## Bernoulli

• Bernoulli
• Volume 22, Number 1 (2016), 143-192.

### Adaptive quantile estimation in deconvolution with unknown error distribution

#### Abstract

Quantile estimation in deconvolution problems is studied comprehensively. In particular, the more realistic setup of unknown error distributions is covered. Our plug-in method is based on a deconvolution density estimator and is minimax optimal under minimal and natural conditions. This closes an important gap in the literature. Optimal adaptive estimation is obtained by a data-driven bandwidth choice. As a side result, we obtain optimal rates for the plug-in estimation of distribution functions with unknown error distributions. The method is applied to a real data example.

#### Article information

Source
Bernoulli, Volume 22, Number 1 (2016), 143-192.

Dates
Revised: April 2014
First available in Project Euclid: 30 September 2015

https://projecteuclid.org/euclid.bj/1443620846

Digital Object Identifier
doi:10.3150/14-BEJ626

Mathematical Reviews number (MathSciNet)
MR3449779

Zentralblatt MATH identifier
06543266

#### Citation

Dattner, Itai; Reiß, Markus; Trabs, Mathias. Adaptive quantile estimation in deconvolution with unknown error distribution. Bernoulli 22 (2016), no. 1, 143--192. doi:10.3150/14-BEJ626. https://projecteuclid.org/euclid.bj/1443620846

#### References

• [1] Carroll, R.J. and Hall, P. (1988). Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 1184–1186.
• [2] Carroll, R.J., Ruppert, D., Stefanski, L.A. and Crainiceanu, C.M. (2006). Measurement Error in Nonlinear Models: A Modern Perspective, 2nd ed. Monographs on Statistics and Applied Probability 105. Boca Raton, FL: Chapman & Hall/CRC.
• [3] Comte, F. and Lacour, C. (2011). Data-driven density estimation in the presence of additive noise with unknown distribution. J. R. Stat. Soc. Ser. B Stat. Methodol. 73 601–627.
• [4] Dattner, I., Goldenshluger, A. and Juditsky, A. (2011). On deconvolution of distribution functions. Ann. Statist. 39 2477–2501.
• [5] Dattner, I. and Reiser, B. (2013). Estimation of distribution functions in measurement error models. J. Statist. Plann. Inference 143 479–493.
• [6] Delaigle, A., Hall, P. and Meister, A. (2008). On deconvolution with repeated measurements. Ann. Statist. 36 665–685.
• [7] Dudley, R.M. (1992). Fréchet differentiability, $p$-variation and uniform Donsker classes. Ann. Probab. 20 1968–1982.
• [8] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257–1272.
• [9] Frese, E.M., Fick, A. and Sadowsky, H.S. (2011). Blood pressure measurement guidelines for physical therapists. Cardiopulm. Phys. Ther. J. 22 5–12.
• [10] Girardi, M. and Weis, L. (2003). Operator-valued Fourier multiplier theorems on Besov spaces. Math. Nachr. 251 34–51.
• [11] Goldenshluger, A. and Nemirovski, A. (1997). On spatially adaptive estimation of nonparametric regression. Math. Methods Statist. 6 135–170.
• [12] Hall, P. and Lahiri, S.N. (2008). Estimation of distributions, moments and quantiles in deconvolution problems. Ann. Statist. 36 2110–2134.
• [13] Johannes, J. (2009). Deconvolution with unknown error distribution. Ann. Statist. 37 2301–2323.
• [14] Johannes, J. and Schwarz, M. (2013). Adaptive circular deconvolution by model selection under unknown error distribution. Bernoulli 19 1576–1611.
• [15] Kannel, W.B. (1995). Framingham study insights into hypertensive risk of cardiovascular disease. Hypertension Research: Official Journal of the Japanese Society of Hypertension 18 181–196.
• [16] Kappus, J. (2014). Adaptive nonparametric estimation for Lévy processes observed at low frequency. Stochastic Process. Appl. 124 730–758.
• [17] Lepskiĭ, O.V. (1990). A problem of adaptive estimation in Gaussian white noise. Teor. Veroyatn. Primen. 35 459–470.
• [18] Lounici, K. and Nickl, R. (2011). Global uniform risk bounds for wavelet deconvolution estimators. Ann. Statist. 39 201–231.
• [19] Massart, P. (2007). Concentration Inequalities and Model Selection. Lecture Notes in Math. 1896. Berlin: Springer.
• [20] Meister, A. (2004). On the effect of misspecifying the error density in a deconvolution problem. Canad. J. Statist. 32 439–449.
• [21] Neumann, M.H. (1997). On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Stat. 7 307–330.
• [22] Neumann, M.H. (2007). Deconvolution from panel data with unknown error distribution. J. Multivariate Anal. 98 1955–1968.
• [23] Neumann, M.H. and Reiß, M. (2009). Nonparametric estimation for Lévy processes from low-frequency observations. Bernoulli 15 223–248.
• [24] Nickl, R. and Reiß, M. (2012). A Donsker theorem for Lévy measures. J. Funct. Anal. 263 3306–3332.
• [25] Rosenthal, H.P. (1970). On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables. Israel J. Math. 8 273–303.
• [26] Söhl, J. and Trabs, M. (2012). A uniform central limit theorem and efficiency for deconvolution estimators. Electron. J. Stat. 6 2486–2518.
• [27] Spokoiny, V.G. (1996). Adaptive hypothesis testing using wavelets. Ann. Statist. 24 2477–2498.
• [28] Stirnemann, J.J., Comte, F. and Samson, A. (2012). Density estimation of a biomedical variable subject to measurement error using an auxiliary set of replicate observations. Stat. Med. 31 4154–4163.
• [29] Triebel, H. (2010). Theory of Function Spaces. Modern Birkhäuser Classics. Basel: Birkhäuser. Reprint of 1983 edition [MR0730762], Also published in 1983 by Birkhäuser Verlag [MR0781540].
• [30] Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. New York: Springer. Revised and extended from the 2004 French original, Translated by Vladimir Zaiats.
• [31] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge: Cambridge Univ. Press.