趙文芝,夏志明

(1.西安工程大學(xué)理學(xué)院,陜西 西安 710048;2.西北大學(xué)數(shù)學(xué)系,陜西 西安 710127)

小波檢測并估計(jì)非參函數(shù)變點(diǎn)

趙文芝1,夏志明2

(1.西安工程大學(xué)理學(xué)院,陜西 西安 710048;2.西北大學(xué)數(shù)學(xué)系,陜西 西安 710127)

研究隨機(jī)設(shè)計(jì)下非參函數(shù)變點(diǎn)的小波檢測與估計(jì)問題.將小波方法與設(shè)計(jì)點(diǎn)轉(zhuǎn)化方法相結(jié)合給出變點(diǎn)的檢測統(tǒng)計(jì)量并研究檢測的一致性.給出了變點(diǎn)個(gè)數(shù)和變點(diǎn)位置的估計(jì)量,證明了變點(diǎn)個(gè)數(shù)估計(jì)量的相合性并得到變點(diǎn)位置估計(jì)量的收斂速度.

變點(diǎn);非參數(shù)回歸模型;小波變換;收斂速度

1 引言

本文考慮非參數(shù)回歸模型

其中εi是均值為0方差為1的i.i.d.序列,Xi是與εi相互獨(dú)立,支撐區(qū)間為[0,1]的設(shè)計(jì)點(diǎn).

考慮如下假設(shè)檢驗(yàn)問題:

H0: f在[0,1]上連續(xù)可微;

H1(m):f存在至少1個(gè)至多m個(gè)不連續(xù)點(diǎn),在其余點(diǎn)是光滑函數(shù).

本文假定回歸函數(shù)f中不連續(xù)點(diǎn)的個(gè)數(shù)和不連續(xù)點(diǎn)的位置都是未知的,但是不連續(xù)點(diǎn)個(gè)數(shù)的上界是已知的.在備擇假設(shè)H1(m)中,m就表示的是這樣的上界.

小波系數(shù)能夠反應(yīng)原函數(shù)的局部特性:在光滑點(diǎn)的小波系數(shù)絕對(duì)值較小而在不連續(xù)點(diǎn)的小波系數(shù)絕對(duì)值較大[1].小波系數(shù)的這種性質(zhì)使之成為處理非參數(shù)回歸模型變點(diǎn)問題的一個(gè)有力工具.選取適當(dāng)?shù)呐R界值,考察小波系數(shù)的絕對(duì)值是否超過給定的臨界值.若小波系數(shù)的絕對(duì)值超過給定的臨界值則認(rèn)為該函數(shù)存在變點(diǎn),否則認(rèn)為該函數(shù)是光滑的.文獻(xiàn)[2]率先用小波方法檢測并估計(jì)非參數(shù)回歸模型變點(diǎn).文獻(xiàn)[3]用小波系數(shù)的累積和對(duì)變點(diǎn)進(jìn)行檢驗(yàn).文獻(xiàn)[4]給出了非參數(shù)回歸模型變點(diǎn)的Minimax估計(jì).上述文獻(xiàn)解決的都是固定等距設(shè)計(jì)且誤差為Gauss過程時(shí)的變點(diǎn)檢測問題.文獻(xiàn)[1]用廣義Pareto分布對(duì)小波系數(shù)進(jìn)行建模,解決了方差有窮的厚尾信號(hào)的變點(diǎn)檢測問題.文獻(xiàn)[5-6]用有界小波分別估計(jì)固定設(shè)計(jì)下以及隨機(jī)設(shè)計(jì)下含有變點(diǎn)的非參函數(shù).文獻(xiàn)[7]用小波方法來檢測異方差自回歸模型的變點(diǎn).文獻(xiàn)[8]用小波方法檢測誤差為 Gauss過程時(shí)分段光滑函數(shù)的變點(diǎn).文獻(xiàn)[9-10]用小波方法檢測非參數(shù)(自)回歸模型的變點(diǎn),文獻(xiàn)[11]用小波方法檢測異方差回歸模型的變點(diǎn),文獻(xiàn)[12]用小波方法檢測回歸函數(shù)導(dǎo)函數(shù)的變點(diǎn).文獻(xiàn)[13]用小波方法檢測并估計(jì)均勻隨機(jī)設(shè)計(jì)下,即模型(1.1)中Xi～U[0,1]時(shí)非參函數(shù)變點(diǎn)的檢測與估計(jì).文獻(xiàn)[14]研究半?yún)?shù)回歸模型估計(jì)的收斂速度.本文研究Xi是一般隨機(jī)設(shè)計(jì)時(shí)非參函數(shù)變點(diǎn)的檢測與估計(jì)問題.

