線性模型中的加權(quán)混合幾乎無偏兩參數(shù)估計(jì)

左衛(wèi)兵， 康萃雯

(華北水利水電大學(xué) 數(shù)學(xué)與信息科學(xué)學(xué)院，河南 鄭州 450046)

線性模型中的加權(quán)混合幾乎無偏兩參數(shù)估計(jì)

左衛(wèi)兵， 康萃雯

(華北水利水電大學(xué) 數(shù)學(xué)與信息科學(xué)學(xué)院，河南 鄭州 450046)

主要討論了隨機(jī)約束線性模型的有偏估計(jì)問題,提出了一種新的加權(quán)混合幾乎無偏兩參數(shù)估計(jì).證明了加權(quán)混合幾乎無偏兩參數(shù)估計(jì)在二次偏差的準(zhǔn)則下優(yōu)于加權(quán)混合兩參數(shù)估計(jì),并給出了在均方誤差矩陣準(zhǔn)則下新估計(jì)優(yōu)于其他相關(guān)估計(jì)的充要條件.

加權(quán)混合兩參數(shù)估計(jì)；加權(quán)混合幾乎無偏兩參數(shù)估計(jì)；二次偏差；均方誤差矩陣

0 引言

考慮含隨機(jī)約束線性模型

(1)

其中y是n維觀測向量,X為n×p的列滿秩設(shè)計(jì)陣,β是p維未知參數(shù)向量,ε是n維誤差向量,且E(ε)=0,E(εεT)=σ2I,I是n階單位矩陣.H為r×p行滿秩矩陣,e為隨機(jī)擾動(dòng)項(xiàng)且E(e)=0,Cov(e)=σ2W,W為已知正定陣.

模型(1)的研究起源于文獻(xiàn)[1]提出的普通混合估計(jì),當(dāng)先驗(yàn)信息和樣本信息重要性不一致時(shí),文獻(xiàn)[2]提出了加權(quán)混合估計(jì).但當(dāng)模型存在多重共線性時(shí),混合估計(jì)存在明顯不足.因此,目前諸多學(xué)者通過把不同估計(jì)結(jié)合起來探討新的估計(jì),這些估計(jì)比原估計(jì)更優(yōu).文獻(xiàn)[3-4]將混合估計(jì)分別與Liu估計(jì)和嶺估計(jì)結(jié)合,提出了隨機(jī)約束Liu估計(jì)和隨機(jī)約束嶺估計(jì),文獻(xiàn)[5-6]將加權(quán)混合估計(jì)分別與Liu和幾乎無偏嶺估計(jì)結(jié)合,得到了加權(quán)混合Liu和加權(quán)混合幾乎無偏嶺估計(jì),文獻(xiàn)[7]在文獻(xiàn)[8]的基礎(chǔ)上將兩參數(shù)估計(jì)算子作用于混合估計(jì),提出了隨機(jī)約束兩參數(shù)估計(jì).本文通過將幾乎無偏兩參數(shù)估計(jì)引入到加權(quán)混合估計(jì)中,提出了加權(quán)混合幾乎無偏兩參數(shù)估計(jì),并在均方誤差矩陣準(zhǔn)則下與現(xiàn)有的加權(quán)混合估計(jì)等進(jìn)行優(yōu)良性比較.

1 新估計(jì)的提出

對(duì)于模型(1),文獻(xiàn)[2]提出了加權(quán)混合估計(jì),其定義為

(2)

其中S=X′X,ω(0<ω<1)為常數(shù)權(quán)重系數(shù).對(duì)于不帶約束的線性模型,文獻(xiàn)[9]通過組合嶺估計(jì)和Liu估計(jì)提出了一種新的兩參數(shù)估計(jì)，

(3)

(4)

其中可定義Skd為幾乎無偏兩參數(shù)估計(jì)算子.

