## Electronic Journal of Statistics

### Adaptive confidence sets for kink estimation

#### Abstract

We consider estimation of the location and the height of the jump in the $\gamma$-th derivative - a kink of order $\gamma$ - of a regression curve, which is assumed to be Hölder smooth of order $s\geq \gamma +1$ away from the kink. Optimal convergence rates as well as the joint asymptotic normal distribution of estimators based on the zero-crossing-time technique are established. Further, we construct joint as well as marginal asymptotic confidence sets for these parameters which are honest and adaptive with respect to the smoothness parameter $s$ over subsets of the Hölder classes. The finite-sample performance is investigated in a simulation study, and a real data illustration is given to a series of annual global surface temperatures.

#### Article information

Source
Electron. J. Statist., Volume 13, Number 1 (2019), 1523-1579.

Dates
First available in Project Euclid: 16 April 2019

https://projecteuclid.org/euclid.ejs/1555380050

Digital Object Identifier
doi:10.1214/19-EJS1555

Mathematical Reviews number (MathSciNet)
MR3939304

Zentralblatt MATH identifier
07056157

#### Citation

Bengs, Viktor; Holzmann, Hajo. Adaptive confidence sets for kink estimation. Electron. J. Statist. 13 (2019), no. 1, 1523--1579. doi:10.1214/19-EJS1555. https://projecteuclid.org/euclid.ejs/1555380050

#### References

• Bengs, V. and H. Holzmann (2019). Uniform approximation in classical weak convergence theory., https://arxiv.org/abs/1903.09864. [Online; accessed 26-March-2019].
• Bull, A. D. (2012). Honest adaptive confidence bands and self-similar functions., Electronic Journal of Statistics 6, 1490–1516.
• Cai, T. and M. Low (2004). An adaptation theory for nonparametric confidence intervals., The Annals of Statistics 32 (5), 1805–1840.
• Card, D., D. S. Lee, Z. Pei, and A. Weber (2015). Inference on causal effects in a generalized regression kink design., Econometrica 83 (6), 2453–2483.
• Carlstein, E. G., H.-G. Müller, and D. Siegmund (1994). Change-point problems., IMS.
• Cheng, M. and M. Raimondo (2008). Kernel methods for optimal change-points estimation in derivatives., Journal of Computational and Graphical Statistics 17 (1), 56–75.
• Dette, H., A. Munk, and T. Wagner (1998). Estimating the variance in nonparametric regression – what is a reasonable choice?, Journal of the Royal Statistical Society: Series B (Statistical Methodology) 60 (4), 751–764.
• Eubank, R. and P. Speckman (1994). Nonparametric estimation of functions with jump discontinuities., Lecture Notes-Monograph Series , 130–144.
• Frick, K., A. Munk, and H. Sieling (2014). Multiscale change point inference., Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76 (3), 495–580.
• Giné, E. and R. Nickl (2010). Confidence bands in density estimation., The Annals of Statistics 38 (2), 1122–1170.
• Goldenshluger, A., A. Juditsky, A. Tsybakov, and A. Zeevi (2008a). Change-point estimation from indirect observations. 1. minimax complexity., Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 44 (5), 787–818.
• Goldenshluger, A., A. Juditsky, A. Tsybakov, and A. Zeevi (2008b). Change-point estimation from indirect observations. 2. adaptation., Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 44 (5), 819–836.
• Goldenshluger, A., A. Tsybakov, and A. Zeevi (2006). Optimal change-point estimation from indirect observations., The Annals of Statistics 34 (1), 350–372.
• Hall, P., J. Kay, and D. Titterinton (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression., Biometrika 77 (3), 521–528.
• Haynes, K., I. A. Eckley, and P. Fearnhead (2017). Computationally efficient changepoint detection for a range of penalties., Journal of Computational and Graphical Statistics 26 (1), 134–143.
• Korostelev, A. (1988). On minimax estimation of a discontinuous signal., Theory of Probability & Its Applications 32 (4), 727–730.
• Korostelev, A. and A. Tsybakov (1993)., Minimax Methods for Image Reconstruction. Springer-Verlag New York, Inc.
• Li, K.-C. (1989). Honest confidence regions for nonparametric regression., The Annals of Statistics 17 (3), 1001–1008.
• Loader, C. (1996). Change point estimation using nonparametric regression., The Annals of Statistics 24 (4), 1667–1678.
• Low, M. (1997). On nonparametric confidence intervals., The Annals of Statistics 25 (6), 2547–2554.
• Mallik, A., M. Banerjee, and B. Sen (2013). Asymptotics for $p$-value based threshold estimation in regression settings., Electronic Journal of Statistics 7 (1), 2477–2515.
• Müller, H.-G. (1992). Change-points in nonparametric regression analysis., The Annals of Statistics 20 (2), 737–761.
• Neumann, M. H. (1997). Optimal change-point estimation in inverse problems., Scandinavian Journal of Statistics 24 (4), 503–521.
• Qiu, P. (2005)., Image Processing and Jump Regression Analysis, Volume 599. John Wiley & Sons.
• Tsybakov, A. B. (2009). Introduction to nonparametric estimation., Springer Series in Statistics .
• Van der Vaart, A. (2000)., Asymptotic statistics, Volume 3. Cambridge university press.
• Vershynin, R. (2018)., High-dimensional probability. Cambridge university press.
• Viens, F. and A. Vizcarra (2007). Supremum concentration inequality and modulus of continuity for sub-nth chaos processes., Journal of Functional Analysis 248 (1), 1–26.
• Von Neumann, J. (1941). Distribution of the ratio of the mean square successive difference to the variance., The Annals of Mathematical Statistics 12 (4), 367–395.
• Wishart, J. (2009). Kink estimation with correlated noise., Journal of the Korean Statistical Society 38 (2), 131–143.
• Wishart, J. (2011). Minimax lower bound for kink location estimators in a nonparametric regression model with long-range dependence., Statistics & Probability Letters 81 (12), 1871–1875.
• Wishart, J. and R. Kulik (2010). Kink estimation in stochastic regression with dependent errors and predictors., Electronic Journal of Statistics 4, 875–913.