多元分層抽樣樣本量的分配-

1、內(nèi)蒙古工業(yè)大學(xué)學(xué)報JOU RNAL O F I NN ER M ON GOL I A 第26卷第2期UN I V ER S IT Y O F T ECHNOLO GYV o l.26N o .22007文章編號:1001-5167(2007022*多元分層抽樣樣本量的分配盧靜莉,閆在在,丁昌江(內(nèi)蒙古工業(yè)大學(xué)理學(xué)院,呼和浩特010051摘要:本文給出了精度意義下多元分層隨機抽樣樣本量的最優(yōu)分配,證明了在不包含不合理分配的情況下,最優(yōu)分配下多元分層隨機抽樣的精度最高,比例分配下多元分層隨機抽樣的精度其次,多元簡單隨機抽樣的精度最低.關(guān)鍵詞:簡單隨機抽樣;分層抽樣;比例分配;最優(yōu)分配中圖分類號:O

2、 212.4文獻標(biāo)識碼:A0引言1多元簡單隨機抽樣定義1設(shè)p 元有限總體7:X 1=(X 11,X 12,X 1p ,X 2=(X 21,X 22,X 2p ,X N =(X N 1,X N 2,X N p 從中不放回地抽取容量為n 的樣本x 1=(x 11,x 12,x 1p ,x 2=(x 21,x 22,x 2p ,x n =(x n 1,x n 2,x np .這種抽樣法稱為多元簡單隨機抽樣.記X =1N6Ni =1X i ,總體均值2=1N -16Ni =1(X i -X (X i -X ,總體協(xié)差陣x =1n6ni =1x i ,樣本均值2=1n -16ni =1(x i -x (

3、x i -x ,樣本協(xié)差陣收稿日期:2006201205作者簡介:盧靜莉(1978,女,講師,研究方向:計算數(shù)學(xué).f =nN,抽樣比容易提到,對于多元簡單隨機抽樣,x是X 的無偏估計,即E (x =X ,并且x具有方差V a r (x =1n(1-f 2.2多元分層隨機抽樣定義2設(shè)p 元有限總體7分成L 層:71,72,7L,從每層抽取簡單隨機樣本,合成總體的一個樣本,這種抽樣法稱為多元分層隨機抽樣.正像一元抽樣,多元分層抽樣主要適用于層內(nèi)單元指標(biāo)差異較小,而層間差異較大時,此時分層抽樣的精度較高.記7i =X i 1,X i 2,X i N i X ij =(X (1ij ,X (2ij ,

4、X (p ij i =1,2,L .j =1,2,N iN =6Li =1NiW i =N iN,i =1,2,L ,稱為層權(quán)f i =n iN i,第i 層抽樣比,i =1,2,L f =nN總抽樣比第i 層簡單隨機樣本為:x i 1,x i 2,x in i ,i =1,2,L ,n =6Li =1n iX i =1Ni 6N ij =1X ij 第i 層總體均值2i =1N i -16N ij =1(Xij-Xi (X ij-X i 第i 層總體方差x i =1n i 6n ij =1x ij 第i 層樣本均值i =1n i -16n ij =1(x ij -x i (x ij -x i

5、 第i 層樣本方差X =1N6Li =16N ij =1Xij=6Li =1W i X i 總體均值x st =1N6Li =1N i x i =6Li =1W i x i 樣本均值定理2.1對于多元分層隨機抽樣,xst 是X 的無偏估計,即E (x st =X (2.1證明因為每層都是簡單隨機抽樣所以有E (xi =X i ,進而E (x st =E (1N6Li =1N i x i =1N6Li =1N i E (xi =1N 6Li =1N i X i =X 證畢.28內(nèi)蒙古工業(yè)大學(xué)學(xué)報2007年定理2.2對于多元分層隨機抽樣,有V (x st =6Li =1W 2i1-fin i2i(

6、2.2證明因為每層都是簡單隨機抽樣所以V (xi =1-f i n i2i ,進而V (x st =V (6Li =1W i x i =6Li =1W 2i v (x i =6Li =1W2i1-fin i2i 證畢當(dāng)抽樣按比例進行時即N i N =n in(2.2式變?yōu)閂 (x st =1-f n6Li =1W i 2i (2.3對于多元隨機抽樣,由于它的方差為一個矩陣,因此它的精度大小常用協(xié)差陣的行列式值或跡來刻畫.3多元分層隨機抽樣的最優(yōu)分配定義3在多元分層隨機抽樣中,對給定費用C ,使精度tr V (xst 達到最小,或?qū)o定的x st 的精度tr V (x st =,使總費用最小的各

7、層樣本量的分配稱為最優(yōu)分配.本文考慮簡單線性費用函數(shù),總費用C =c 0+6Lh =1c h n h(3.1其中c 0是與單元抽取量無關(guān)的費用,c h 是在第h 層中抽取一個單元的調(diào)查費用.定理3.1對于多元分層隨機抽樣,費用函數(shù)C =c 0+6Lh =1c h n h 則最優(yōu)分配是n hn=Whtr (2h c h6Lh =1Whtr (2h c h=Nhtr (2h c h 6Lh =1Nhtr (2h c h(h =1,2,L (3.2證明令C =C -c 0=6Lh =1c hn h由于V (x st =6Lh =1W2h1-fhn h2h =6Lh =1W2h2hn h-6Lh =1

