Asymptotic properties of the partition function and applications in tail index inference of heavy-tailed data

This paper considers stationary autoregressive (AR) models with heavy-tailed, general GARCH (G-GARCH) or augmented GARCH noises. Limit theory for the least squares estimator (LSE) of autoregression coefficient ρ=ρn is derived uniformly over stationary values in [0,1), focusing on ρn→1 as sample size n tends to infinity. For tail index α∈(0,4) of G-GARCH innovations, asymptotic distributions of the LSEs are established, which are involved with the stable distribution. The convergence rate of the LSE depends on 1−ρn2, but no condition on the rate of ρn is required. It is shown that, for the tail index α∈(0,2), the LSE is inconsistent, for α=2, logn/(1−ρn2)-consistent, and for α∈(2,4), n1−2/α/(1−ρn2)-consistent. Proofs are based on the point process and the asymptotic properties in AR models with G-GARCH errors. However, this present work provides a bridge between pure stationary and unit-root processes. This paper extends the existing uniform limit theory with three issues: the errors have conditional heteroscedastic variance; the errors are heavy-tailed with tail index α∈(0,4); and no restriction on the rate of ρn is necessary.

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A Method for Confidence Intervals of High Quantiles

Entropy ◽

10.3390/e23010070 ◽

2021 ◽

Vol 23 (1) ◽

pp. 70

Author(s):

Mei Ling Huang ◽

Xiang Raney-Yan

Keyword(s):

Confidence Intervals ◽

Computational Algorithm ◽

Geometric Mean ◽

Asymptotic Properties ◽

Quantile Estimation ◽

Heavy Tailed Distributions ◽

High Quantiles ◽

High Quantile ◽

Heavy Tailed ◽

Infinite Moments

The high quantile estimation of heavy tailed distributions has many important applications. There are theoretical difficulties in studying heavy tailed distributions since they often have infinite moments. There are also bias issues with the existing methods of confidence intervals (CIs) of high quantiles. This paper proposes a new estimator for high quantiles based on the geometric mean. The new estimator has good asymptotic properties as well as it provides a computational algorithm for estimating confidence intervals of high quantiles. The new estimator avoids difficulties, improves efficiency and reduces bias. Comparisons of efficiencies and biases of the new estimator relative to existing estimators are studied. The theoretical are confirmed through Monte Carlo simulations. Finally, the applications on two real-world examples are provided.

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From partitions to Hodge numbers of Hilbert schemes of surfaces

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2018.0435 ◽

2019 ◽

Vol 378 (2163) ◽

pp. 20180435 ◽

Cited By ~ 1

Author(s):

Nate Gillman ◽

Xavier Gonzalez ◽

Ken Ono ◽

Larry Rolen ◽

Matthew Schoenbauer

Keyword(s):

Partition Function ◽

Royal Society ◽

Asymptotic Properties ◽

Circle Method ◽

100Th Anniversary ◽

Hilbert Schemes ◽

Hodge Numbers ◽

Projective Surfaces

We celebrate the 100th anniversary of Srinivasa Ramanujan's election as a Fellow of the Royal Society, which was largely based on his work with G. H. Hardy on the asymptotic properties of the partition function. After recalling this revolutionary work, marking the birth of the ‘circle method’, we present a contemporary example of its legacy in topology. We deduce the equidistribution of Hodge numbers for Hilbert schemes of suitable smooth projective surfaces. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

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Estimating Systematic Risk under Extremely Adverse Market Conditions*

Journal of Financial Econometrics ◽

10.1093/jjfinec/nbx033 ◽

2017 ◽

Vol 17 (3) ◽

pp. 432-461 ◽

Cited By ~ 2

Author(s):

Maarten R C van Oordt ◽

Chen Zhou

Keyword(s):

Mean Squared Error ◽

Explanatory Variable ◽

Asymptotic Properties ◽

Estimation Method ◽

Tail Dependence ◽

Stock Portfolios ◽

Squared Error ◽

Regression Approach ◽

Model Coefficient ◽

Heavy Tailed

AbstractThis paper considers the problem of estimating a linear model between two heavy-tailed variables if the explanatory variable has an extremely low (or high) value. We propose an estimator for the model coefficient by exploiting the tail dependence between the two variables and prove its asymptotic properties. Simulations show that our estimation method yields a lower mean-squared error than regressions conditional on tail observations. In an empirical application, we illustrate the better performance of our approach relative to the conditional regression approach in projecting the losses of industry-specific stock portfolios in the event of a market crash.

