scholarly journals Consistency of Hill's estimator for dependent data

1995 ◽  
Vol 32 (1) ◽  
pp. 139-167 ◽  
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.

scholarly journals Consistency of Hill's estimator for dependent data

1995 ◽  
Vol 32 (01) ◽  
pp. 139-167 ◽  
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 &lt; 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.

On the asymptotic normality of Hill's estimator

1995 ◽  
Vol 118 (2) ◽  
pp. 375-382 ◽  
Author(s):  
Sándor Csörgő ◽  
László Viharos
Keyword(s):  
Distribution Function ◽  
Asymptotic Normality ◽  
Order Statistics ◽  
Random Variables ◽  
Fixed Number ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Hill’S Estimator ◽  
Common Distribution ◽  
Common Distribution Function

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .

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 Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

2021 ◽  
Vol 9 (1) ◽  
pp. 424-438
Author(s):  
Guillaume Boglioni Beaulieu ◽  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Marginal Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Sample Mean ◽  
Non Gaussian ◽  
Mutually Independent ◽  
Specific Sequences

Abstract We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of F). This allows us to illustrate the extent of the ‘failure’ of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer K, new K-tuplewise independent sequences that are not mutually independent. For K [four.tf], it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.

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.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

Methodology of Calculation the Terminal Settling Velocity Distribution of Irregular Particles for High Values of the Reynold’s Number/Metodologia Wyliczania Rozkładu Granicznej Prędkości Opadania Ziaren Nieregularnych Dla Wysokich Wartości Liczb Reynoldsa

2014 ◽  
Vol 59 (2) ◽  
pp. 553-562 ◽  
Author(s):  
Agnieszka Surowiak ◽  
Marian Brożek
Keyword(s):  
Velocity Distribution ◽  
Settling Velocity ◽  
Random Variables ◽  
Independent Random Variables ◽  
Calculation Algorithm ◽  
Shape Coefficient ◽  
Geometrical Properties ◽  
Size And Shape ◽  
Physical Density ◽  
Settling Velocity Distribution

Abstract Settling velocity of particles, which is the main parameter of jig separation, is affected by physical (density) and the geometrical properties (size and shape) of particles. The authors worked out a calculation algorithm of particles settling velocity distribution for irregular particles assuming that the density of particles, their size and shape constitute independent random variables of fixed distributions. Applying theorems of probability, concerning distributions function of random variables, the authors present general formula of probability density function of settling velocity irregular particles for the turbulent motion. The distributions of settling velocity of irregular particles were calculated utilizing industrial sample. The measurements were executed and the histograms of distributions of volume and dynamic shape coefficient, were drawn. The separation accuracy was measured by the change of process imperfection of irregular particles in relation to spherical ones, resulting from the distribution of particles settling velocity.

Asymptotic for LS estimators in the EV regression model for dependent errors

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4845-4856
Author(s):  
Konrad Furmańczyk
Keyword(s):  
Regression Model ◽  
Functional Dependence ◽  
Dependent Data ◽  
Errors In Variables ◽  
Mixing Conditions ◽  
Asymptotic Results ◽  
Dependence Measure ◽  
Wide Range ◽  
Ev Regression Model ◽  
Linear And Nonlinear

We study consistency and asymptotic normality of LS estimators in the EV (errors in variables) regression model under weak dependent errors that involve a wide range of linear and nonlinear time series. In our investigations we use a functional dependence measure of Wu [16]. Our results without mixing conditions complete the known asymptotic results for independent and dependent data obtained by Miao et al. [7]-[10].

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