摘要:在隨機(jī)設(shè)計(jì)(模型中所有變量為隨機(jī)變量)下,提出了非參數(shù)計(jì)量經(jīng)濟(jì)模型的變窗寬核估計(jì),并利用概率論中大數(shù)定理和中心極限定理,在內(nèi)點(diǎn)處證明了它的一致性和漸近正態(tài)性.它在內(nèi)點(diǎn)處的收斂速度達(dá)到了非參數(shù)函數(shù)估計(jì)的最優(yōu)收斂速度.

關(guān)鍵詞:非參數(shù)計(jì)量經(jīng)濟(jì)模型 變窗寬 核估計(jì)

Abstract:This paper presents kernel estimators with variable bandwidth for nonparametric regression e-conometric models in the random design case that all variables in models are stochastic. We prove itsconsistency and asymptotic normality in interior by using laws of large numbers and central limittheoremsin probability. Its rates of convergence in interior points equal the optimal rate convergence for estimatingnonparametric function.

Keywords:nonparametric regression econometric model; variable bandwidth; kernel estimation

當(dāng)多元非參數(shù)計(jì)量經(jīng)濟(jì)模型解釋變量的分布不是均勻分布時(shí),若采用不變窗寬核估計(jì),則在密度大的點(diǎn)處由于窗寬相對(duì)較大,過多的觀察點(diǎn)進(jìn)行局部回歸將導(dǎo)致估計(jì)的精度下降,在密度小的點(diǎn)處由于窗寬相對(duì)較小,過少的觀察點(diǎn)進(jìn)行局部回歸也將導(dǎo)致估計(jì)的精度下降.若掌握解釋變量分布的一些信息,對(duì)密度大的點(diǎn)取較小的窗寬,對(duì)密度小的點(diǎn)取較大的窗寬,這樣采用與掌握的信息有關(guān)的變窗寬核估計(jì)將會(huì)提高估計(jì)的效率[1-3].非參數(shù)計(jì)量經(jīng)濟(jì)模型核估計(jì)是一個(gè)深受歡迎的估計(jì)方法[4].研究多元非參數(shù)回歸模型變窗寬核估計(jì)的性質(zhì),得到了變窗寬核估計(jì)的條件漸近偏和方差.在內(nèi)點(diǎn)處證明了它的一致性和漸近正態(tài)性,它在內(nèi)點(diǎn)處的收斂速度達(dá)到了非參數(shù)函數(shù)估計(jì)的最優(yōu)收斂速度.變窗寬核估計(jì)在邊界點(diǎn)處的性質(zhì)將另文討論.

1　非參數(shù)計(jì)量經(jīng)濟(jì)模型的變窗寬核估計(jì)

設(shè)(X1, Y1),…, (Xn, Yn)是Rd+1維獨(dú)立同分布的隨機(jī)變量向量序列,考慮非參數(shù)計(jì)量經(jīng)濟(jì)模型:Yi= m(Xi)+ ui(1)其中:隨機(jī)誤差項(xiàng)序列{ui}是條件均值E(uiXi) =0,條件方差為σ2(x) =Var(uiXi=x)的相互獨(dú)立隨機(jī)變量序列,于是m(Xi) = E(YiXi). 設(shè)f(x)是X1的密度函數(shù),假定inff(x)>0, m(x)的二階偏導(dǎo)數(shù)連續(xù),σ2(x)連續(xù)有界.設(shè)K(·)是d維對(duì)稱密度函數(shù), K(u)≥0,∫K(u)du=1;令Kh(u) = h-dK(h-1u).假設(shè)∫uuTK(u)du=μ2(K)I,其中μ2(K)≠0,I為d×d單位陣;假設(shè)當(dāng)l1+…+ld為奇數(shù)時(shí),∫ul11…ulddK(u)du=0,其中l(wèi)i為非負(fù)整數(shù);假定K(·)的支撐是有界閉集,假設(shè)hn= cn-1/(d+4).m(x)的變窗寬核估計(jì)為:m^n(x, hn,α) =∑ni=1Khn/α(Xi)(Xi-x)Yi∑ni=1Khn/α(Xi)(Xi-x) (2)其中:hn為不變窗寬;α(·)為變窗寬函數(shù).假設(shè)α(·)連續(xù)可微.

2　主要結(jié)論

首先給出了非參數(shù)計(jì)量經(jīng)濟(jì)模型變窗寬核估計(jì)的逐點(diǎn)條件漸近偏和漸近方差.其次,給出了變窗寬核估計(jì)的漸近正態(tài)性的結(jié)論.

