抽样调查第二章

第二单随机抽

简单估差估计与回归简单随机抽样的基本从有U(N)=U1U2,UN中，抽取n个单元作样D

第i个单元Yi被抽0 第i个单元Yi未被抽0N(1的向量，它指示了一个具体的样全部这样的D组成一个样本空间方案确定上的一个概率分布N2.1.对简单随机抽样

1}n,

1,

1}n(n1),ij,i,j1,2,...,NN(N1)证明

1,

Yi被抽Y未被抽 P(Di 1)

nn NnANnANN证明

1,

Yi被抽Y未被抽 P(Di 1,D

1)

n(n1)AnA NA N2

n(nN(N2.1.对简单随机抽样Nii

ii (1n),ij,i,j N(N 证明

1,

Yi被抽Y未被抽 E(D)P( 1) Var(D)P( 1)P( 0)n(1n 证明Cov(Di,DjE(DiDj)E(Di)E(DjP(Di1,Dj1)P(

1)P(Djn(n1) nnN(N (1nN(N 2.1.设y1y2yn)是来自总体Y1Y2的简单随机样本Yi有界,即存在一个与N无关MYN

Mi1,2,...,N 1i ii

0E(y)Y

O(1E( ) n

N)SY (Nn)(N2n

3O(1E( )

n2N(N1)(N

Y ninE(y)Y

O(1E( ) n

N)SY 3E( )3

(Nn)(N2nn2N(N1)(N2

NNi

3O(1 n4(4)E( )4

(Nn)[N2(6n1)N6n2n3N(N1)(N2)(N3

NNiYii3(n1)(Nn)(Nn1) n3N(N1)(N2)(N3

Y2

O(n2ii i其中y

1 ni

y,S

N

i

i证明 y

yi nin

YiDininNE(y)1N

YE(D)1

n1

YYNiiniNii

ni

i

NiN

1 n

Y

DiYiYjDiDj i

i

i 1 E(y)

n

E(Di)YiYjE(DiDji i 1

n(n1)2Yi i1

i

YiY

N(N1)证明E(y2)

1

n(n1)n2niN

YiYji N N

N(N1)i0(NY)2(ii

Y)

YiiYiN

YiYji Y

Yi

ii1E(y2)

1

n

n(n1)nN nNi

i

N(N1)(11

O(1 N

Yii1

n

N)SY 证明

3

YiDii 1 n3

Di3

YjDiD

YiYjYkDiDjDki 1

i

i E(y)

3E(D)3

2YE(DDn3

i

i YiYjYkE(DiDjDk)i NN NN

n(nn3 N

YiY

N(NNNNN

i i YY n(n1)(n2) i

N(N1)(N证明0(NY

(Y

NYN

3Y

2Y

YY NiNi N

iN

i

i

202

Yi

YiYi2

Yi3

Yi2Yji i i i i YY 2Yi

iNY

2YY i

i 证 YiYjYkij k

NN2 YiiYi3N3i

Yi2Y iiiE(y3)

n(n1n Y

N3

YiY

N(N1)i i n(n1)(n2) i

Y

jY

N(N1)(N2

n(

2

3 3

(1

n

)2(

in (n1)(n2 n N

N

1(N1)(N2))证明3E(y n n

n (n1)(n iin3

(1

)N

N (

1)(N2))Yi N (n1)(Nn) n2N

N

2(N1)(N

Yii1 (N

(n1) 1n2N(N1)1

2(N

ninY(Nn)(N Y

3O(1n2N(N1)(N

Y i证明 1

iN1

n n i

i

6Y2YYDDDYYYYDDDD ij ij

4

nN nN

i

j

i

j

n(n1)(n2)(n3)

N(N1)(N2)

ij ij i

i

N ki ij ij i N

N

i

i

ij

iiii iiii a+4b+3c+6d+e=0a+2b+c+d a+b

i

b=-c+d=ae=-3db=- c+d= e=-

2

)

faidac2a即e3d3fN

i1

4E(y)4

1Y4

4

3Y2Y2nN nN

i

jN(N

i

jN(N n(n1)(n2)

NYYYYn(n1)(n2)(n3)

ij ij1an4bn(n1)3cn(n1) N(Ne n(n1)(n2)(n

N(N N(N1)(NN(N1)(N2)(Nbcfadac2a即E(y4)

1

n

N(N N(N N(N1)(Ne N(N1)(N2)(N 1 N(N N(N n4 n(n1)(n2)(n3) N(N1)(N2)(2af)N(N1)(N2)(N3)(3f E(y

f

N(N1)(N2) )f N(N1)(N2)a的系数1[(17(n1)12(n1)(n2) N(N N(N1)(N N(N1)(N2)(N1[(1

n1

)(

6(n1)(n2) N(N N(N N(N1)(N(N(N1)(N

N(N1)(N2)(N [(1n1)6n1(1n2)6(n1)(n2)(1n3 N N N2 (N1)(N N3 [Nn6n1Nn6(n1)(n2) N1 N1N2 (N1)(N2)N N [16n16(n1)(n2)n3N(N N2 (N2)(Na的系数N

[16n1

(n1)(n n3N(N N (N2)(N3) Nn3N(N

[16n1(1n2N N N

[16n

Nn1n3N(N N N(Nn)[N2(6n1)N6n2n3N(N1)(N2)(N3)f的系数3(n1)[ 2(n (n2)(n N(N N(N1)(N N(N1)(N2)(N 3(nn3N(N 3(nn3N(N 3(n

