Weak Convergence of the Hill Estimator Process

1994 ◽  
pp. 419-431 ◽  
Author(s):  
David M. Mason ◽  
Tatyana S. Turova
Keyword(s):  
Weak Convergence ◽  
Hill Estimator ◽  
The Hill

scholarly journals Rapid variation with remainder and rates of convergence

1990 ◽  
Vol 22 (04) ◽  
pp. 787-801 ◽  
Author(s):  
J. Beirlant ◽  
E. Willekens
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Rates Of Convergence ◽  
Convergence Result ◽  
Hill Estimator ◽  
Rapid Variation ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
The Hill

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

scholarly journals Rapid variation with remainder and rates of convergence

1990 ◽  
Vol 22 (4) ◽  
pp. 787-801 ◽  
Author(s):  
J. Beirlant ◽  
E. Willekens
Keyword(s):  
Weak Convergence ◽  
Asymptotic Behaviour ◽  
Rates Of Convergence ◽  
Convergence Result ◽  
Hill Estimator ◽  
Rapid Variation ◽  
Double Exponential Distribution ◽  
Double Exponential ◽  
The Hill

In this paper, we refine the concept of Γ-variation up to second order, and we give a characterization of this type of asymptotic behaviour. We apply our results to obtain uniform rates of convergence in the weak convergence of renormalised sample maxima to the double exponential distribution. In a second application we derive a rate of convergence result for the Hill estimator.

scholarly journals Smoothing the Hill Estimator

1997 ◽  
Vol 29 (1) ◽  
pp. 271-293 ◽  
Author(s):  
Sidney Resnick ◽  
Cătălin Stărică
Keyword(s):  
Weak Convergence ◽  
Order Statistics ◽  
Empirical Process ◽  
Asymptotic Variance ◽  
Random Variables ◽  
Process Approach ◽  
Hill Estimator ◽  
Direct Result ◽  
The Hill ◽  
Tail Empirical Process

For sequences of i.i.d. random variables whose common tail 1 – F is regularly varying at infinity wtih an unknown index –α < 0, it is well known that the Hill estimator is consistent for α–1 and usually asymptotically normally distributed. However, because the Hill estimator is a function of k = k(n), the number of upper order statistics used and which is only subject to the conditions k →∞, k/n → 0, its use in practice is problematic since there are few reliable guidelines about how to choose k. The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function of k decreases and the successful use of the esimator is made less dependent on the choice of k. A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.

scholarly journals Smoothing the Hill Estimator

1997 ◽  
Vol 29 (01) ◽  
pp. 271-293 ◽  
Author(s):  
Sidney Resnick ◽  
Cătălin Stărică
Keyword(s):  
Weak Convergence ◽  
Order Statistics ◽  
Empirical Process ◽  
Asymptotic Variance ◽  
Random Variables ◽  
Process Approach ◽  
Hill Estimator ◽  
Direct Result ◽  
The Hill ◽  
Tail Empirical Process

For sequences of i.i.d. random variables whose common tail 1 – F is regularly varying at infinity wtih an unknown index –α &lt; 0, it is well known that the Hill estimator is consistent for α–1 and usually asymptotically normally distributed. However, because the Hill estimator is a function of k = k(n), the number of upper order statistics used and which is only subject to the conditions k →∞, k/n → 0, its use in practice is problematic since there are few reliable guidelines about how to choose k. The purpose of this paper is to make the use of the Hill estimator more reliable through an averaging technique which reduces the asymptotic variance. As a direct result the range in which the smoothed estimator varies as a function of k decreases and the successful use of the esimator is made less dependent on the choice of k. A tail empirical process approach is used to prove the weak convergence of a process closely related to the Hill estimator. The smoothed version of the Hill estimator is a functional of the tail empirical process.

