Electronic Journal of Statistics

Nonparametric Laguerre estimation in the multiplicative censoring model

Denis Belomestny, Fabienne Comte, and Valentine Genon-Catalot

Full-text: Open access

Abstract

We study the model $Y_{i}=X_{i}U_{i},\;i=1,\ldots,n$ where the $U_{i}$’s are i.i.d. with $\beta(1,k)$ density, $k\ge1$, $k$ integer, the $X_{i}$’s are i.i.d., nonnegative with unknown density $f$. The sequences $(X_{i}),(U_{i}),$ are independent. We aim at estimating $f$ on ${\mathbb{R}}^{+}$ from the observations $(Y_{1},\dots,Y_{n})$. We propose projection estimators using a Laguerre basis. A data-driven procedure is described in order to select the dimension of the projection space, which performs automatically the bias variance compromise. Then, we give upper bounds on the ${\mathbb{L}}^{2}$-risk on specific Sobolev-Laguerre spaces. Lower bounds matching with the upper bounds within a logarithmic factor are proved. The method is illustrated on simulated data.

Article information

Source
Electron. J. Statist. Volume 10, Number 2 (2016), 3114-3152.

Dates
Received: May 2016
First available in Project Euclid: 10 November 2016

Permanent link to this document
http://projecteuclid.org/euclid.ejs/1478747031

Digital Object Identifier
doi:10.1214/16-EJS1203

Zentralblatt MATH identifier
06673438

Subjects
Primary: 62G07: Density estimation

Keywords
Adaptive estimation lower bounds model selection multiplicative censoring projection estimator

Citation

Belomestny, Denis; Comte, Fabienne; Genon-Catalot, Valentine. Nonparametric Laguerre estimation in the multiplicative censoring model. Electron. J. Statist. 10 (2016), no. 2, 3114--3152. doi:10.1214/16-EJS1203. http://projecteuclid.org/euclid.ejs/1478747031.


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References

  • Abbaszadeh, M., Chesneau, C. and Doosti, H. (2012). Nonparametric estimation of density under bias and multiplicative censoring via wavelet methods., Statist. Probab. Lett. 82, 932–941.
  • Abbaszadeh, M., Chesneau, C. and Doosti, H. (2013). Multiplicative censoring: estimation of a density and its derivatives under the Lp-risk., REVSTAT 11, 255–276.
  • Abramowitz, M. and Stegun, I. A. (1964)., Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C.
  • Asgharian, M., Carone, M., Fakoor, V. (2012). Large-sample study of the kernel density estimators under multiplicative censoring., Ann. Statist. 40, 159–187.
  • Andersen, K. E. and Hansen, M. B. (2001). Multiplicative censoring: density estimation by a series expansion approach., J. Statist. Plann. Inference 98, 137–155.
  • Balabdaoui, F. and Wellner, J. A. (2007). Estimation of a k-monotone density: limit distribution theory and the spline connection., Ann. Statist. 35, 2536–2564.
  • Balabdaoui, F. and Wellner, J. A. (2010). Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds., Stat. Neerl. 64, 45–70.
  • Baudry, J.-P., Maugis, C. and Michel, B. (2012). Slope Heuristics: overview and implementation., Statistics and Computing, 22, 455–470.
  • Belomestny, D., Comte, F. and Genon-Catalot, V. (2016). Laguerre estimation for $k$-monotone densities observed with noise. Preprint MAP5 2016-01, https://hal.archives-ouvertes.fr/hal-01252143/.
  • Birgé, L. and Massart, P. (2007). Minimal penalties for Gaussian model selection., Probab. Theory Related Fields 138, 33–73.
  • Bongioanni, B. and Torrea, J. L. (2009). What is a Sobolev space for the Laguerre function system?, Studia Mathematica 192 (2), 147–172.
  • Brunel, E., Comte, F. and Genon-Catalot, V. (2016). Nonparametric density and survival function estimation in the multiplicative censoring model., Test 25, 570–590.
  • Butucea, C. and Matias, C. (2005). Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution model., Bernoulli 11, 309–340.
  • Chee, C.-S. and Wang, Y. (2014). Least squares estimation of a k-monotone density function., Comput. Statist. Data Anal. 74, 209–216.
  • Chesneau, C. (2013). Wavelet estimation of a density in a GARCH-type model., Comm. Statist. Theory Methods 42, 98–117.
  • Comte, F., Cuenod, C.-A., Pensky, M. and Rozenholc, Y. (2013). Laplace deconvolution and its application to Dynamic Contrast Enhanced imaging. To appear in, J. R. Stat. Soc. Ser. B Stat. Methodol. DOI: 10.1111/rssb.12159
  • Comte, F. and Genon-Catalot, V. (2015) Adaptive Laguerre density estimation for mixed Poisson models., Electron. J. Stat. 9, 1113–1149.
  • van Es, B., Klaassen, C. A. J. and Oudshoorn, K. (2000). Survival analysis under cross-sectional sampling: length bias and multiplicative censoring. Prague Workshop on Perspectives in Modern Statistical Inference: Parametrics, Semi-parametrics, Non-parametrics (1998)., J. Statist. Plann. Inference 91, 295–312.
  • van Es, B., Spreij, P. and van Zanten, H. (2003). Nonparametric volatility density estimation., Bernoulli 9, 451–465.
  • van Es, B., Jongbloed, G. and van Zuijlen, M. (1998). Isotonic inverse estimators for nonparametric deconvolution., Ann. Statist. 26, 2395–2406.
  • Jirak, M., Meister, A. and Reiss, M. (2014). Adaptive function estimation in nonparametric regression with one-sided errors., Ann. Statist. 42, 1970–2002.
  • Juditsky, A., Lambert-Lacroix, S. (2004). On minimax density estimation on R., Bernoulli 10, 187–220.
  • Jongbloed, G. (1998). Exponential deconvolution: two asymptotically equivalent estimators., Statist. Neerlandica 52, 6–17.
  • Mabon, G. (2015). Adaptive deconvolution on the nonnegative real line. Preprint HAL hal-01076927, version, 2.
  • Magnus, J. R. and Neudecker, H. (1999)., Matrix differential calculus with applications in statistics and econometrics. Revised reprint of the 1988 original. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd., Chichester.
  • Muckenhoupt, B. (1970). Asymptotic forms for Laguerre polynomials., Proceedings of the American mathematical Society, 288–292.
  • Shen, J. (2000). Stable and efficient spectral methods in unbounded domains using Laguerre functions., SIAM J. Numer. Anal. 38, 1113–1133.
  • Szegö, G. (1975)., Orthogonal polynomials. Fourth edition. American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I.
  • Tsybakov, A. B. (2009)., Introduction to nonparametric estimation. Springer Series in Statistics. Springer, New York, 2009.
  • Vardi, Y. (1989). Multiplicative censoring, renewal processes, deconvolution and decreasing density: nonparametric estimation., Biometrika 76, 751–761.
  • Vardi, Y. and Zhang, C.-H. (1992). Large sample study of empirical distributions in a random-multiplicative censoring model., Ann. Statist. 20, 1022–1039.
  • Williamson R. E. (1956). Multiply monotone functions and their Laplace transforms., Duke Mathematical Journal 23, 189–207.