程 靖, 岳榮先

(1.安徽農(nóng)業(yè)大學(xué) 理學(xué)院,安徽 合肥 230036; 2.上海師范大學(xué) 數(shù)理學(xué)院,上海 200234)

二次隨機(jī)系數(shù)回歸模型的A-最優(yōu)設(shè)計(jì)

程 靖1, 岳榮先2

(1.安徽農(nóng)業(yè)大學(xué) 理學(xué)院,安徽 合肥 230036; 2.上海師范大學(xué) 數(shù)理學(xué)院,上海 200234)

研究了單位設(shè)計(jì)域上二次隨機(jī)系數(shù)回歸模型在恒等設(shè)計(jì)類(lèi)中的A-最優(yōu)設(shè)計(jì).首先證明了A-最優(yōu)設(shè)計(jì)準(zhǔn)則滿(mǎn)足洛納偏序性質(zhì);利用A-最優(yōu)設(shè)計(jì)準(zhǔn)則的洛納偏序性質(zhì),證明了單位設(shè)計(jì)域上的二次隨機(jī)系數(shù)回歸模型在恒等設(shè)計(jì)類(lèi)中的A-最優(yōu)設(shè)計(jì)可以在包含設(shè)計(jì)域兩個(gè)端點(diǎn)0,1在內(nèi)的3個(gè)設(shè)計(jì)點(diǎn)上獲得;進(jìn)而給出了二次隨機(jī)系數(shù)回歸模型A-最優(yōu)恒等設(shè)計(jì)的精確結(jié)果.結(jié)果表明,二次隨機(jī)系數(shù)回歸模型在恒等設(shè)計(jì)類(lèi)中的A-最優(yōu)設(shè)計(jì)不受到隨機(jī)效應(yīng)項(xiàng)的方差影響,其A-最優(yōu)設(shè)計(jì)的第三個(gè)譜點(diǎn)位置接近單位設(shè)計(jì)域的中點(diǎn),且在該點(diǎn)處的A-最優(yōu)設(shè)計(jì)權(quán)重接近于0.5.

A-最優(yōu)設(shè)計(jì); 二次回歸; 隨機(jī)系數(shù); 恒等設(shè)計(jì)

0 引 言

隨機(jī)系數(shù)模型在生物學(xué)、心理學(xué)、經(jīng)濟(jì)學(xué)和藥代動(dòng)力學(xué)等領(lǐng)域的研究中被廣泛應(yīng)用,這些研究的目的往往不是了解個(gè)體自身的特性,而是個(gè)體所在總體的特性,此時(shí)將個(gè)體效應(yīng)引入模型可大大提升模型的精度.早在20世紀(jì)90年代Liski等[1]就應(yīng)用隨機(jī)系數(shù)模型描述了林地類(lèi)型、林分密度、氣候以及遺傳等因素對(duì)木材的影響,Pena等[2]則將隨機(jī)系數(shù)模型應(yīng)用于質(zhì)量檢測(cè).近期也有很多關(guān)于隨機(jī)系數(shù)回歸模型的研究成果,李拂曉等[3]將隨機(jī)系數(shù)自回歸模型的變值點(diǎn)應(yīng)用于股票價(jià)格的監(jiān)測(cè);Tjiong[4]采用隨機(jī)系數(shù)Logit回歸模型重新評(píng)估了英國(guó)非工作行程時(shí)間節(jié)省的評(píng)估指標(biāo);Feng & Zhang[5]采用隨機(jī)系數(shù)的隨機(jī)前沿方法研究了1997～2010年間美國(guó)大型銀行的規(guī)模收益.