scholarly journals Threshold selection in univariate extreme value analysis

Extremes ◽  
2021 ◽  
Author(s):  
Laura Fee Schneider ◽  
Andrea Krajina ◽  
Tatyana Krivobokova
Keyword(s):  
Mean Squared Error ◽  
Extreme Value ◽  
Hill Estimator ◽  
Small Samples ◽  
Extreme Value Analysis ◽  
Threshold Selection ◽  
Value Analysis ◽  
Wide Range ◽  
Asymptotic Mean Squared Error ◽  
The Hill

AbstractThreshold selection plays a key role in various aspects of statistical inference of rare events. In this work, two new threshold selection methods are introduced. The first approach measures the fit of the exponential approximation above a threshold and achieves good performance in small samples. The second method smoothly estimates the asymptotic mean squared error of the Hill estimator and performs consistently well over a wide range of processes. Both methods are analyzed theoretically, compared to existing procedures in an extensive simulation study and applied to a dataset of financial losses, where the underlying extreme value index is assumed to vary over time.

Almost sure convergence of the Hill estimator

1988 ◽  
Vol 104 (2) ◽  
pp. 371-381 ◽  
Author(s):  
Paul Deheuvels ◽  
Erich Haeusler ◽  
David M. Mason
Keyword(s):  
Order Statistics ◽  
Almost Sure Convergence ◽  
Tail Index ◽  
Hill Estimator ◽  
Type Distribution ◽  
The Hill ◽  
Strongly Consistent

AbstractIn this note we characterize those sequences kn such that the Hill estimator of the tail index based on the kn upper order statistics of a sample of size n from a Pareto-type distribution is strongly consistent.

scholarly journals Data-adaptive trimming of the Hill estimator and detection of outliers in the extremes of heavy-tailed data

2019 ◽  
Vol 13 (1) ◽  
pp. 1872-1925
Author(s):  
Shrijita Bhattacharya ◽  
Michael Kallitsis ◽  
Stilian Stoev
Keyword(s):  
Hill Estimator ◽  
Heavy Tailed ◽  
Data Adaptive ◽  
The Hill ◽  
Detection Of Outliers

scholarly journals EXTREME VALUE ANALYSIS WITHOUT THE LARGEST VALUES: WHAT CAN BE DONE?

2019 ◽  
Vol 34 (2) ◽  
pp. 200-220
Author(s):  
Jingjing Zou ◽  
Richard A. Davis ◽  
Gennady Samorodnitsky
Keyword(s):  
Order Statistics ◽  
Extreme Values ◽  
Sample Path ◽  
Estimation Procedure ◽  
Extreme Value ◽  
Hill Estimator ◽  
Extreme Value Analysis ◽  
Value Analysis ◽  
Heavy Tailed ◽  
The Hill

AbstractIn this paper, we are concerned with the analysis of heavy-tailed data when a portion of the extreme values is unavailable. This research was motivated by an analysis of the degree distributions in a large social network. The degree distributions of such networks tend to have power law behavior in the tails. We focus on the Hill estimator, which plays a starring role in heavy-tailed modeling. The Hill estimator for these data exhibited a smooth and increasing “sample path” as a function of the number of upper order statistics used in constructing the estimator. This behavior became more apparent as we artificially removed more of the upper order statistics. Building on this observation we introduce a new version of the Hill estimator. It is a function of the number of the upper order statistics used in the estimation, but also depends on the number of unavailable extreme values. We establish functional convergence of the normalized Hill estimator to a Gaussian process. An estimation procedure is developed based on the limit theory to estimate the number of missing extremes and extreme value parameters including the tail index and the bias of Hill's estimator. We illustrate how this approach works in both simulations and real data examples.

scholarly journals Tail prepivoting for the Hill estimator

2016 ◽  
Vol 49 (19) ◽  
pp. 194004
Author(s):  
Margarida Brito ◽  
Ana Cristina Moreira Freitas ◽  
Jorge Milhazes Freitas
Keyword(s):  
Hill Estimator ◽  
The Hill
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