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
https://projecteuclid.org/euclid.ejs/1478747031

Digital Object Identifier
doi:10.1214/16-EJS1203

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. https://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.REVSTAT11, 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. Inference98, 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 Fields138, 33–73.
  • Bongioanni, B. and Torrea, J. L. (2009). What is a Sobolev space for the Laguerre function, system?Studia Mathematica192(2), 147–172.
  • Brunel, E., Comte, F. and Genon-Catalot, V. (2016). Nonparametric density and survival function estimation in the multiplicative censoring, model.Test25, 570–590.
  • Butucea, C. and Matias, C. (2005). Minimax estimation of the noise level and of the deconvolution density in a semiparametric convolution, model.Bernoulli11, 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 Methods42, 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, inJ. 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. Inference91, 295–312.
  • van Es, B., Spreij, P. and van Zanten, H. (2003). Nonparametric volatility density, estimation.Bernoulli9, 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.Bernoulli10, 187–220.
  • Jongbloed, G. (1998). Exponential deconvolution: two asymptotically equivalent, estimators.Statist. Neerlandica52, 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.Biometrika76, 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 Journal23, 189–207.