On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise
-
Yuncai Yu
,
Ling Liu
Abstract
This paper considers the nonparametric regression model with negatively super-additive dependent (NSD) noise and investigates the convergence rates of thresholding estimators. It is shown that the term-by-term thresholding estimator achieves nearly optimal and the block thresholding estimator attains optimal (or nearly optimal) convergence rates over Besov spaces. Additionally, some numerical simulations are implemented to substantiate the validity and adaptivity of the thresholding estimators with the presence of NSD noise.
Communicated by Gejza Wimmer
Acknowledgement
This work is supported by National Natural Science Foundation of China [No. 61374183, 51535005], and the Postgraduate Research & Practice Innovation Program of Jiangsu Province [No. KYCX19_0149].
References
[1] Cai, T. T.: Adaptive wavelet estimation: a block thresholding and oracle inequality approach, Ann. Statist. 27 (1999), 898–924.10.1214/aos/1018031262Search in Google Scholar
[2] Christofides, T. C.—Vaggelatou, E.: A connection between supermodular ordering and positive/negative association, J. Multivarite Anal. 88 (2004), 138–151.10.1016/S0047-259X(03)00064-2Search in Google Scholar
[3] Ding, L.—Chen, P.—Li, Y.: Consistency for Wavelet Estimator in Nonparametric Regression Model with Extended Negatively Dependent Sample. Statistical Papers, Springer, 2018.10.1007/s00362-018-1050-9Search in Google Scholar
[4] Donoho, D. L.—Johnstone, I. M.—Kerkyacharian, G.—Picard, D.: Wavelet shrinkage, asymptopia, J. R. Stat. Soc. Ser. B Stat. Methodol. 57 (1995), 301–369.10.1111/j.2517-6161.1995.tb02032.xSearch in Google Scholar
[5] Doosti, H.—Iranmanesh, A.—Arashi, M.—Hosseinioun, N.: On minimaxity of block thresholded wavelets under elliptical symmetry, J. Statist. Plann. Inference 141 (2011), 1526–1534.10.1016/j.jspi.2010.11.011Search in Google Scholar
[6] Eghbal, N.—Amini, M.—Bozorgnia, A.: Some maximal inequalities for quadratic forms of negative superadditive dependence random variables, Statist. Probab. Lett. 80 (2010), 587–591.10.1016/j.spl.2009.12.014Search in Google Scholar
[7] Gao, C.—Zhou, H. H.: Rate exact Bayesian adaptation with modified block priors, Ann. Statist. 44 (2016), 318–345.10.1214/15-AOS1368Search in Google Scholar
[8] Hall, P.—Kerkyacharian, G.—Picard, D.: On the minimax optimality of block thresholded wavelet estimators, Statist. Sin. 9 (1999), 33–50.Search in Google Scholar
[9] Hall, P.—Patil, P.: On the choice of smoothing parameter threshold, and truncation in nonparametric regression by nonlinear wavelet methods, J. R. Stat. Soc. Ser. B Stat. Methodol. 58 (1996), 361–377.10.1111/j.2517-6161.1996.tb02087.xSearch in Google Scholar
[10] Hoffmann, M.—Rousseau, J.—Schmidthieber, J.: On adaptive posterior concentration rates, Ann. Statist. 43 (2015), 2259–2295.10.1214/15-AOS1341Search in Google Scholar
[11] Hu, T. Z.: Negatively super-additive dependence of random variables with applications, Chinese J. Appl. Probab. Statist. 16 (2000), 133–144.Search in Google Scholar
[12] Li, L.—Xiao, Y.: On the minimax optimality of block thresholding wavelet estimators with long memory data, J. Statist. Plann. Inference 137 (2007), 2850–2869.10.1016/j.jspi.2006.10.009Search in Google Scholar
[13] Li, L.—Xiao, Y.: A note on block-thresholded wavelet estimators with correlated noise, Comm. Statist. Theory Methods 39 (2010), 1111–1128.10.1080/03610920902846570Search in Google Scholar
[14] Li, Y. M.—Yang, S. C.—Zhou, Y.—Datta, S.—Koul, H. L.: Consistency and uniformly asymptotic normality of wavelet estimator in regression model with associated samples, Statist. Probab. Lett. 78 (2008), 2947–2956.10.1016/j.spl.2008.05.004Search in Google Scholar
[15] Shen, A. T.—Xue, M. X.—Volodin, A.: Complete moment convergence for arrays of rowwise NSD random variables, Stochastics 88 (2016), 606–621.10.1080/17442508.2015.1110153Search in Google Scholar
[16] Shen, A. T.—Zhang, Y.—Volodin, A.: Applications of the Rosenthal-type inequality for negatively super-additive dependent random variables, Metrika 78 (2015), 295–311.10.1007/s00184-014-0503-ySearch in Google Scholar
[17] Shen, Y.—Wang, X. J.—Yang, W. Z.—Hu, S. H.: Almost sure convergence theorem and strong stability for weighted sums of NSD random variables, Acta Math. Sin. 29 (2013), 743–756.10.1007/s10114-012-1723-6Search in Google Scholar
[18] Tang, X. F.—Xi, M. M.—Wu, Y.—Wang, X. J.: Asymptotic normality of a wavelet estimator for asymptotically negatively associated errors, Statist. Probab. Lett. 140 (2018), 191–201.10.1016/j.spl.2018.04.024Search in Google Scholar