定理1　設(shè)x為supp(f) = {xf(x)≠0}的內(nèi)點(diǎn),則1) E{m^n(x,α)X1,…,Xn}-m(x) = h2na(x,α,K)+ op(h2n)其中: a(x,α, K) =α(x)-3∫supp(K)uTDm(x)DTα(x)uuTDK(u)du+μ2(K)[dDTα(x) +α(x)f(x)-1DTf(x)]Dm(x)+12μ2(K)(α(x))-2s(Hm(x))　　; Hm(x) = 2m(x) xi xj d×d, s(·)為矩陣的所有元素之和.2)Var[m^n(x, hn,α)X1,…,Xn] = n-1h-dnR(K)(α(x))dσ2(x)f(x)-1+ op(n-1h-dn)其中R(K) =∫(K(u))2du.由定理1知,變窗寬核估計(jì)的漸近偏和漸近方差將趨于零.

定理2　設(shè)x為supp(f) = {xf(x)≠0}的內(nèi)點(diǎn),則n2/(d+4)[m^n(x,α)-m(x)]dN(c2a(x,α, K), c-dR(K)(α(x))dσ2(x)f(x)-1)　　

由定理2知,變窗寬核估計(jì)具有漸近正態(tài)性.由于漸近方差趨于零,利用大數(shù)定律可知,變窗寬局部線性估計(jì)是一致估計(jì).易見,其收斂速度為O(n-2/(d+4)),該收斂速度達(dá)到了Stone[5]的非參數(shù)函數(shù)估計(jì)的最優(yōu)收斂速度.

3　主要結(jié)論的證明

因?yàn)閅i= m(x)+(Xi-x)TDm(x)+1/2Qmi(x)+ ui(3)其中:Dm(x) = m(x)/ x1　…　 m(x)/ xdT,Qmi(x) = (Xi-x)THm(zi(x,Xi))(Xi-x),zi(x,Xi)-x≤Xi-x,所以,m^n(x, hn,α)-m(x) =∑ni=1Khn/α(Xi)(Xi-x)[(Xi-x)TDm(x)+12Qmi(x)+ ui]∑ni=1Khn/α(Xi)(Xi-x)(4)由Xi相互獨(dú)立,可知zi(x,Xi)相互獨(dú)立.

引理1　①n-1∑ni=1Khn/α(Xi)(Xi-x) =f(x)+ op(1)②　n-1∑ni=1Khn/α(Xi)(Xi-x)(Xi-x)　= h2nα(x)-3f(x)∫supp(K)uDTα(x)uuTDK(u)du+μ2(K)[df(x)Dα(x)+α(x)Df(x)] +op(h2ni) 其中i為元素全為1的列向量、行向量或矩陣(下同).③n-1∑ni=1Khn/α(Xi)(Xi-x)Qmi(x) = h2nf(x)μ2(K)(α(x))-2s(Hm(x))+ op(h2n)④[n-1∑ni=1Khn/α(Xi)(Xi-x)]-1=f(x)-1+ op(1)⑤E[n-1∑ni=1Khn/α(Xi)(Xi-x)ui] =0(nhdn)1/2n-1∑ni=1Khn/α(Xi)(Xi-x)uidN(0, R(K))(α(x))dσ2(x)f(x))只證明引理1②和⑤,其它類似可證.

3.1

引理1②的證明因?yàn)閚-1∑ni=1Khn/α(Xi)(Xi-x)(Xi-x) = E[Khn/α(Xi)(Xi-x)(Xi-x)]+Opn-1Ψ,其中Ψ是VarKhn/α(Xi)(Xi-x)(Xi-x)的對(duì)角元素組成的列向量.因x為內(nèi)點(diǎn),則當(dāng)hn充分小時(shí),supp(K) {z:(x+ hn(α(x))-1z)∈supp(f)}由f、K和α的連續(xù)性,可得到:　E[Khn/α(Xi)(Xi-x)(Xi-x)]=∫supp(f)h-dn(α(X1))dK(h-1n(α(X1)(X1-x))(X1-x)f(X1)dX1=∫Ωn(α(x+ hnQ))dK(Qα(x+ hnQ))f(x+ hnQ)hnQdQ= h2n(α(x))-3{f(x)∫supp(K)DTα(x)uuTDK(u)udu+μ2(K)[df(x)Dα(x)+α(x)Df(x))]+o(h2n)其中Ωn= {Q:x+ hnQ∈supp(f)}.因?yàn)?　VarKhn/α(Xi)(Xi-x)(Xi-x)= E Khn/α(Xi)(Xi-x)(Xi-x)-E[Khn/α(Xi)(Xi-x)(Xi-x)]　Khn/α(Xi)(Xi-x)(Xi-x)-E[Khn/α(Xi)(Xi-x)(Xi-x)]T= E [Khn/α(Xi)(Xi-x)(Xi-x)][Khn/α(Xi)(Xi-x)(Xi-x)]T　- E[Khn/α(Xi)(Xi-x)(Xi-x)] E[Khn/α(Xi)(Xi-x)(Xi-x)]T由f、K和α的連續(xù)性,可得到:　E [Khn/α(Xi)(Xi-x)(Xi-x)][Khn/α(Xi)(Xi-x)(Xi-x)]T= E[(Khn/α(Xi)(Xi-x))2(Xi-x)(Xi-x)T]=∫supp(f)[h-dn(α(X1))dK(h-1α(X1)(X1-x))]2f(X1)(X1-x)(X1-x)TdX1= h-d+2n∫Ωn((α(x+ hnQ))dK(Qα(x+ hnQ)))2f(x+ hnQ)QQTdQ= h-d+2n∫Ωn((α(x))dK(Qα(x)))2f(x)QQTdQ+ o(h-d+2ni) = O(h-d+2ni)易見: Opn-1Ψ= op(h2ni)綜合上述結(jié)論,可知引理1②成立.