8、W2h2hNh=6Lh =1W2h2hn h-6Lh =1Wh2hN所以tr V (x st =tr 6Lh =1W2h2hn h-tr 6Lh =1Wh2hN則tr V (x st 最小等價于tr 6Lh =1W 2h2hn h最小令V =6Lh =1W 2h2hn h在給定費用C 下極小化tr V (xst 與給定tr V (x st =下極小化C 兩者都等價于極小化:tr (V C =tr (6Lh =1W2h2hn h (6Lh =1c h n h =(6Lh =1W2htr (2h n h (6Lh =1c h n h =6Lh =1(Whtr (2h n h26Lh =1c h n

9、 h 2由Cauchy 2Schw arz 不等式,可知38第2期盧靜莉等多元分層抽樣樣本量的分配tr (V C =6Lh =1(Whtr (2h n h26L h =1c h n h 2(6Lh =1Whtr (2h n hc h n h 2等號當(dāng)且僅當(dāng)c h n hWhtr (2h n h=K (K 為常數(shù)時才成立.即當(dāng)tr (V C 達到極小值時n h =KWhtr (2h c h(h =1,2,L 則由n =6Lh =1n h=K 6L h =1Whtr (2h c h得出K =n 6L h =1Whtr (2h c h所以使tr (V C 達到極小值的最優(yōu)分配為:n h n=Whtr

10、 (2h c h6Lh =1Whtr (2h c h=Nhtr (2h c h 6Lh =1Nhtr (2h c h(h =1,2,L 如果各層單位抽樣費用相等,即c h =c 時最優(yōu)分配簡化為:n hn=Whtr (2h 6Lh =1Whtr (2h =Nhtr (2h 6Lh =1Nhtr (2h (h =1,2,L (3.3把n h =Whtr (2h 6Lh =1Whtr (2h n 代入(2.2式得到最優(yōu)分配下的精度為:m intr V (xst =6Lh =1W htr (2h 2n-6Lh =1W h tr (2h N(3.44多元簡單隨機抽樣與相同樣本量的多元分層隨機抽樣的比例

11、分配和最優(yōu)分配的精度比較在通常情況下,多元分層隨機抽樣的精度要比簡單隨機抽樣的精度高.由于分層隨機抽樣的精度與樣本量的分配有密切關(guān)系,因此這里不包括明顯不合理分配的多元分層隨機抽樣.定理4.1若1N h1,(h =1,2,L 則最優(yōu)分配(各層費用相等的情形下多元分層隨機抽樣x st 的方差V op t 、比例分配多元分層隨機抽樣xst 的方差V p rop 與多元簡單隨機抽樣x 的方差V srs 之間有如下關(guān)系:tr (V op t tr (V p rop tr (V srs 證明(N -12=6Lh =16N hi =1(Xh i-X(X hi-X=6Lh =16N hi =1(X h i-

12、Xh +(X h -X (Xh i -X h +(X h -X =6Lh =16N hi =1(Xh i-X h (Xh i-X h +6Lh =16N hi =1(Xh i-Xh (X h -X 48內(nèi)蒙古工業(yè)大學(xué)學(xué)報2007年+6L h=16N h i=1(Xh-X(X h i-Xh+6L h=16N h i=1(Xh-X(Xh-X其中6L h=16N hi=1(X h i-Xh(Xh-X=0,6Lh=16N hi=1(Xh-X(X h i-Xh=0所以(N-12=6L h=16N h i=1(X h i-Xh(X h i-Xh+6L h=1N h(Xh-X(Xh-X=6L h=1(N h

13、-12h+6L h=1N h(Xh-X(Xh-X兩端除以N-1有2=6L h=1N h-1N-12h+6L h=1N h N-1(Xh-X(Xh-X由于對于所有的h,1N h1所以N h N-1N h-1N-1W h從而有26L h=1W h2h+6L h=1W h(Xh-X(Xh-Xtr(2tr6L h=1W h2h+6L h=1W h(Xh-X(Xh-X=tr(6L h=1W h2h+tr6L h=1W h(Xh-X(Xh-X由于tr6L h=1W h(Xh-X(Xh-X0所以tr(2tr(6L h=1W h2h即tr(V p roptr(V srs根據(jù)最優(yōu)分配的定義,tr(V op tt

14、r(V p rop因此有tr(V op ttr(V p roptr(V srs證畢.參考文獻:1馮士雍.倪加勛,鄒國華.抽樣調(diào)查理論與方法M.北京:中國統(tǒng)計出版社,1998.2許寶馬錄.抽樣論M.北京:北京大學(xué)出版社,1982.3范金城,閆在在.多元抽樣技術(shù)( J.工程數(shù)學(xué)學(xué)報,1998,15(4:97105.4閆在在,范金城.多元抽樣技術(shù)( J.內(nèi)蒙古工業(yè)大學(xué)學(xué)報,1999,18(1:1621.5M ing T an,Hong2B in Fang,Guo2L iang T ian,etc.T esting M ultivariate N o r m ality in Incomp lete

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16、ied M a them a tics and Com p u ta tion, 2004, 153: 505 511. 7 于秀林, 任雪松. 多元統(tǒng)計分析 M . 北京: 中國統(tǒng)計出版社, 1999. TH E ALLO CA T I ON O F SAM PL E S IZE O F M U L T I R I T E VA A STRA T IF IED SAM PL I G N (S chool of B asic S ciences, I nner M ong olia U n iv ersity of T echnology , H ohhot 010051, C h ina L

17、U J ing 2li, YAN Za i2za i, D I G Chang 2jiang N Abstract: In th is p ap er, the a lloca t ion p rob lem of sam p le size of random m u lt iva ria te st ra t ified sam p ling is stud ied. It is p roved tha t, in ca se tha t im p rop er a lloca t ion is excluded, the efficiency of op t im um a lloca t ion of random m u lt iva ria te st ra t ified sam p ling app ea rs to be the h ighest, nex t