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Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data

  • Xingcai Zhou EMAIL logo , Beibei Ni , Hongxia Wang and Xingfang Huang
Published/Copyright: October 5, 2019
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 69 Issue 5

Abstract

In this paper, we discuss wavelet estimation of time-varying coefficient models based on censored data where the survival and the censoring times are from a stationary α-mixing sequence. Under the appropriate conditions, the Berry-Esseen bouds of wavelet estimators are established. For the purpose of statistical inference, a random weighted wavelet estimator of the time-varying coefficient is also constructed, and some approximation rates are given.

MSC 2010: Primary 62G05; Secondary 60G07
Keywords: Berry-Esseen bounds; wavelet estimation; time-varying coefficient; strong mixing; censored data

The work was supported by The National Social Science Fund of China (No. 19BTJ034).

  1. (Communicated by Gejza Wimmer)

Acknowledgement

We thank the Associate Editor and the reviewers for comments that greatly improved the paper.

Appendix

Lemma A.1

(Antoniadis et al. [1]). Suppose that Assumption (A3)(iii) holds. We have

  1. sup0≤t,s≤1|Em(t,s)|=O(2m).

  2. sup0≤t≤1∫01|Em(t,s)|ds≤c,wherecis a positive constant.

Lemma A.2

(Antoniadis et al. [1]). Suppose that Assumptions (A3)(iii)–(iv) hold andh(t) satisfies Assumptions (A3)(i)–(ii). Then

sup0≤t≤1|h(t)−∑i=1nh(ti)∫AiEm(t,s)ds|=O(n−υ)+O(τm),

where

τm=(1/2m)γ−1/2if1/2<γ<3/2,m/2mifγ=3/2,1/2mifγ>3/2.

Lemma A.3

(Li, et al. [19]). Suppose that {ζn, n ≥ 1}, {ηn, n ≥ 1} and {ξn, n ≥ 1} are three random variable sequences, {γn, n ≥ 1} is a positive constant sequence, andγn → 0. Ifsupu|Fζn(u)−Φ(u)|≤Cγn,then for anyϵ1 > 0 andϵ2 > 0

supuFζn+ηn+ξn(u)−Φ(u)≤Cγn+ϵ1+ϵ2+P(|ηn|≥ϵ1)+P(|ξn|≥ϵ2).

Lemma A.4

(Hall and Heyde [14]). Suppose thatXandYare random variables such that E|X|p < ∞, E|Y|q < ∞, wherep, q > 1, p−1 + q−1 < 1. Then

|EXY−EXEY|≤8E|X|p1/pE|X|q1/q(supA∈σ(X),B∈σ(Y)|P(AB)−P(A)P(B)|)1−1/p−1/q.

In the following section, let {Xi, i ≥ 1} be an α-mixing sequence of random variables with EXi = 0 and mixing coefficient α(i).

Lemma A.5

(Yang [36]).

  1. Suppose that E|Xi|2+δ < ∞ forδ > 0. Then

    E|∑i=1nXi|2=(1+20∑j=1nαδ/(2+δ)(j))∑i=1nE|Xi|2+δ2/(2+δ);

  2. Let E|Xi|r+δ < ∞ forr > 2 andδ > 0. Suppose thatθ > r(r + δ)/(2δ) andα(n) = O(n–θ). Then for any givenϵ > 0, there exists a positive constantsC = C(r, δ, ϵ, θ) such that

    Emax1≤j≤n|∑i=1jXi|r≤C{nϵ∑i=1nE|Xi|r+(∑i=1nE|Xi|r+δ2/(r+δ))r/2}.

Lemma A.6

(Yang and Li [1]). Letrandsare two positive integers. Letξd=∑j=(d−1)(r+s)+1(d−1)(r+s)+rXjfor 1 ≤ d ≤ k. Ifp > 0, q > 0 and1p+1q=1,then

|Eexp⁡(iu∑d=1kξd)−∏d=1kEexp⁡(iuξd)|≤C|u|α1/p(s)∑d=1k(E|ξd|q)1/q.

Lemma A.7

(Fan and Yao [12]). Let {Xi} be a zero-mean real-valuedα-mixing process satisfyingP(|Xi| ≤ b) = 1 for alli ≥ 1. Then for each integerq ∈ [1, n/2] and eachϵ > 0, we have

P(|∑i=1nXi|>nϵ)≤4exp⁡(−ϵ2q8v2(q))+22(1+4bϵ)1/2qα(n2q),

wherev2(q) = 2σ2(q)/p2 + bϵ/2 withp = n/(2q) and

σ2(q)=max1≤j≤2q−1E{([jp]+1−jp)X[jp]+1+X[jp]+2+⋯+X[(j+1)p]+((j+1)p−[(j+1)p])X[(j+1)p]+1}2.

Lemma A.8

(Cai [5]). LetF̂n(x) be the Kaplan-Meier estimator of the common distributionF(x) of {Xi, i ≥ 1} in the censored setup. Suppose that theα-mixing coefficient ofξksatisfiesα(k) = O(k–λ), for λ > 3. Then, for anyT ∈ (0, τH), we have

supx∈[0,T]|F^n(x)−F(x)|=O((loglog⁡n/n)1/2),a.s..

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Received: 2018-05-10
Accepted: 2019-01-11
Published Online: 2019-10-05
Published in Print: 2019-10-25

© 2019 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2017-0301
Keywords for this article
Berry-Esseen bounds; wavelet estimation; time-varying coefficient; strong mixing; censored data

Articles in the same Issue

  1. Regular papers
  2. Professor Anatolij Dvurečenskij
  3. A generating theorem of punctured surface triangulations with inner degree at least 4
  4. Natural extension of EF-spaces and EZ-spaces to the pointfree context
  5. On weak isometries in directed groups
  6. On self-injectivity of the f-ring Frm(𝓟(ℝ), L)
  7. On the extensions of discrete valuations in number fields
  8. On certain classes of generalized derivations
  9. Nilpotent elements in medial semigroups
  10. Power of meromorphic function sharing polynomials with derivative of it’s combination with it’s shift
  11. Sharp coefficient bounds for starlike functions associated with the Bell numbers
  12. Sufficient conditions for Carathéodory functions and applications to univalent functions
  13. Blending type approximation by complex Szász-Durrmeyer-Chlodowsky operators in compact disks
  14. Linear directional differential equations in the unit ball: solutions of bounded L-index
  15. On oscilatory fourth order nonlinear neutral differential equations – IV
  16. Oscillation criteria for second-order half-linear delay differential equations with mixed neutral terms
  17. Lerch’s theorem on nabla time scales
  18. IK-convergence of sequences of functions
  19. Convolution theorems related with the solvability of Wiener-Hopf plus Hankel integral equations and Shannon’s sampling formula
  20. Stabilization of third order differential equation by delay distributed feedback control with unbounded memory
  21. On the amenability of a class of Banach algebras with application to measure algebra
  22. A bivariate Kumaraswamy-exponential distribution with application
  23. Berry-Esseen bounds for wavelet estimator in time-varying coefficient models with censored dependent data
  24. Corrigendum to ”on the congruent number problem over integers of cyclic number fields”, Math. Slovaca 66(3) (2016), 561–564
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