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Bootstrap approximation of wavelet estimates in a semiparametric

regression model

Liugen Xue Qiang Liu

College of Applied Sciences School of Statistics

Beijing University of Technology Capital University of Economics and Business

Beijing, P. R. China 100124 Beijing, P. R. China 100070

lgxue@bjut.edu.cn liuqiang2008@emails.bjut.edu.cn

Abstract

The inference for the parameters in a semiparametric regression model is stud-

ied by using the wavelet and the bootstrap methods. The bootstrap statistics

are constructed by using Efron’s resampling technique, and the strong uniformly

convergency of the bootstrap approximation is proved. Our results can be used

to construct the large sample conﬁdence intervals for the parameters of interest.

A simulation study is conducted to evaluate the ﬁnite-sample performance of

the bootstrap method and to compare it with the normal approximation-based

method.

Keywords: Bootstrap approximation; Conﬁdence interval; Semiparametric re-

gression model; Strong uniformly convergency; Wavelet estimate.

1 Introduction

The wavelet smoothing is an useful method for constructing the nonparametric

estimates. It has many nice properties, the main advantage is that it has the low

demand for the function of interest and obtains nice properties of large sample. Many

authors have applied the wavelet smoothing to the estimation of density function, the

estimation of nonparametric regression function and the semiparametric regression

models. Some related works include: Antoniads Gregorie (1994), Donob (1996),

Speckman (1988), Chai and Xu (1999), Qian and Chai (1999) and Xue (2002).

Consider the semiparamertic regression model

= x

β + g(t

) + e

, i = 1, . . . , n, (1.1)

The research was supported by the National Natural Science Foundation of China

(No.10571008), the Natural Science Foundation of Beijing (No.1072004) and Ph. D. Program

Foundation of Ministry of Education of China (No. 20070005003).

http://www.paper.edu.cn

where β = (β

, . . . , β

)

is an unknown parametric vector, g(·) is an unknown func-

tion on [0, 1], {x

= (x

, . . . , x

)

; 1 ≤ i ≤ n} are the random design sequences,

; 1 ≤ i ≤ n} are the constant sequences on [0,1], and the errors e

are independent

and identically distributed (i.i.d.) random variables with mean 0 and variance σ

Model (1.1) was proposed by Engle et al. (1986), and has been widely investi-

gated in recent years. Some estimation methods have been developed, such as the

kernel smoothing,the wavelet smoothing, the piecewise polynomial method, and so

on. See, for example, Speckman (1988), Chai and Xu (1999) and Chen (1988). They

obtained that the estimator of β achieved the convergence rate O

−1/2

). However,

it is very important to study the bootstrap approximation of the estimator.

Efron (1979) introduced the b ootstrap method for estimating the distribution

of statistic based on independent observations. The distribution is obtained by

replacing the unknown distribution with the empirical distribution of the data in the

deﬁnition of the statistical function, and then resample the data to obtain a Monte

Carlo distribution for the resulting random variable. This method would probably

be used in practice only when the distribution could not be estimated analytically.

The purpose of the present investigation is to construct the bootstrap statistics of the

wavelet estimators of β and σ

for model (1.1) using Efron’s resampling procedure.

We also prove the consistency of the bo otstrap approximation. Our results can be

used to construct the large sample conﬁdence intervals for β and σ

The rest of this paper is organized as follows. In section 2, the wavelet estimators

of β and σ

β and ˆσ

say, are constructed by combining the wavelet smoothing and

the least square method, and the bootstrap statistics of

β and ˆσ

are given by using

bootstrap method. Section 3 gives the main results. In section 4, a simulation study

is carried out to compare the proposed method with the normal approximation-based

method. The proofs of theorems appear in the appendix.

2 Estimation method

Using the condition of Speckman (1988), we assume, as is common in the setting

of semiparametric regression model, that {x

} and {t

} are related via

= f

) + η

, 1 ≤ r ≤ d, i = 1, . . . , n,

where f

(t) is a given function on [0,1], {η

= (η

, . . . , η

)

; 1 ≤ i ≤ n} are i.i.d.

random variables with independent of {e

; 1 ≤ i ≤ n}, and satisfy E(η

) = 0 and

Cov(η

) = V . Here V = (v

) is a d × d positive deﬁnite matrix.

To construct the wavelet estimators of β and σ

, we take the scalar function φ(·)

on the Schwarz space S

of order l. The multi-resolving analysis of connection L

(R)

is {V

, m ∈ Z} , the regenerated kernel of V

is E

(t, s), where

(t, s) = 2

t, 2

s) = 2

k∈Z

φ(2

t − k)φ(2

s − k).

http://www.paper.edu.cn

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