周 玲,何道江

(安徽師范大學(xué) 數(shù)學(xué)計(jì)算機(jī)科學(xué)學(xué)院,安徽 蕪湖 241003)

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相依誤差線性模型中的主成分s-K估計(jì)

周 玲,何道江

(安徽師范大學(xué) 數(shù)學(xué)計(jì)算機(jī)科學(xué)學(xué)院,安徽 蕪湖 241003)

為同時(shí)克服線性回歸模型的自相關(guān)性和回歸變量間的復(fù)共線性,通過融合主成分回歸估計(jì)和s-K估計(jì),提出一類新估計(jì),稱為主成分s-K估計(jì);并在均方誤差陣意義下,得到了這類估計(jì)分別優(yōu)于廣義最小二乘估計(jì)、主成分估計(jì)、r-k和s-K估計(jì)的充要條件.Monto Carlo數(shù)值模擬表明,新估計(jì)是一種同時(shí)克服自相關(guān)性和復(fù)共線性的有效方法.

自相關(guān)性;復(fù)共線性;主成分回歸估計(jì);s-K估計(jì);均方誤差陣

為了克服統(tǒng)計(jì)學(xué)中線性模型的復(fù)共線性問題,常用的方法是使用有偏估計(jì).如Stein估計(jì)[1]、主成分回歸(PCR)估計(jì)[2]、普通嶺(ORR)估計(jì)[3]、Liu估計(jì)[4]和s-K估計(jì)[5]等.此外,融合兩種不同估計(jì)可能會保留這兩種估計(jì)的優(yōu)點(diǎn).Baye等[6]將PCR估計(jì)與ORR估計(jì)融合,提出了r-k估計(jì);Chang等[7]將PCR估計(jì)與兩參數(shù)估計(jì)[8]融合,提出了主成分兩參數(shù)估計(jì)(PCTP).為了克服模型中自相關(guān)的影響,Aitken[9]運(yùn)用OLS技術(shù)引入了廣義最小二乘(GLS)估計(jì);吳燕等[10]基于模型的參數(shù)信息提出了一類新的s-K估計(jì).但此時(shí)模型中的復(fù)共線性可能仍然存在,進(jìn)而GLS估計(jì)由于具有很大的方差而給出不可靠的估計(jì).目前,同時(shí)解決自相關(guān)和復(fù)共線性問題的研究已有許多結(jié)果[11-17].本文為同時(shí)克服自相關(guān)誤差和復(fù)共線性問題,通過融合PCR估計(jì)和s-K估計(jì),提出一類新的估計(jì),稱為主成分s-K估計(jì),并進(jìn)一步考察新估計(jì)相對于這些現(xiàn)有估計(jì)的優(yōu)良性.

1 新估計(jì)量的定義

考慮如下線性回歸模型:

(1)

其中:Y是n×1維可觀測隨機(jī)向量;X是n×p維列滿秩陣;β是p×1維未知參數(shù)向量;ε是n×1維誤差向量;V是一個(gè)已知的n×n階正定矩陣.于是,存在一個(gè)n×n階非奇異陣P,使得P′P=V-1.用P左乘式(1),則模型(1)可寫成

(2)

記Y*=PY,X*=PX,ε*=Pε,則式(2)可表達(dá)為

(3)

式(3)即為轉(zhuǎn)換模型[11].

Λr=diag(λ1,λ2,…,λr),Λp-r=diag(λr+1,λr+2,…,λp).

對于轉(zhuǎn)換模型,由文獻(xiàn)[18]可知,r-k估計(jì)[6]可寫為

(4)

(5)

其中k≥0和0

(6)

將X*和Y*分別代換成X和Y的關(guān)系式,則模型(1)的s-K估計(jì)可寫成

(7)

其中:s≥1;K=diag(k1,k2,…,kp),且ki≥0,i=1,2,…,p.

下面給出β的一個(gè)新估計(jì),它由PCR估計(jì)和s-K估計(jì)融合而成,形式如下:

(8)

(9)

(10)

(11)

(12)

2 新估計(jì)量在均方誤差陣意義下的優(yōu)良性

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

證明:由式(14),(15)得

(22)

且C可寫為

(23)

因此,有

(24)

(25)

等價(jià)于式(21).證畢.

在式(21)中,取r=p,可得:

(26)

(27)

此為文獻(xiàn)[16]的結(jié)論.

(28)

此為文獻(xiàn)[11]的結(jié)論.

這里(U?v)是一個(gè)酉矩陣(U可能不存在),Δ是一個(gè)正定對角陣(當(dāng)U存在時(shí)才出現(xiàn)),且λ是一個(gè)正數(shù).進(jìn)一步,條件1)～3)均不依賴于廣義逆D-∈G(D)的選擇.