隨機(jī)系數(shù)模型的最優(yōu)設(shè)計(jì)近十幾年來(lái)受到越來(lái)越多的重視,Schmelter等[6],Schwabe & Schmelter[7]分別討論了隨機(jī)截距模型和隨機(jī)斜率模型在單位設(shè)計(jì)域上的最優(yōu)設(shè)計(jì);Debusho & Haines[8]針對(duì)具有隨機(jī)截距的回歸模型給出了均值參數(shù)在離散設(shè)計(jì)域上的D-最優(yōu)和V-最優(yōu)設(shè)計(jì);程靖和岳榮先[9-10]討論了兩變量隨機(jī)系數(shù)回歸模型的最優(yōu)設(shè)計(jì);Cheng & Yue[11]獲得了一階異方差隨機(jī)系數(shù)回歸模型的幾類(lèi)最優(yōu)設(shè)計(jì);Grabhoff等[12]研究了由隨機(jī)系數(shù)引起異方差的線性回歸模型的最優(yōu)設(shè)計(jì)軌跡的幾何形狀;Liu等[13]研究了隨機(jī)系數(shù)回歸模型的R-最優(yōu)設(shè)計(jì).

隨機(jī)系數(shù)回歸模型存在較為復(fù)雜的方差-協(xié)方差結(jié)構(gòu),已有的成果多集中在對(duì)低階隨機(jī)系數(shù)回歸模型最優(yōu)設(shè)計(jì)的研究.本文作者以一元二次隨機(jī)系數(shù)回歸模型為研究對(duì)象,討論其在單位設(shè)計(jì)域上的A-最優(yōu)設(shè)計(jì).首先證明了A-最優(yōu)設(shè)計(jì)準(zhǔn)則滿(mǎn)足偏序性質(zhì),進(jìn)而證明了一元二次隨機(jī)系數(shù)回歸模型的A-最優(yōu)設(shè)計(jì)可以在包含設(shè)計(jì)域兩個(gè)端點(diǎn)0,1的3個(gè)設(shè)計(jì)點(diǎn)上獲得,并給出了A-最優(yōu)設(shè)計(jì)的精確結(jié)果.結(jié)論表明,一元二次隨機(jī)系數(shù)回歸模型的A-優(yōu)設(shè)計(jì)不依賴(lài)與隨機(jī)效應(yīng)項(xiàng)的方差.

1 模型分析

考慮單位設(shè)計(jì)域[0,1]上的一元二次隨機(jī)系數(shù)回歸模型

(1)

Cov(Yi)

模型(1)可以表示為

由此得到參數(shù)向量θ的最佳線性無(wú)偏估計(jì)量及估計(jì)量的協(xié)方差陣分別為

2 A-最優(yōu)設(shè)計(jì)

在許多實(shí)際試驗(yàn)中出于實(shí)際需要和操作的要求,都采用恒等設(shè)計(jì),即每個(gè)個(gè)體觀測(cè)次數(shù)和觀測(cè)點(diǎn)都相同.對(duì)模型(1)也考慮恒等設(shè)計(jì),即mi=m,Xi=X,i=1,2,…,n,進(jìn)而有Fi=F,Vi=V,i=1,2,…,n.此時(shí)

由此可得,在恒等設(shè)計(jì)下,對(duì)任意一近似設(shè)計(jì)ξ滿(mǎn)足

(2)

上式中M0(ξ)為普通一元二次回歸模型在近似設(shè)計(jì)ξ下的信息陣.不失一般性,令σ2=1.

引理 A-最優(yōu)設(shè)計(jì)準(zhǔn)則Φ滿(mǎn)足洛納偏序,即對(duì)?M(ξ1)≥M(ξ2),有

Φ(M(ξ1))≤Φ(M(ξ2)).

對(duì)A-最優(yōu)設(shè)計(jì),Φ(M(ξ))=Tr(M-1(ξ)),很容易發(fā)現(xiàn)上述引理成立.對(duì)于普通的一元二次回歸模型,其A-最優(yōu)設(shè)計(jì)可以在包含設(shè)計(jì)域端點(diǎn)的3個(gè)設(shè)計(jì)點(diǎn)上獲得,下面將這一結(jié)論推廣至二次隨機(jī)系數(shù)回歸的情形.

定理 模型(1)在恒等設(shè)計(jì)下,對(duì)任意一個(gè)p(3≤p≤m)點(diǎn)設(shè)計(jì)

(3)

存在一個(gè)形如下式的三點(diǎn)設(shè)計(jì):

(4)

使得M(ξ*)≥M(ξ).

