Non-linear wavelet-based density estimators under random censorship

doi:10.1016/S0378-3758(02)00366-X

Journal of Statistical Planning and Inference

Volume 117, Issue 1, 1 November 2003, Pages 35-58

https://doi.org/10.1016/S0378-3758(02)00366-X Get rights and content

Abstract

We provide an asymptotic expansion for the mean integrated squared error (MISE) of non-linear wavelet-based density estimators with randomly censored data. Our technique is facilitated by a result of Stute (Ann. Statist. 23 (1995) 422) that approximates the Kaplan–Meier integrals by an average of i.i.d. random variables with a certain rate. We show this MISE expansion, when the underlying survival density function and censoring distribution function are only piecewise smooth, is the same as analogous expansion for the kernel density estimators. However, for the kernel estimators, this MISE expansion holds only under the additional smoothness assumption.

Introduction

The mathematical theory of wavelets and their applications in statistics have become a well-known technique for non-parametric curve estimation: See e.g. Meyer (1990), Daubechies (1992), Chui (1992), Mallat (1989), Donoho and Johnstone (1994), Donoho 1995, Donoho 1996 and Kerkyacharian and Picard 1992, Kerkyacharian and Picard 1993. For a systematic discussion of wavelets and their applications see the recent monograph by Härdle et al. (1998). The major advantage of the wavelet method is its adaptation to the erratic behavior of the density and local adaptation to the degree of smoothness of the unknown density. These wavelet estimators typically achieve the optimal convergence rates over exceptionally large function spaces. They do an excellent job of taking care of discontinuities in the target function, and in consequence they enjoy a very good convergence rate even if smoothness conditions are imposed only in a piecewise sense.

Hall and Patil (1995) first explicitly demonstrated that, in the case of no censorship, the discontinuities of densities have a negligible effect on the performance of the non-linear wavelet density estimators. The mean integrated squared error (MISE) of the kernel estimator of a density function f has the form $MISE ∼c_{1} (nh)^{−1} +c_{2} h^{2r},$ where “∼” means that the ratio of the left- and right-hand sides converges to 1 as n→∞ and n denotes the sample size, h is the bandwidth of the kernel estimator, r is the order of the kernel and c₁ and c₂ are constants depending on both the kernel and unknown density. The first term derives from the variance and the second from the squared bias. This expansion for the kernel estimators generally fails if the underlying density function does not have r derivatives (Hall and Patil, 1995, p. 906). However, the MISE expansion of the non-linear wavelet estimators is still valid for only piecewise smooth density function, and even has the same constants c₁ and c₂. Patil (1997) provided similar results for non-linear wavelet hazard rate estimators with complete data.

In industrial life-testing, medical follow-up research and other studies, the observation of the occurrence of the failure event may be prevented by the previous occurrence of the censoring event. So only part of the observations are real failure times. Formally, let X₁,X₂,…,X_n be i.i.d. survival times with a common distribution function F and density function f. Also let Y₁,Y₂,…,Y_n be i.i.d. censoring times with a common distribution function G. It is assumed that X_i is independent of Y_i for every i. Rather than observing X₁,X₂,…,X_n, the variables of interest, in the randomly right-censored model, one observes Z_i=min(X_i,Y_i)=X_i∧Y_i and δ_i=I(X_i⩽Y_i), i=1,2,…,n, where I(A) denotes the indicator function of the set A.

Antoniadis et al. (1999) describe a wavelet method for the estimation of density and hazard rate functions from randomly right-censored data. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and survival function of the observations are then separately smoothed over time via linear wavelet smoothers. They provide asymptotic normality of the estimator and obtain best possible asymptotic MISE convergence rate under the assumption that survival time density function f is r-times continuously differentiable and the censoring density g is continuous.

The objective of this paper is to propose a non-linear wavelet estimator of a density function with censored data and derive a result similar to the main result, Theorem 2.1, of Hall and Patil (1995). One of the consequence of this extension is that we can show that MISE has the analogous expansion $MISE ∼k_{1} n^{−1} p+k_{2} p^{−2r},$ where n denotes the sample size, p is the smoothing parameter depending on n, a wavelet analog of the bandwidth h⁻¹ for kernel estimators and k₁ and k₂ are constants depending on the wavelet, unknown density and censoring distribution.

Wu and Wells (1999) provided hazard rate estimation by non-linear wavelet methods in the left truncation and right censoring model. They applied counting process techniques and obtained analogous expansion. They provide a wavelet-based estimator for the hazard rate function over a bounded interval [ι,τ] which is chosen such that the size of risk population satisfies some additional conditions.