[12(n2)(n2)(n3)N (N2)(N[(1n2)(n2(n2)(n3)N N (N2)(N[Nnn2(1n3n3N(N1)N N N 3(n1) [Nnn2Nn]n3N(N1)N2 N2N3 3(n1)(N [1n2]3(n1)(Nn)(Nnn3N(N1)(N N n3N(N1)(N2)(N4E(y

(Nn)[N2(6n1)Nn3N(N1)(N2)(N

a3(n1)(Nn)(Nn1)fn3N(N1)(N2)(N3)(Nn)[N2(6n1)Nn3N(N1)(N2)(N

n3N(N1)(N2)(N3) iO(1)O(1)O(1 iai

fii注:对有限总体Y1Y2作简单随机抽样则Yi必有界,即存在一个与N无关的数M,使YiM(i1,2,...,N ZiYiY i1,,N则 N则N

Zi

(Y

Y)N1i iN1Zi2M(i1,2,...,N推 设(y1,y2,...,yn)是来自有限总Y1Y2,...,YN的简单随机样本E(yY)E(yY

(1

1)S

(Nn)(N E(yY

n2N(N1)(N

i

Y)O(n2E(yY)4

(Nn)[N2(6n1)N6n2n3N(N1)(N2)(N

(YiYN4iN43(n1)(Nn)(Nn1)

n3N(N1)(N2)(N

(YYiii

简单随机抽样的对一个总体进行简单随机抽样首先要有一个抽样框,将总体的N单元与N个形成一一对应概率从N个单元中抽取第一个样本单元,取目前利随机数表、计算机、掷随机等例1现利用下表从N=345的总体中无放回地抽取一个n=15的简单随机样本。先将总体单元与001-0345对应。当 例2(例1在例1中可以看到在抽取随机数时经常 例3随机数方法用该方法.抽样者给总体的每一个赋予一完成.即对i单元对应一个0,1间的随机数Ri,Ri与i单元对应,在以后各次抽样中不变。抽样设计时,确定好抽样比f,则Rif的例3随机数方法抽样设计时,确定好抽样比f,则Rif的样比基本上会是f，但会有一些摆动。例一商店为了了解顾客对商店服意见,在商店门口对走出商店的顾客 简单估值一般对一个抽样方案给出的抽样归估计等2.3.设y1y2yn)是来自U=Y1,Y2,...,YN的一个样本量为n的简单随机样本,n(1)样本均yn

y是总体均 i1

Y Ni1

的无偏估计

y的均方偏差

V(y)E(yY)

NS2N

Y).证明: 1y yi Yini niE(y)

NN1nYi1n

E(Dii1Y nni iNi1YNi1i证明(2Vy)EyYE(y)E(y)又y

nNininNi

NN1n Yi 1nNiNV(y)Var(n

i

YiDi)

Var(2 i2

YiDin

iN

Var(YiDi)

i ji

Cov(YiDi,YjDjii iin i

Y2Var(D)

i ji

YiYjCov(Di,Dj1 ii iin i

Y2Var(D)

i ji

YiYjCov(Di,DjNi 1n证明Ni 1nV(y)

i1

YiYjCov(Di,Dji1 ijN212

Y2n(1n)

YY

(1ni i N

i i

jN(N 1n(1n

Y2

YY1

i

i1 ij

jN1 1

iY2i

YYY2

1 i

Ni N

i

N1 N N

N N Ni

NiN

证明V(1f

2

YiYjN N1i N1i N 1fn(N1)

NNi

2NY2Yi(11)S21(1n)SYi fNiNiS2 (YY)2

Y2NY2N

i

N1i

fN

称为抽样V(y)1(1n)S21fS 1NY Yii1N

称为总体均值2

Y)称为总体方差N

i注:(2)有放回简单随机抽样时 V(y) (YiY NiN1S不放回简单随机抽样时V(y)1(1n)S NnS由此可见,不放回简单随机抽样优于有简单随机抽样,二者Nn1fN2.3在简单随机抽样下，样本方 s2

(nn

y)2v(y)1(1n 是估y的均Vy的无偏估计证明

2 n

(i

n

i

ny n

Y Y

Dny2

NE(s2)i N Yn1i

iE(Di)nE(yVar(y)1(1n Y2n

n1i

2n

2n i1

Y2i nY2in1i

(1N

)S2nY21 2 21 N

(Yii

Y

N

Yii

E(s2)

n

i

(1

n)SN

nY2 2 n Yn i

NnYn1(1N

2 N(n

Yii

NY n1(1Nn(N1)S2

(1n)SN(n n nNNS2SN(n例某一社区居民用于食物消费序户人数(x)(人人均月收入户月食物支出12233444534例 n=35,N=平均每月每户食物消费的支出的估yn

nni

y

3135035

895.71(元上述估计的标准差的估计值v(yn1(1v(yn1(1n)sN1(1 35)s例2(部分估米的土地内,蝗蝻的数量时,往往要就该查单位,全部共有108个单位,其中属于耕 当第i 单位是耕 当第i 单位是荒NZZi即为所需估计的子总体i 按简单估值可用N

nin

zi估算总和.其均方偏差为ENzZ2

N2(1n (

Z)2i)iN(Nn) 1

N