證明 記在設(shè)計(jì)ξ,ξ*下,普通一元二次回歸模型的信息陣分別為M0(ξ),M0(ξ*),則有

令

解得

ω1=1-ω2-ω3.

容易看出0<λ<1,ω2>0,ω3<1.由柯西不等式可得

⑤蔣介石自己也承認(rèn)，“國(guó)府成立以來(lái)，各種設(shè)施，百分之九十九悉依漢民之主張”(《國(guó)府紀(jì)念周蔣主席報(bào)告胡辭職經(jīng)過(guò)》，《大公報(bào)》1931年3月6日，第1張第3版)。

ΣδixiΣδixi(1-xi)2≥[Σδixi(1-xi)]2.

故有ω3>0.而

故有ω3+ω2<1,綜上所述,有0<ω1,ω2,ω3<1.此時(shí)

由柯西不等式知上式中

故有

結(jié)合(2)式的結(jié)論有M-1(ξ*)≤M-1(ξ)?M(ξ*)≥M(ξ).

由定理容易得出結(jié)論:對(duì)于滿(mǎn)足偏序性質(zhì)的A-最優(yōu)設(shè)計(jì)一定可以在形如(4)的設(shè)計(jì)類(lèi)中獲得.在設(shè)計(jì)(4)下,信息陣滿(mǎn)足

對(duì)于A-最優(yōu)設(shè)計(jì),有

其中A=ω2λ+ω3,B=ω2λ2+ω3,C=ω2λ3+ω3,D=ω2λ4+ω3.運(yùn)用模擬退火優(yōu)化程序得到ω1=0.3277,ω2=0.4934,ω3=0.1789,λ=0.4898時(shí)Tr(M-1(ξ*))取得最小值,因此得到二次回歸系數(shù)方程在[0,1]2上的A-最優(yōu)設(shè)計(jì)為

在二次隨機(jī)系數(shù)回歸模型中A-最優(yōu)設(shè)計(jì)不依賴(lài)于隨機(jī)效應(yīng)項(xiàng)的方差,可以在包含0,1兩個(gè)端點(diǎn)在內(nèi)的3個(gè)設(shè)計(jì)點(diǎn)上獲得,且第三個(gè)設(shè)計(jì)點(diǎn)在中點(diǎn)0.5附近,在該點(diǎn)上有接近0.5的權(quán)重.

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(責(zé)任編輯：馮珍珍)

A-optimal design of quadratic random coefficient regression model

Cheng Jing1, Yue Rongxian2

(1.College of Science,Anhui Agricultural University,Hefei,Anhui 230036,China 2.College of Mathematics and Science,Shanghai Normal University,Shanghai 200234,China)

A-optimal identical design of quadratic random coefficient regression model on the design domain of [0,1] is constructed in this paper.Loewner order character of A-optimal criterion is proved in the paper and it is proved that A-optimal identical design of quadratic random coefficient regression model could be obtained on three design points including the two extreme settings of the design domain of [0,1].Accurate result of A-optimal identical design of quadratic random coefficient regression model is given in the paper.The result shows that A-optimal identical design of quadratic random coefficient regression model does not depend on the variances of the random effects.The result also shows that the third spectral point of A-optimal design is near the midpoint of the design domain of [0,1] and it′s weight coefficient of A-optimal design is near 0.5.

A-optimal design; quadratic regression; random coefficient; identical design

2015-11-30

國(guó)家自然科學(xué)基金(11401056,11471216);安徽省環(huán)保廳項(xiàng)目(2015-5);安徽農(nóng)業(yè)大學(xué)穩(wěn)定與引進(jìn)人才項(xiàng)目(yj2015-27)

程 靖(1979-),女,副教授,主要從事試驗(yàn)設(shè)計(jì)與分析方面的研究.E-mail:chengjing@ahau.edu.cn

O 212.6

A

1000-5137(2017)02-0195-05