On Functions of Order Statistics for Mixing Processes

Mehra, K. L.; Rao, M. Sudhakara

doi:10.1214/aos/1176343188

Cited by 11 publications

(3 citation statements)

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“…Tsukahara (2013) obtained the same result. A similar result can also be derived from an earlier work by Mehra and Rao (1975), but under a faster decay of the mixing coefficients and under an additional assumption on the dependence structure. We emphasize that the method of proof proposed by Beutner and Zähle is rather flexible, because it easily extends to other weak and strong dependence concepts and other risk measures (see Zähle 2010, 2016;Krätschmer et al 2013;Krätschmer and Zähle 2017).…”

Section: Introductionsupporting

confidence: 81%

Bootstrapping Average Value at Risk of Single and Collective Risks

Beutner

Zähle

2018

Risks

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Almost sure bootstrap consistency of the blockwise bootstrap for the Average Value at Risk of single risks is established for strictly stationary β -mixing observations. Moreover, almost sure bootstrap consistency of a multiplier bootstrap for the Average Value at Risk of collective risks is established for independent observations. The main results rely on a new functional delta-method for the almost sure bootstrap of uniformly quasi-Hadamard differentiable statistical functionals, to be presented here. The latter seems to be interesting in its own right.

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Section: Introductionsupporting

confidence: 81%

Bootstrapping Average Value at Risk of Single and Collective Risks

Beutner

Zähle

2018

Risks

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“…Using the same argument as Mehra and Rao (1975), we can show that I T,2 , I T,3 , I T,4 , and I T,5 converges to zero with probability 1.…”

Section: Appendicesmentioning

confidence: 84%

Testing Distributional Assumptions: A L-Moment Approach

Chu

Salmon

2008

SSRN Journal

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“…Nelson, 1990;Carrasco and Chen, 2002), hence y [αn] = q α + O p (1/n 1/2 ). See, e.g., Mehra and Rao (1975). Define…”

Section: Profile-weighted Tail-trimmed Esmentioning

confidence: 99%

GEL estimation for heavy-tailed GARCH models with robust empirical likelihood inference

Hill

Prokhorov

2016

Journal of Econometrics

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We construct a Generalized Empirical Likelihood estimator for a GARCH(1,1) model with a possibly heavy tailed error. The estimator imbeds tail-trimmed estimating equations allowing for over-identifying conditions, asymptotic normality, efficiency and empirical likelihood based confidence regions for very heavy-tailed random volatility data. We show the implied probabilities from the tail-trimmed Continuously Updated Estimator elevate weight for usable large values, assign large but not maximum weight to extreme observations, and give the lowest weight to nonleverage points. We derive a higher order expansion for GEL with imbedded tailtrimming (GELITT), which reveals higher order bias and efficiency properties, available when the GARCH error has a finite second moment. Higher order asymptotics for GEL without tail-trimming requires the error to have moments of substantially higher order. We use first order asymptotics and higher order bias to justify the choice of the number of trimmed observations in any given sample. We also present robust versions of Generalized Empirical Likelihood Ratio, Wald, and Lagrange Multiplier tests, and an efficient and heavy tail robust moment estimator with an application to expected shortfall estimation. Finally, we present a broad simulation study for GEL and GELITT, and demonstrate profile weighted expected shortfall for the Russian Ruble -US Dollar exchange rate. We show that tail-trimmed CUE-GMM dominates other estimators in terms of bias, mse and approximate normality.Key words and phrases: GEL, GARCH, tail trimming, heavy tails, robust inference, efficient moment estimation, expected shortfall, Russian Ruble. AbstractWe construct a Generalized Empirical Likelihood estimator for a GARCH(1,1) model with a possibly heavy tailed error. The estimator imbeds tail-trimmed estimating equations allowing for over-identifying conditions, asymptotic normality, efficiency and empirical likelihood based confidence regions for very heavy-tailed random volatility data. We show the implied probabilities from the tail-trimmed Continuously Updated Estimator elevate weight for usable large values, assign large but not maximum weight to extreme observations, and give the lowest weight to non-leverage points. We derive a higher order expansion for GEL with imbedded tail-trimming (GELITT), which reveals higher order bias and efficiency properties, available when the GARCH error has a finite second moment. Higher order asymptotics for GEL without tail-trimming requires the error to have moments of substantially higher order. We use first order asymptotics and higher order bias to justify the choice of the number of trimmed observations in any given sample. We also present robust versions of Generalized Empirical Likelihood Ratio, Wald, and Lagrange Multiplier tests, and an efficient and heavy tail robust moment estimator with an application to expected shortfall estimation. Finally, we present a broad simulation study for GEL and GELITT, and demonstrate profile weighted expected shortfall for the ...

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On Functions of Order Statistics for Mixing Processes

Cited by 11 publications

References 0 publications

Bootstrapping Average Value at Risk of Single and Collective Risks

Bootstrapping Average Value at Risk of Single and Collective Risks

Testing Distributional Assumptions: A L-Moment Approach

GEL estimation for heavy-tailed GARCH models with robust empirical likelihood inference

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