Geodesic
Incomplete least squared regression function estimator based on wavelets
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 2, pp. 204-215.

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In this paper, we introduce an estimator of the least squares regression function, for $Y$ right censored by $R$ and $\min (Y, R)$ left censored by $L$. It is based on ideas derived from the context of wavelet estimates and is constructed by rigid thresholding of the coefficient estimates of a series development of the regression function. We establish convergence in norm $L_2$. We give enough criteria for the consistency of this estimator. The result shows that our estimator is able to adapt to the local regularity of the related regression function and distribution.
Keywords: non-parametric regression, $L_2$ error, least squares estimators, orthogonal series estimates, convergence in the $L_2$-norm, twice censored data, hard thresholding.
Mots-clés : regression estimation 
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Ryma Douas; Ilhem Laroussi; Soumia Kharfouchi. Incomplete least squared regression function estimator based on wavelets. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 16 (2023) no. 2, pp. 204-215. http://geodesic.mathdoc.fr/item/JSFU_2023_16_2_a5/

[1] L.Devroye, L.Györfi, A.Krzyżak, G.Lugosi, “On the strong universal consistency of nearest neighbor regression function estimates”, Annals of Statistics, 22 (1994), 1371–1385 | MR | Zbl

[2] L. Devroye, L.Györfi, G. Lugosi, A probabilist theory of pattern Recognition, Springer Verlag, 1996 | MR

[3] N.R.Draper, H.Smith, Applied Regression Analysis, 2nd ed., Wiley, New York, 1981 | MR | Zbl

[4] D.Donoho, I.M.Johnstone, “Ideal spatial adaptation by wavelet shrinkage”, Biometrika, 81 (1994), 425–455 | DOI | MR | Zbl

[5] D.Donoho, “Adapting to unknown smoothness via wavelet shrinkage”, J. Am. Statist. Ass., 90 (1995), 1200–1224 | DOI | MR | Zbl

[6] D.Donoho, I.M.Johnstone, “Minimax estimation via wavelet shrinkage”, Annals of Statistics, 26 (1998), 879–921 | DOI | MR | Zbl

[7] L.Györfi, M.Kohler, A.Krzyżak, H.Walk, A Distribution Free theory of Non parametric Regression, Springer–Verlag, New York, 2002 | DOI | MR

[8] D.J.Hart, Non parametric Smoothing and Lack-of-Fit Tests, Springer–Verlag, New York, 1997 | MR

[9] T.Hastie, R.J.Tibshirani, Generalized Additive Models, Chapman and Hall, London, UK, 1990 | MR | Zbl

[10] D.Haussler, “Decision theoretic generalizations of the PAC model for neural net and other learning applications”, Inform. Comput., 100 (1992), 78–150 | DOI | MR | Zbl

[11] E.L.Kaplan, P.Meier, “Nom parametric estimation from incomplete observations”, J. Amer. Statist. Assoc., 53 (1958), 457–481 | DOI | MR | Zbl

[12] K.Kebabi, I.Laroussi, F.Messaci, “Least squares estimators of the regression function with twice censored data”, Statist. Probab. Lett., 81 (2011), 1588–1593 | DOI | MR | Zbl

[13] K.Kebabi, F.Messaci, “Rate of the almost complete convergence of a kernel regression estimate with twice censored data”, Statist. Probab. Lett., 82:11 (2012), 1908–1913 | DOI | MR | Zbl

[14] M.Kohler, “On the universal consistency of a least squares spline regression estimator”, Math. Methods Statist., 6 (1997), 349–364 | MR | Zbl

[15] M.Kohler, “Universally consistent regression function estimation using hierarchical b-splines”, J. Multivariate Anal., 67 (1999), 138–164 | DOI | MR

[16] M.Kohler, A.Krzyżak, “Nonparametric regression estimation using penalized least squares”, IEEE Trans. Inform. Theory, 47 (2001), 3054–3058 | DOI | MR | Zbl

[17] M.Kohler, K.Mȧthé, M. Pintér, “Prediction from randomly right censored data”, J. Multivariate Anal., 80 (2002), 73–100 | DOI | MR | Zbl

[18] F.Messaci, “Local averaging estimates of the regression function with twice censored data”, Statist. Probab. Lett., 80 (2010), 1508–1511 | DOI | MR | Zbl

[19] D.Morales, L.Pardo, V.Quesada, “Bayesian survival estimation for incomplete data when the life distribution is proportionally related to the censoring time distribution”, Comm. Statist. Theory Methods, 20 (1991), 831–850 | DOI | MR | Zbl

[20] E.A.Nadaraya, “On estimating regression”, Theory of Probability and Its Applications, 9:1 (1964), 141–142 | DOI

[21] A.Nobel, “Histogram Regression Estimation Using Data-dependent Partitions”, Ann. Statist, 24 (1996), 1084–1105 | DOI | MR | Zbl

[22] V.Patilea, J.M.Rolin, “Product-limit estimators of the survival function with twice censored data”, Ann. Statist., 34:2 (2006), 925–938 | DOI | MR | Zbl

[23] V.N.Vapnik, A.Y.Chervonenkis, “On the uniform convergence of relative frequencies of events to their probabilities”, Theory of Probability and its Applications, 16 (1971), 264–280 | DOI | MR | Zbl

[24] V.N.Vapnik, Estimation of Dependencies Based on Empirical Data, Springer-Verlag, New York, 1982 | MR

[25] V.N.Vapnik, Statistical Learning Theory, Wiley, New York, 1998 | MR | Zbl