對(duì)于帶隨機(jī)約束的線性模型,本文將定義的新的兩參數(shù)估計(jì)算子和幾乎無偏兩參數(shù)估計(jì)算子分別作用于加權(quán)混合估計(jì),提出了如下的加權(quán)混合兩參數(shù)估計(jì)和加權(quán)混合幾乎無偏兩參數(shù)估計(jì)，

(5)

(6)

(7)

(8)

(9)

2 偏差比較

(10)

(11)

βT[(TkFd-I)T(TkFd-I)-(Skd-I)T(Skd-I)]β.

(12)

由上述定理知,加權(quán)混合幾乎無偏兩參數(shù)估計(jì)是對(duì)加權(quán)混合兩參數(shù)估計(jì)的偏差進(jìn)行矯正的估計(jì).

3 均方誤差矩陣比較

為了進(jìn)一步考慮在偏差和方差共同作用下新估計(jì)的優(yōu)良性,我們利用均方誤差矩陣準(zhǔn)則MSEM,將其與相關(guān)估計(jì)進(jìn)行比較.記A=(S+ωHTW-1H)-1,

(13)

(14)

(15)

(16)

其中b1=[(2I-Tk)Tk-I]β=-(Tk-I)2β,b2=[(2I-TkFd)TkFd-I]β=-(TkFd-I)2β,b3=b2.

證明 由式(13)和式(16)可得

證明 由式(14)和式(16)可得

4 數(shù)據(jù)模擬

為了說明上述估計(jì)的性質(zhì),這一節(jié)我們用具體的數(shù)據(jù)來分析.本文進(jìn)行如下數(shù)據(jù)模擬,數(shù)據(jù)來源于文獻(xiàn)[14],數(shù)據(jù)如下：

表和的均方誤差 (ω=0.2)

表和的均方誤差 (ω=0.4)

表和的均方誤差 (ω=0.7)

5 結(jié)論

本文針對(duì)帶有隨機(jī)約束線性模型的參數(shù)估計(jì)問題,在加權(quán)混合估計(jì)和幾乎無偏兩參數(shù)估計(jì)的基礎(chǔ)上,提出了加權(quán)混合幾乎無偏兩參數(shù)估計(jì),并在二次偏差和均方誤差矩陣準(zhǔn)則下,對(duì)新估計(jì)和相關(guān)估計(jì)進(jìn)行了比較分析,結(jié)果表明加權(quán)混合幾乎無偏兩參數(shù)估計(jì)在二次偏差準(zhǔn)則下一致優(yōu)于加權(quán)混合兩參數(shù)估計(jì),而在均方誤差矩陣準(zhǔn)則下分別優(yōu)于加權(quán)混合估計(jì),加權(quán)混合幾乎無偏嶺估計(jì),幾乎無偏兩參數(shù)估計(jì)的充要條件,最后通過數(shù)據(jù)模擬計(jì)算驗(yàn)證了我們的結(jié)論.

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Weighted Mixed almost Unbiased Two Parameter Estimator in Linear Regression Model

ZUO Weibing, KANG Cuiwen

(CollegeofMathematicsandInformationScience，NorthChinaUniversityofWaterResourcesandElectricPower，Zhengzhou450046，China)

Some biased estimators of random restricted linear regression model were discussed and a new weighted mixed almost unbiased two parameter estimator was proposed. Furthermore, proved that the new estimator performed better than weighted mixed two parameter estimator under the rule of quadratic deviation. Under the mean square error matrix criterion, the necessary and sufficient conditions for the new estimator to be superior to other estimators were given out.

weighted mixed two parameter estimator; weighted mixed almost unbiased two parameter estimator; quadratic deviation; mean square error matrix

2016-10-26

河南省基礎(chǔ)與前沿技術(shù)研究項(xiàng)目(142300410401)

左衛(wèi)兵(1976—)，男，河南內(nèi)黃人，華北水利水電大學(xué)數(shù)學(xué)與信息科學(xué)學(xué)院教授，主要研究方向：數(shù)理統(tǒng)計(jì).

10.3969/j.issn.1007-0834.2017.01.001

O212.1

A

1007-0834(2017)01-0001-05