本文需要如下假設(shè)條件:

2 變點(diǎn)的檢測

3 變點(diǎn)的估計(jì)

該收斂速度與文獻(xiàn)[2]的收斂速度一致.文獻(xiàn)[2]的結(jié)論是在i.i.d.正態(tài)隨機(jī)誤差下得到的,且數(shù)據(jù)是在固定設(shè)計(jì)下得到的,而本文結(jié)論是在一般的i.i.d.誤差下得到,且數(shù)據(jù)是在隨機(jī)設(shè)計(jì)下獲得的.

[1]Raimondo M,Tajvidi T.A peaks over threshold model for change point detection by wavelets[J].Statistica Sinica,2004,14:395-412.

[2]Wang Y.Jump and sharp cusp detection by wavelets[J].Biometrika,1995,82(2):385-397.

[3]Odgen T,Parzen O.Change point approach to data analytic thresholding[J].Statistics and Computing, 1996,6:93-99.

[4]Raimondo M.Minimax estimation of sharp change points[J].The Annals of Statistics,1998,26:1379-1397.

[5]Park C,Kim W.Estimation of a regression function with a sharp change point using boundary wavelets[J]. Statistics and Probability Letters,2004,66:435-448.

[6]Park C,Kim W.Wavelet estimation of a regression function with a sharp change point in a random design[J]. Journal of Statistical Planning and Inference,2006,136:2381-2394.

[7]Wong H,Ip W,Li Y.Detection of jumps by wavelets in a heteroscedastic aotogressive model[J].Statistics and Probability Letters,2001,52:365-372.

[8]Antoniadis A,Gijbels I.Detection abrupt changes by wavelet methods[J].Journal of Nonparametric Statistics,2002,14(1-2):7-29.

[9]Li Y,Xie Z.Jump detection by wavelet in nonlinear autogressive models[J].ACTA Mathematica Scientia, 1999,19(3):261-271.

[10]Li Y,Xie Z.The wavelet detection of the jump and cusp points of a regression function[J].ACTA Mathematicae Applicatae Sinica,2000,16(3):283-291.

[11]Zhao Y,Li Y.Detection of the jump points of a heteroscedastic regression model by wavelets[J].ACTA Mathematicae Applicatae Sinica,2001,16(4):420-429.

[12]Luan Y,Xie Z.The wavelet identi fi cation for jump points of derivative in regression model[J].Statistics and Probability Letters,2001,53:167-180.

[13]Zhao W Z,Tian Z,Xia Z M.Wavelet detection and estimation of change points in nonparametric regression models under random design[J].Journal of Mathematical Research and Exposition,2009,29(2):247-256.

[14]朱春浩.誤差為鞅差序列的半?yún)?shù)回歸模型參數(shù)估計(jì)的收斂速度[J].純粹數(shù)學(xué)與應(yīng)用數(shù)學(xué),2008,24(2):306-310.

[15]H¨ardle W,Kerkyacharian G,Picard D,et al.Approximation,and Statistical Application.Lecture Notes in Statistics[M].New York:Springer-Verlag,1998.

[16]David F N,Johnson N L.Statistical treatment of censored data:part I fundamental formulae[J].Biometrika, 1954,41:228-240.

Wavelet detection and estimation change point in nonparametric regression model

Zhao Wenzhi1,Xia Zhiming2
(1.School of Science,Xi′an Polytechnic University,Xi′an 710048,China;
2.Department of Mathematics,Northwest University,Xi′an 710127,China)

This paper considers the detection and estimation problem of change point in nonparametric regression model in a random design.The test statistics is proposed by method of wavelet and design point transformation.The consistence of the test is proved.The consistence of estimator for numbers of change point while the convergence rate for location of change points are given.

change point,nonparametric regression model,wavelet transformation,convergence rate

O212.1

A

1008-5513(2012)01-0041-06

2011-10-01.

陜西省教育廳基金(2010JK561);西安工程大學(xué)基礎(chǔ)研究基金(2010JC07);

國家自然科學(xué)專項(xiàng)基金(數(shù)學(xué)天元)(11026135,11126312);教育部人文社科基金(10YJC910007).

趙文芝(1979-),博士,講師,研究方向：非線性時(shí)間序列分析及應(yīng)用.

2010 MSC:62G08