(29)

有時(shí)候也會想，其實(shí)現(xiàn)實(shí)世界并不是全然美好的，而是曲折、復(fù)雜的，要不要把這樣的面貌如實(shí)呈現(xiàn)在小人兒面前呢？可就好像蓋樓房，首先要做的是打地基，你可以說樓房是高高地往上去蓋的，可是地基卻得深深地向下去打啊！2歲多的孩子，還處于主要是模仿、重復(fù)大人的語言，而自己的思考能力才剛剛起步的階段，我選擇先用那些光明、美好、積極的材料為他打下地基，為他將來面對世界的復(fù)雜性準(zhǔn)備下基本的心理和情感資源。

(30)

另一方面,

因此,充分條件化為

類似地,可得:

3 數(shù)值模擬

為了進(jìn)一步考察所提估計(jì)類的均方誤差,下面進(jìn)行Monte Carlo數(shù)值模擬.設(shè)計(jì)矩陣X=(xij)n×p由下式給出:

(31)

其中ωij(i=1,2,…,n;j=1,2,…,p+1)是獨(dú)立的標(biāo)準(zhǔn)正態(tài)偽隨機(jī)數(shù),且γ是給定的數(shù),γ2表示任意兩個(gè)解釋變量之間的相關(guān)系數(shù).響應(yīng)變量由下式給出:

(32)

這里ε=(ε1,ε2,…,εn)′是均值為0、協(xié)方差陣為σ2V的正態(tài)隨機(jī)變量.

(33)

分別取ρ=0.5,0.8.與文獻(xiàn)[12,16]一致,取β的真實(shí)值為X′V-1X最大特征值所對應(yīng)的標(biāo)準(zhǔn)化特征向量.此外,取s=1.01,1.001.為方便,K=diag(k1,k2,k3,k4,k5)分別取為A1,A2,A3,B1,B2,B3,其中:

A1=diag(0.1,0.1,0.1,0.1,0.1);A2=diag(0.1,0.1,1,1,1);A3=diag(0.1,1,1,1,1);

B1=diag(1.5,1.5,1.5,1.5,1.5);B2=diag(1.5,1.5,15,15,15);B3=diag(1.5,15,15,15,15).

表1 當(dāng)s=1.01,ρ=0.5,k=0.1時(shí)各估計(jì)的均方誤差Table 1 Estimated MSE values with s=1.01,ρ=0.5,k=0.1

表2 當(dāng)s=1.01,ρ=0.5,k=1.5時(shí)各估計(jì)的均方誤差Table 2 Estimated MSE values with s=1.01,ρ=0.5,k=1.5

表3 當(dāng)s=1.01,ρ=0.8,k=0.1時(shí)各估計(jì)的均方誤差Table 3 Estimated MSE values with s=1.01,ρ=0.8,k=0.1

表4 當(dāng)s=1.01,ρ=0.8,k=1.5時(shí)各估計(jì)的均方誤差Table 4 Estimated MSE values with s=1.01,ρ=0.8,k=1.5

表5 當(dāng)s=1.001,ρ=0.5,k=0.1時(shí)各估計(jì)的均方誤差Table 5 Estimated MSE values with s=1.001,ρ=0.5,k=0.1

表6 當(dāng)s=1.001,ρ=0.5,k=1.5時(shí)各估計(jì)的均方誤差Table 6 Estimated MSE values with s=1.001,ρ=0.5,k=1.5

表7 當(dāng)s=1.001,ρ=0.8,k=0.1時(shí)各估計(jì)的均方誤差Table 7 Estimated MSE values with s=1.001,ρ=0.8,k=0.1

表8 當(dāng)s=1.001,ρ=0.8,k=1.5時(shí)各估計(jì)的均方誤差Table 8 Estimated MSE values with s=1.001,ρ=0.8,k=1.5

綜上,本文提出了一個(gè)新的估計(jì)量同時(shí)克服模型的自相關(guān)性和復(fù)共線性.在均方誤差陣意義下,比較了新估計(jì)量與GLS,PCR,r-k和s-K估計(jì)量,并給出了新估計(jì)量優(yōu)于其他估計(jì)量的條件.數(shù)值模擬表明,新估計(jì)是一種同時(shí)克服自相關(guān)性和復(fù)共線性的有效方法.

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(責(zé)任編輯：趙立芹)

PrincipalComponentss-KClassEstimatorintheLinearModelwithCorrelatedErrors

ZHOU Ling,HE Daojiang

(SchoolofMathematicsandComputerScience,AnhuiNormalUniversity,Wuhu241003,AnhuiProvince,China)

To combat autocorrelation in errors and multicollinearity among the regressors in linear regression model,we proposed a new estimator by combining the principal components regression (PCR)estimator and thes-Kestimator.Then necessary and sufficient conditions for the superiority of the new estimator over the GLS,the PCR,ther-kand thes-Kestimators were derived by the mean squared error matrix criterion.Finally,a Monte Carlo simulation study was carried out to investigate the performance of the proposed estimator.

autocorrelation;multicollinearity;principal components regression estimator;s-Kestimator;mean squared error matrix

10.13413/j.cnki.jdxblxb.2015.03.17

2014-07-16.

周 玲(1989—),女,漢族,碩士研究生,從事數(shù)理統(tǒng)計(jì)的研究,E-mail:lingzhou1989@163.com.通信作者:何道江(1980—),男,漢族,博士,教授,從事數(shù)理統(tǒng)計(jì)的研究,E-mail:djheahnu@163.com.

安徽省自然科學(xué)基金(批準(zhǔn)號:1308085QA13).

O212.2

：A

：1671-5489(2015)03-0444-07