[19] Wang, Y.: Function estimation via wavelet shrinkage for long-memory data, Ann. Statist. 24 (1996) 466–484.10.1214/aos/1032894449Search in Google Scholar
[20] Wang, X. J.—Deng, X —Zheng, L. L.—Hu S. H.: Complete convergence for arrays of rowwise negatively superadditive dependent random variables and its applications, Statistics 48 (2014), 834–850.10.1080/02331888.2013.800066Search in Google Scholar
[21] Wang, X. J.—Shen, A. T.—Chen, Z. Y.: Complete convergence for weighted sums of NSD random variables and its application in the EV regression model, Test 24 (2015), 166–184.10.1007/s11749-014-0402-6Search in Google Scholar
[22] Wang, X. J.—Wu, Y —Hu, S. H.: Strong and weak consistency of LS estimators in the EV regression model with negatively superadditive-dependent errors, AStA Adv. Statist. Anal. 102 (2018), 41–65.10.1007/s10182-016-0286-8Search in Google Scholar
[23] Wu, Y.—Wang, X. J.—Hu, S. H.: Complete convergence for arrays of rowwise negatively super-additive dependent random variables and its applications, Appl. Math. J. Chinese Univ. 31 (2016), 439–457.10.1007/s11766-016-3406-zSearch in Google Scholar
[24] Yu, Y. C.—Hu, H. C.—Liu, L.—Huang, S. Y.: M-test in linear models with negatively super-additive dependent errors, J. Inequal. Appl. 2017 (2017), Art. ID 235, 21 pages.10.1186/s13660-017-1509-6Search in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- RNDr. Kvetoslava Dvořáková passed away
- On the Riesz structures of a lattice ordered abelian group
- On Diophantine equation x4 + y4 = n(u4 + v4)
- On a Waring-Goldbach problem involving squares and cubes
- D(n)-quadruples in the ring of integers of ℚ(√2, √3)
- Geometry of ℙ2 blown up at seven points
- Preservation of Rees exact sequences
- Pointwise multipliers between weighted copson and cesàro function spaces
- Some properties associated to a certain class of starlike functions
- Asymptotic properties of noncanonical third order differential equations
- Existence and regularity results for unilateral problems with degenerate coercivity
- Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M
- Some approximation properties of a kind of (p, q)-Phillips operators
- Best proximity points for a new type of set-valued mappings
- Weakly demicompact linear operators and axiomatic measures of weak noncompactness
- Fixed point results for F𝓡-generalized contractive mappings in partial metric spaces
- Einstein-Weyl structures on trans-Sasakian manifolds
- Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space
- ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds
- Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants
- A note on the consistency of wavelet estimators in nonparametric regression model under widely orthant dependent random errors
- On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise
- Generalized Meir-Keeler type contractions and discontinuity at fixed point II
Articles in the same Issue
- Regular papers
- RNDr. Kvetoslava Dvořáková passed away
- On the Riesz structures of a lattice ordered abelian group
- On Diophantine equation x4 + y4 = n(u4 + v4)
- On a Waring-Goldbach problem involving squares and cubes
- D(n)-quadruples in the ring of integers of ℚ(√2, √3)
- Geometry of ℙ2 blown up at seven points
- Preservation of Rees exact sequences
- Pointwise multipliers between weighted copson and cesàro function spaces
- Some properties associated to a certain class of starlike functions
- Asymptotic properties of noncanonical third order differential equations
- Existence and regularity results for unilateral problems with degenerate coercivity
- Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M
- Some approximation properties of a kind of (p, q)-Phillips operators
- Best proximity points for a new type of set-valued mappings
- Weakly demicompact linear operators and axiomatic measures of weak noncompactness
- Fixed point results for F𝓡-generalized contractive mappings in partial metric spaces
- Einstein-Weyl structures on trans-Sasakian manifolds
- Characterizations of linear Weingarten space-like hypersurface in a locally symmetric Lorentz space
- ∗-Ricci solitons and gradient almost ∗-Ricci solitons on Kenmotsu manifolds
- Uncorrelatedness sets of discrete random variables via Vandermonde-type determinants
- A note on the consistency of wavelet estimators in nonparametric regression model under widely orthant dependent random errors
- On adaptivity of wavelet thresholding estimators with negatively super-additive dependent noise
- Generalized Meir-Keeler type contractions and discontinuity at fixed point II
