scholarly journals Consistency of the Hill Estimator for Time Series Observed with Measurement Errors

2020 ◽  
Vol 41 (3) ◽  
pp. 421-435 ◽  
Author(s):  
Mihyun Kim ◽  
Piotr Kokoszka
Keyword(s):  
Time Series ◽  
Measurement Errors ◽  
Hill Estimator ◽  
The Hill

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.

Predicting in multivariate incomplete time series. Application of the expectation-maximisation algorithm supplemented by the Newton-Raphson method

2021 ◽  
Vol 68 (1) ◽  
pp. 17-46
Author(s):  
Adam Korczyński
Keyword(s):  
Time Series ◽  
Missing Data ◽  
Em Algorithm ◽  
Measurement Errors ◽  
Parameters Estimation ◽  
Maximum Likelihood Estimates ◽  
Original Algorithm ◽  
Expectation Maximisation ◽  
Newton Raphson ◽  
Raphson Method

Statistical practice requires various imperfections resulting from the nature of data to be addressed. Data containing different types of measurement errors and irregularities, such as missing observations, have to be modelled. The study presented in the paper concerns the application of the expectation-maximisation (EM) algorithm to calculate maximum likelihood estimates, using an autoregressive model as an example. The model allows describing a process observed only through measurements with certain level of precision and through more than one data series. The studied series are affected by a measurement error and interrupted in some time periods, which causes the information for parameters estimation and later for prediction to be less precise. The presented technique aims to compensate for missing data in time series. The missing data appear in the form of breaks in the source of the signal. The adjustment has been performed by the EM algorithm to a hybrid version, supplemented by the Newton-Raphson method. This technique allows the estimation of more complex models. The formulation of the substantive model of an autoregressive process affected by noise is outlined, as well as the adjustment introduced to overcome the issue of missing data. The extended version of the algorithm has been verified using sampled data from a model serving as an example for the examined process. The verification demonstrated that the joint EM and Newton-Raphson algorithms converged with a relatively small number of iterations and resulted in the restoration of the information lost due to missing data, providing more accurate predictions than the original algorithm. The study also features an example of the application of the supplemented algorithm to some empirical data (in the calculation of a forecasted demand for newspapers).

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 Technical note: Analytical sensitivity analysis and uncertainty estimation of baseflow index calculated by a two-component hydrograph separation method with conductivity as a tracer

2019 ◽  
Vol 23 (2) ◽  
pp. 1103-1112 ◽  
Author(s):  
Weifei Yang ◽  
Changlai Xiao ◽  
Xiujuan Liang
Keyword(s):  
Time Series ◽  
Sensitivity Analysis ◽  
Measurement Errors ◽  
Uncertainty Estimation ◽  
Separation Method ◽  
Analytical Sensitivity ◽  
Hydrograph Separation ◽  
Time Step ◽  
Sensitivity Indices ◽  
Two Component

Abstract. The two-component hydrograph separation method with conductivity as a tracer is favored by hydrologists owing to its low cost and easy application. This study analyzes the sensitivity of the baseflow index (BFI, long-term ratio of baseflow to streamflow) calculated using this method to errors or uncertainties in two parameters (BFC, the conductivity of baseflow, and ROC, the conductivity of surface runoff) and two variables (yk, streamflow, and SCk, specific conductance of streamflow, where k is the time step) and then estimates the uncertainty in BFI. The analysis shows that for time series longer than 365 days, random measurement errors in yk or SCk will cancel each other out, and their influence on BFI can be neglected. An uncertainty estimation method of BFI is derived on the basis of the sensitivity analysis. Representative sensitivity indices (the ratio of the relative error in BFI to that of BFC or ROC) and BFI′ uncertainties are determined by applying the resulting equations to 24 watersheds in the US. These dimensionless sensitivity indices can well express the propagation of errors or uncertainties in BFC or ROC into BFI. The results indicate that BFI is more sensitive to BFC, and the conductivity two-component hydrograph separation method may be more suitable for the long time series in a small watershed. When the mutual offset of the measurement errors in conductivity and streamflow is considered, the uncertainty in BFI is reduced by half.

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.

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