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Estimation of the tail-index in a conditional location-scale family of heavy-tailed distributions

Dependence Modeling ◽

10.1515/demo-2019-0021 ◽

2019 ◽

Vol 7 (1) ◽

pp. 394-417

Author(s):

Aboubacrène Ag Ahmad ◽

El Hadji Deme ◽

Aliou Diop ◽

Stéphane Girard

Keyword(s):

Asymptotic Properties ◽

Real Data ◽

Scale Model ◽

Extreme Value Index ◽

Finite Sample ◽

Scale Functions ◽

Nonparametric Estimators ◽

Heavy Tailed Distributions ◽

Finite Sample Properties ◽

Heavy Tailed

AbstractWe introduce a location-scale model for conditional heavy-tailed distributions when the covariate is deterministic. First, nonparametric estimators of the location and scale functions are introduced. Second, an estimator of the conditional extreme-value index is derived. The asymptotic properties of the estimators are established under mild assumptions and their finite sample properties are illustrated both on simulated and real data.

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Maxima of Sums of Heavy-Tailed Random Variables

Astin Bulletin ◽

10.2143/ast.32.1.1013 ◽

2002 ◽

Vol 32 (1) ◽

pp. 43-55 ◽

Cited By ~ 37

Author(s):

K.W. Ng ◽

Q.H. Tang ◽

H. Yang

Keyword(s):

Asymptotic Properties ◽

Random Variables ◽

Independent Random Variables ◽

Partial Sums ◽

Tail Probabilities ◽

Large Classes ◽

Heavy Tailed Distributions ◽

Heavy Tailed ◽

The Individual ◽

Risk Processes

AbstractIn this paper, we investigate asymptotic properties of the tail probabilities of the maxima of partial sums of independent random variables. For some large classes of heavy-tailed distributions, we show that the tail probabilities of the maxima of the partial sums asymptotically equal to the sum of the tail probabilities of the individual random variables. Then we partially extend the result to the case of random sums. Applications to some commonly used risk processes are proposed. All heavy-tailed distributions involved in this paper are supposed on the whole real line.

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Characterizing Heavy-Tailed Distributions Induced by Retransmissions

Advances in Applied Probability ◽

10.1017/s0001867800006212 ◽

2013 ◽

Vol 45 (01) ◽

pp. 106-138 ◽

Cited By ~ 3

Author(s):

Predrag R. Jelenković ◽

Jian Tan

Keyword(s):

Heavy Tails ◽

Functional Relationship ◽

Asymptotic Properties ◽

Asymptotic Equivalence ◽

Total Transmission ◽

Heavy Tailed Distributions ◽

Generic Data ◽

Different Types ◽

Heavy Tailed ◽

Completion Times

Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel availability dynamic is modeled as an independent and identically distributed sequence {A, A i } i≥1 that is independent of L. During each period of time that the channel becomes available, say A i , we attempt to transmit the data unit. If L <A i , the transmission is considered successful; otherwise, we wait for the next available period A i+1 and attempt to retransmit the data from the beginning. We investigate the asymptotic properties of the number of retransmissions N and the total transmission time T until the data is successfully transmitted. In the context of studying the completion times in systems with failures where jobs restart from the beginning, it was first recognized by Fiorini, Sheahan and Lipsky (2005) and Sheahan, Lipsky, Fiorini and Asmussen (2006) that this model results in power-law and, in general, heavy-tailed delays. The main objective of this paper is to uncover the detailed structure of this class of heavy-tailed distributions induced by retransmissions. More precisely, we study how the functional relationship ℙ[L>x]-1 ≈ Φ (ℙ[A>x]-1) impacts the distributions of N and T; the approximation ‘≈’ will be appropriately defined in the paper based on the context. Depending on the growth rate of Φ(·), we discover several criticality points that separate classes of different functional behaviors of the distribution of N. For example, we show that if log(Φ(n)) is slowly varying then log(1/ℙ[N>n]) is essentially slowly varying as well. Interestingly, if log(Φ(n)) grows slower than e√(logn) then we have the asymptotic equivalence log(ℙ[N>n]) ≈ - log(Φ(n)). However, if log(Φ(n)) grows faster than e√(logn), this asymptotic equivalence does not hold and admits a different functional form. Similarly, different types of distributional behavior are shown for moderately heavy tails (Weibull distributions) where log(ℙ[N>n]) ≈ -(log Φ(n))1/(β+1), assuming that log Φ(n) ≈ nβ, as well as the nearly exponential ones of the form log(ℙ[N>n]) ≈ -n/(log n)1/γ, γ>0, when Φ(·) grows faster than two exponential scales log log (Φ(n)) ≈ n γ.

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