AStA Advances in Statistical Analysis

, Volume 96, Issue 4, pp 517–541 | Cite as

Simultaneous confidence bands for expectile functions

  • Mengmeng GuoEmail author
  • Wolfgang Karl Härdle
Original Paper


Expectile regression, as a general M smoother, is used to capture the tail behaviour of a distribution. Let (X 1,Y 1),…,(X n ,Y n ) be i.i.d. rvs. Denote by v(x) the unknown τ-expectile regression curve of Y conditional on X, and by v n (x) its kernel smoothing estimator. In this paper, we prove the strong uniform consistency rate of v n (x) under general conditions. Moreover, using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1|v n (x)−v(x)|. According to the asymptotic theory, we construct simultaneous confidence bands around the estimated expectile function. Furthermore, we apply this confidence band to temperature analysis. Taking Berlin and Taipei as an example, we investigate the temperature risk drivers to these two cities.


Expectile regression Consistency rate Simultaneous confidence bands Asymmetric least squares Kernel smoothing 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Institute for Statistics and EconometricsHumboldt-Universität zu BerlinBerlinGermany
  2. 2.C.A.S.E.—Center for Applied Statistics and EconomicsHumboldt-Universität zu BerlinBerlinGermany

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