When is tail mean estimation more efficient than tail median? Answers and implications for quantitative risk management

Roger W. Barnard, Kent Pearce and A. Alexandre Trindade ()
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Roger W. Barnard: Texas Tech University
Kent Pearce: Texas Tech University
A. Alexandre Trindade: Texas Tech University

Annals of Operations Research, 2018, vol. 262, issue 1, No 4, 47-65

Abstract: Abstract We investigate the relative efficiency of the empirical “tail median” versus “tail mean” as estimators of location when the data can be modeled by an exponential power distribution (EPD), a flexible family of light-tailed densities. By considering appropriate probabilities so that the quantile of the untruncated EPD (tail median) and mean of the left-truncated EPD (tail mean) coincide, limiting results are established concerning the ratio of asymptotic variances of the corresponding estimators. The most remarkable finding is that in the limit of the right tail, the asymptotic variance of the tail median estimate is approximately 36% larger than that of the tail mean, irrespective of the EPD shape parameter. This discovery has important repercussions for quantitative risk management practice, where the tail median and tail mean correspond to value-at-risk and expected shortfall, respectively. To this effect, a methodology for choosing between the two risk measures that maximizes the precision of the estimate is proposed. From an extreme value theory perspective, analogous results and procedures are discussed also for the case when the data appear to be heavy-tailed.

Keywords: Value-at-risk; Expected shortfall; Generalized quantile function; Exponential power distribution; Extreme value theory; Asymptotic relative efficiency; Rate of convergence (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s10479-017-2547-7

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