Smoothing the Hill Estimator

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.

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Almost sure convergence of the Hill estimator

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100065531 ◽

1988 ◽

Vol 104 (2) ◽

pp. 371-381 ◽

Cited By ~ 69

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.

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A point-process approach to almost-sure behaviour of record values and order statistics

Advances in Applied Probability ◽

10.1017/s0001867800048564 ◽

1996 ◽

Vol 28 (02) ◽

pp. 426-462 ◽

Cited By ~ 3

Author(s):

Charles M. Goldie ◽

Ross A. Maller

Keyword(s):

Order Statistics ◽

Point Process ◽

Relative Stability ◽

Asymptotic Properties ◽

Random Variables ◽

Record Values ◽

Process Approach ◽

Integral Conditions ◽

Comprehensive Investigation ◽

Trimmed Sums

Point-process and other techniques are used to make a comprehensive investigation of the almost-sure behaviour of partial maxima(the rth largest among a sample ofni.i.d. random variables), partial record valuesand differences and quotients involving them. In particular, we obtain characterizations of such asymptotic properties asa.s. for some finite constantc, ora.s. for some constantcin [0,∞], which tell us, in various ways, how quickly the sequences increase. These characterizations take the form of integral conditions on the tail ofF,which furthermore characterize such properties as stability and relative stability of the sequence of maxima. We also develop their relation to the large-sample behaviour of trimmed sums, and discuss some statistical applications.

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

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964818000542 ◽

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.

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Rapid variation with remainder and rates of convergence

Advances in Applied Probability ◽

10.1017/s0001867800023132 ◽

1990 ◽

Vol 22 (04) ◽

pp. 787-801 ◽

Cited By ~ 2

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.

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Rapid variation with remainder and rates of convergence

Advances in Applied Probability ◽

10.2307/1427562 ◽

1990 ◽

Vol 22 (4) ◽

pp. 787-801 ◽

Cited By ~ 3

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.

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A tail empirical process approach to some nonstandard laws of the iterated logarithm

Journal of Theoretical Probability ◽

10.1007/bf01046994 ◽

1991 ◽

Vol 4 (1) ◽

pp. 53-85 ◽

Cited By ~ 14

Author(s):

Paul Deheuvels ◽

David M. Mason

Keyword(s):

Empirical Process ◽

Process Approach ◽

Iterated Logarithm ◽

Tail Empirical Process

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A point-process approach to almost-sure behaviour of record values and order statistics

Advances in Applied Probability ◽

10.2307/1428066 ◽

1996 ◽

Vol 28 (2) ◽

pp. 426-462 ◽

Cited By ~ 6

Author(s):

Charles M. Goldie ◽

Ross A. Maller

Keyword(s):

Order Statistics ◽

Point Process ◽

Relative Stability ◽

Asymptotic Properties ◽

Random Variables ◽

Record Values ◽

Process Approach ◽

Integral Conditions ◽

Comprehensive Investigation ◽

Trimmed Sums

Point-process and other techniques are used to make a comprehensive investigation of the almost-sure behaviour of partial maxima (the rth largest among a sample of n i.i.d. random variables), partial record values and differences and quotients involving them. In particular, we obtain characterizations of such asymptotic properties as a.s. for some finite constant c, or a.s. for some constant c in [0,∞], which tell us, in various ways, how quickly the sequences increase. These characterizations take the form of integral conditions on the tail of F, which furthermore characterize such properties as stability and relative stability of the sequence of maxima. We also develop their relation to the large-sample behaviour of trimmed sums, and discuss some statistical applications.

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Consistency of Hill's estimator for dependent data

Journal of Applied Probability ◽

10.1017/s0021900200102621 ◽

1995 ◽

Vol 32 (01) ◽

pp. 139-167 ◽

Cited By ~ 14

Author(s):

Sidney Resnick ◽

Cătălin Stărică

Keyword(s):

Infinite Order ◽

Empirical Process ◽

Marginal Distribution ◽

Random Variables ◽

Ar Model ◽

Independent Random Variables ◽

Dependent Data ◽

Mixing Conditions ◽

Hill’S Estimator ◽

Tail Empirical Process

Consider a sequence of possibly dependent random variables having the same marginal distribution F, whose tail 1−F is regularly varying at infinity with an unknown index − α < 0 which is to be estimated. For i.i.d. data or for dependent sequences with the same marginal satisfying mixing conditions, it is well known that Hill's estimator is consistent for α−1 and asymptotically normally distributed. The purpose of this paper is to emphasize the central role played by the tail empirical process for the problem of consistency. This approach allows us to easily prove Hill's estimator is consistent for infinite order moving averages of independent random variables. Our method also suffices to prove that, for the case of an AR model, the unknown index can be estimated using the residuals generated by the estimation of the autoregressive parameters.

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Limit laws for kth order statistics from conditionally mixing arrays of random variables

Journal of Applied Probability ◽

10.1017/s0021900200111830 ◽

1986 ◽

Vol 23 (03) ◽

pp. 679-687

Author(s):

W. Dziubdziela

Keyword(s):

Weak Convergence ◽

Order Statistics ◽

Poisson Distribution ◽

Sufficient Conditions ◽

Random Variables ◽

Compound Poisson Distribution ◽

Limit Laws ◽

Compound Poisson ◽

Mixed Compound

We present sufficient conditions for the weak convergence of the distributions of thekth order statistics from a conditionally mixing array of random variables to limit laws which are represented in terms of a mixed compound Poisson distribution.

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