3.2　引理1⑤的證明顯然E n-1∑ni=1Khn/α(Xi)(Xi-x)ui= n-1∑ni=1E E Khn/α(Xi)(Xi-x)uiXi=0由f、K、σ2和α的連續(xù)性,可得到:　Varn-1∑ni=1Khn/α(Xi)(Xi-x)ui= n-1Var[Khn/α(Xi)(Xi-x)ui]= n-1∫supp(f)[h-dn(α(X1))dK(h-1nα(X1)(X1-x))]2σ2(X1)f(X1)dX1= n-1h-dn∫Ω2((α(x+ hQ))dK(Qα(x+ hQ)))2σ2(x+ hQ)f(x+ hQ)dQ= n-1h-dn∫Ωn((α(x))dK(Qα(x)))2σ2(x)f(x)dQ+ o(n-1h-dn)= n-1h-dn(α(x))dσ2(x)R(K)f(x)+ o(n-1h-dn)綜合上述結(jié)論,可知引理1⑤成立.

3.3　定理1的證明由引理1②、③,有:　E{m^n(x, hn,α)X1,…,Xn}-m(x) =∑ni=1Khn/α(Xi)(Xi-x)[(Xi-x)TDm(x)+12Qmi(x)]∑ni=1Khn/α(Xi)(Xi-x)= h2nα(x)-3∫supp(K)uTDm(x)DTα(x)uuTDK(u)du+μ2(K)[dDTα(x)+α(x)f(x)-1DTf(x)]Dm(x)　+12μ2(K)(α(x))-2s(Hm(x)) + op(h2n)易見:Var{m^n(x, hn,α)X1,…,Xn} =∑ni=1[Khn/α(Xi)(Xi-x)]2σ2(Xi)∑ni=1Khn/α(Xi)(Xi-x)2容易證明: n-1∑ni=1[Khn/α(Xi)(Xi-x)]2σ2(Xi) = h-dnR(K)(α(x))dσ2(x)f(x)+ op(h-dn)綜合上述結(jié)論和引理1④,可得到:Var{m^n(x, hn,α)X1,…,Xn} = n-1h-dnR(K)(α(x))dσ2(x)f(x)-1+ op(n-1h-dn)

3.4　定理2的證明由引理1②、③、⑤和中心極限定理,易見:n2/(d+4)n-1∑ni=1Khn/α(Xi)(Xi-x)[(Xi-x)TDm(x)+12Qmi(x)+ ui]dN(c2f(x)a(x,α, K), c-dR(K)(α(x))dσ2(x)f(x))再由引理1④,可推得該定理成立.

參考文獻(xiàn):

[1] Fan J, Gjibels I. Local polynomial modeling and its applications[M]. London: Chapman&Hall, 1996.http://www.51lunwen.com/jiliangjingji/

[2] Pagan A, Ullah A. Nonparametric econometrics[M]. Cambridge : Cambridge University Press, 1999.

[3]葉阿忠.非參數(shù)計(jì)量經(jīng)濟(jì)學(xué)[M].天津:南開大學(xué)出版社, 2003.

[4] Wang M P, Jones M C. Kernel smoothing[M]. London: Chapman&Hall, 1995.

[5] Stone C J. Optimal global rates of convergence for nonparametric regression[J]. Annals Statistics, 1982(10): 1 040-1 053.