment estimates. The reason for this is the bounded t-score of heavy-tailed distributions (and, consequently, bounded influence functions of the estimators). We illustrate the developed methodology on a small sample data set of stake measurements from Guanaco glacier in Chile.

Statistical Challenges of Administrative and Transaction Data

“…Many data are aggregated by an inappropriate statistic (e.g. the arithmetic mean) which fails to address extremes severely (Beran et al, 2014;Stehlík et al, 2016a, b;Jordanova et al, 2016). Aggregation data statistics can have very different properties for sparse and dense data (Stehlík, 2016), which in the case of for example complicated medical images needs a very careful choice of method (Hermann et al, 2015).…”

Priority statement and some properties of t-lgHill estimator

“…Motivated by the fact that asymptotic bias and variance of existing estimators of ρ are high, the authors of Gomes et al (2000) considered the estimator γ G k n ,n (−1) ≡ H L k n ,n . We observed that Theorem 1 in Jordanova et al (2016) is a direct consequence of the distributional representation of H L k n ,n provided in Gomes et al (2000), see (2.11) therein. We also recognize a numerical typo in variance in Theorem 1 (Jordanova et al 2016), where the asymptotic variance 8 should be replaced by 5.…”

“…We have found that the estimator H L k n ,n was firstly developed in Gomes et al (2000), see (2.10) therein. We have not been aware of this fact in time of publication of Jordanova et al 2016, where we developed the estimator H L k n ,n , see Theorem 1 in Jordanova et al (2016).…”

“…The main purpose of this priority letter is to demonstrate that the estimator H L k n ,n was introduced in Gomes et al (2000) and Jordanova et al (2016) by applying different approaches. In Gomes et al (2000) the estimator H L k n ,n has been built by assuming the following second regular variation condition:…”

Estimation for heavy tailed moving average process

“…Other consistent estimator of α for the moving average process (4) is the t-lgHill estimator introduced in Jordanova et al [14]. It is given by:…”