In this paper, we apply the method of Stute (1995) that approximates a Kaplan–Meier integrals by an average of i.i.d. random variables with a certain small rate. We provide an MISE expansion similar to that of Hall and Patil (1995) for density function over (−∞,T], for any fixed T<τ_H, where $τ_{H} = inf {x : H(x)=1}⩽∞$ is the least upper bound for the support of H, the distribution function of Z₁.

In the next section, we give the elements of wavelet transform and provide non-linear wavelet-based density estimators. The main results are described in Section 3, while their proofs appear in Sections 4 and 5.

Section snippets

Notations and estimators

This section contains some facts about wavelets that will be used in the sequel. Let φ(x) and ψ(x) be father and mother wavelets, having the properties: φ and ψ are bounded and compactly supported; ∫φ²=∫ψ²=1, $μ_{k} ≡∫y^{k} ψ(y) d y=0$ for 0⩽k⩽r−1 and μ_r=r!κ≠0, where $κ=(r!)^{−1} ∫y^{r} ψ(y) d y$ . Let $φ_{j} (x)=p^{1/2} φ(px−j), ψ_{ij} (x)=p_{i}^{1/2} ψ(p_{i} x−j), x∈ R$ for arbitrary p>0,−∞<j<∞ and $p_{i} =p2^{i}, i⩾0$ . Then $∫φ_{j_{1}} φ_{j_{2}} =δ_{j_{1}j_{2}}, ∫ψ_{i_{1}j_{1}} ψ_{i_{2}j_{2}} =δ_{i_{1}i_{2}} δ_{j_{1}j_{2}}, ∫φ_{j_{1}} ψ_{ij_{2}} =0,$ where δ_ij denotes the Kronecker delta, i.e. δ_ij=1, if i=j; 0, otherwise. For more on

Main results

We assume that the smoothing parameters $p, q$ and δ satisfy the following condition: $(SP): p→∞, q→∞, p_{q} δ^{2} →0, p^{2r+1} δ^{2} →∞, δ⩾C n^{−1} ln n,$ where $C>C_{0} ≡2{r(2r+1)^{−1} sup f_{1} (1−G)^{−1}}^{1/2} .$

Theorem 3.1

In addition to the conditions on φ and ψ stated in Section 2, assume that the rth derivative f^(r) is continuous on (−∞,∞) and is bounded, monotone on (−∞,−u) for a sufficiently large positive u and the censoring distribution function G is continuous. Also assume condition (SP) holds. Then $E ∫(f_{1} ̂ −f_{1})^{2} − n^{−1} p∫ f_{1} 1−G +p^{−2r} κ^{2} (1−2^{−2r})^{−1} ∫f_{1}^{(r)^{2}} = o (n^{}$

Proofs

The proof of the above theorem follows along the lines in Hall and Patil (1995), combined with Stute (1995) which establishes an approximation for the Kaplan–Meier integral $∫ϕ d F_{n} ̂$ as an average of i.i.d. random variables with a sufficiently small error. This allows for a more traditional and direct approach to the density estimation problem for censored data, compared to the martingale approach as used, e.g. in Wu and Wells (1999). We begin with some lemmas.

Lemma 4.1

Let $b ̂_{j}$ and $b ̂_{ij}$ be defined as in Eqs.

Acknowledgements

The author expresses his deep gratitude to his advisor Professor Hira L. Koul for his constant advice, valuable suggestion and careful reading which greatly improve the presentation of this paper. The author also appreciates the constructive suggestion from Professor Winfried Stute on Lemma 4.1 and is very grateful to one referee for his pointing out errors and typos and providing many insightful comments.

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¹: Research partly supported by the NSF Grant DMS 0071619.

View full text

Non-linear wavelet-based density estimators under random censorship

Abstract

Introduction

Section snippets

Notations and estimators

Main results

Proofs

Acknowledgements

Statist. Probab. Lett.

Statist. Probab. Lett.

J. Statist. Plann. Inference

Density and hazard rate estimation for right-censored data by using wavelet methods

J. Roy. Statist. Soc. B

Wavelets: A Tutorial in Theory and Applications

Ten Lectures on Wavelets

Ideal spatial adaptation by wavelet shrinkage

Biometrika

Wavelet shrinkageasymptopia?

J. Roy. Statist. Soc. Ser. B

Density estimation by wavelet thresholding

Ann. Statist.