TY - JOUR

T1 - When is tail mean estimation more efficient than tail median? Answers and implications for quantitative risk management

AU - Barnard, Roger W.

AU - Pearce, Kent

AU - Trindade, A. Alexandre

N1 - Publisher Copyright:
© 2017, Springer Science+Business Media New York.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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.

AB - 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.

KW - Asymptotic relative efficiency

KW - Expected shortfall

KW - Exponential power distribution

KW - Extreme value theory

KW - Generalized quantile function

KW - Rate of convergence

KW - Value-at-risk

UR - http://www.scopus.com/inward/record.url?scp=85020642728&partnerID=8YFLogxK

U2 - 10.1007/s10479-017-2547-7

DO - 10.1007/s10479-017-2547-7

M3 - Article

AN - SCOPUS:85020642728

VL - 262

SP - 47

EP - 65

JO - Annals of Operations Research

JF - Annals of Operations Research

SN - 0254-5330

IS - 1

ER -