# Semiparametric deconvolution with unknown noise variance

ESAIM: Probability and Statistics (2002)

- Volume: 6, page 271-292
- ISSN: 1292-8100

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topMatias, Catherine. "Semiparametric deconvolution with unknown noise variance." ESAIM: Probability and Statistics 6 (2002): 271-292. <http://eudml.org/doc/245543>.

@article{Matias2002,

abstract = {This paper deals with semiparametric convolution models, where the noise sequence has a gaussian centered distribution, with unknown variance. Non-parametric convolution models are concerned with the case of an entirely known distribution for the noise sequence, and they have been widely studied in the past decade. The main property of those models is the following one: the more regular the distribution of the noise is, the worst the rate of convergence for the estimation of the signal’s density $g$ is [3]. Nevertheless, regularity assumptions on the signal density $g$ improve those rates of convergence [15]. In this paper, we show that when the noise (assumed to be gaussian centered) has a variance $\sigma ^2$ that is unknown (actually, it is always the case in practical applications), the rates of convergence for the estimation of $g$ are seriously deteriorated, whatever its regularity is supposed to be. More precisely, the minimax risk for the pointwise estimation of $g$ over a class of regular densities is lower bounded by a constant over $\log n$. We construct two estimators of $\sigma ^2$, and more particularly, an estimator which is consistent as soon as the signal has a finite first order moment. We also mention as a consequence the deterioration of the rate of convergence in the estimation of the parameters in the nonlinear errors-in-variables model.},

author = {Matias, Catherine},

journal = {ESAIM: Probability and Statistics},

keywords = {convolution; deconvolution; density estimation; mixing distribution; normal mean mixture model; semiparametric mixture model; noise; variance estimation; minimax risk},

language = {eng},

pages = {271-292},

publisher = {EDP-Sciences},

title = {Semiparametric deconvolution with unknown noise variance},

url = {http://eudml.org/doc/245543},

volume = {6},

year = {2002},

}

TY - JOUR

AU - Matias, Catherine

TI - Semiparametric deconvolution with unknown noise variance

JO - ESAIM: Probability and Statistics

PY - 2002

PB - EDP-Sciences

VL - 6

SP - 271

EP - 292

AB - This paper deals with semiparametric convolution models, where the noise sequence has a gaussian centered distribution, with unknown variance. Non-parametric convolution models are concerned with the case of an entirely known distribution for the noise sequence, and they have been widely studied in the past decade. The main property of those models is the following one: the more regular the distribution of the noise is, the worst the rate of convergence for the estimation of the signal’s density $g$ is [3]. Nevertheless, regularity assumptions on the signal density $g$ improve those rates of convergence [15]. In this paper, we show that when the noise (assumed to be gaussian centered) has a variance $\sigma ^2$ that is unknown (actually, it is always the case in practical applications), the rates of convergence for the estimation of $g$ are seriously deteriorated, whatever its regularity is supposed to be. More precisely, the minimax risk for the pointwise estimation of $g$ over a class of regular densities is lower bounded by a constant over $\log n$. We construct two estimators of $\sigma ^2$, and more particularly, an estimator which is consistent as soon as the signal has a finite first order moment. We also mention as a consequence the deterioration of the rate of convergence in the estimation of the parameters in the nonlinear errors-in-variables model.

LA - eng

KW - convolution; deconvolution; density estimation; mixing distribution; normal mean mixture model; semiparametric mixture model; noise; variance estimation; minimax risk

UR - http://eudml.org/doc/245543

ER -

## References

top- [1] R.J. Carroll and P. Hall, Optimal rates of convergence for deconvolving a density. J. Amer. Statist. Assoc. 83 (1988) 1184-1186. Zbl0673.62033MR997599
- [2] L. Devroye, Consistent deconvolution in density estimation. Canad. J. Statist. 17 (1989) 235-239. Zbl0679.62029MR1033106
- [3] J. Fan, Asymptotic normality for deconvolution kernel density estimators. Sankhya Ser. A 53 (1991) 97-110. Zbl0729.62034MR1177770
- [4] J. Fan, Global behavior of deconvolution kernel estimates. Statist. Sinica 1 (1991) 541-551. Zbl0823.62032MR1130132
- [5] J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 (1991) 1257-1272. Zbl0729.62033MR1126324
- [6] J. Fan, Adaptively local one-dimensional subproblems with application to a deconvolution problem. Ann. Statist. 21 (1993) 600-610. Zbl0785.62038MR1232507
- [7] W. Feller, An introduction to probability theory and its applications, Vol. II. John Wiley & Sons Inc., New York (1971). Zbl0219.60003MR270403
- [8] R.D. Gill and B.Y. Levit, Applications of the Van Trees inequality: A Bayesian Cramér-Rao bound. Bernoulli 1 (1995) 59-79. Zbl0830.62035MR1354456
- [9] H. Ishwaran, Information in semiparametric mixtures of exponential families. Ann. Statist. 27 (1999) 159-177. Zbl0932.62039MR1701106
- [10] B.G. Lindsay, Exponential family mixture models (with least-squares estimators). Ann. Statist. 14 (1986) 124-137. Zbl0587.62057MR829558
- [11] M.C. Liu and R.L. Taylor, A consistent nonparametric density estimator for the deconvolution problem. Canad. J. Statist. 17 (1989) 427-438. Zbl0694.62017MR1047309
- [12] C. Matias and M.-L. Taupin, Minimax estimation of some linear functionals in the convolution model, Manuscript. Université Paris-Sud (2001). Zbl1130.62323
- [13] P. Medgyessy, Decomposition of superposition of density functions on discrete distributions. II. Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 21 (1973) 261-382. Zbl0275.60023MR440660
- [14] M.H. Neumann, On the effect of estimating the error density in nonparametric deconvolution. J. Nonparametr. Statist. 7 (1997) 307-330. Zbl1003.62514MR1460203
- [15] M. Pensky and B. Vidakovic, Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 (1999) 2033-2053. Zbl0962.62030MR1765627
- [16] L. Stefanski and R.J. Carroll, Deconvoluting kernel density estimators. Statistics 21 (1990) 169-184. Zbl0697.62035MR1054861
- [17] L.A. Stefanski, Rates of convergence of some estimators in a class of deconvolution problems. Statist. Probab. Lett. 9 (1990) 229-235. Zbl0686.62026MR1045189
- [18] M.L. Taupin. Semi-parametric estimation in the non-linear errors-in-variables model. Ann. Statist. 29 (2001) 66-93. Zbl1029.62039MR1833959
- [19] A.W. van der Vaart, Asymptotic statistics. Cambridge University Press, Cambridge (1998). Zbl0910.62001MR1652247
- [20] A.W. van der Vaart and J.A. Wellner, Weak convergence and empirical processes. Springer-Verlag, New York (1996). With applications to statistics. Zbl0862.60002MR1385671
- [21] C.-H. Zhang, Fourier methods for estimating mixing densities and distributions. Ann. Statist. 18 (1990) 806-831. Zbl0778.62037MR1056338

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