11

§§ 11§§ .2.2§§ .3.3§§ 7.47.4

22

••

••

33

•• XX11, X, X22,,……,, XXnn XX1

ˆ ˆ( , , )nX X

ˆ

ˆ

44

§§ .1.1

1 2 ,

1

(1 ) , , ,

1( ) , 1 , 2 , ,

m

nk k

ii

E X X k mn

1, 2ˆ ˆ ˆ, , m

55

ˆ( )g ( )g

2

( ) ,( )

E X XD X B

66

.1.1 2020((kmkm))

29.8 27.6 28.3 27.9 30.1 28.729.8 27.6 28.3 27.9 30.1 28.729.9 28.029.9 28.0

27.9 28.7 28.4 27.2 29.5 28.527.9 28.7 28.4 27.2 29.5 28.528.0 30.028.0 30.0

29.1 29.8 29.6 26.929.1 29.8 29.6 26.9

::28.695, 0.918528.695, 0.9185

228.695, 0.9185,x b

77

2

2 ( ),0( ; )

0,

x xf x

88

1( ) ,3

E X

ˆ( ) , 3E X X X

99

3

1̂ 1/ X

2 2ˆ 1/ B

BB22

XEX

2( )D X B

1010

xx11, x, x22,, ……,, xxnn ((a,ba,b))UU((a,ba,b)) aa bb

kk=2=2

a, ba, b

2( ), ( ) ,2 12

a b b aEX D X

3 ( ), 3 ( ),a EX D X b EX D X

2 2ˆˆ 3 , 3a X B b X B

1111

(( ))

ff((xx;; )) xx11,,xx22 ,, ……,, xxnn

LL(( ;; xx11, x, x22,, ……,, xxnn))LL(( ))

1 1 2( ) ( ; , , ) ( ; ) ( ; ) ( ; )n nL L x x f x f x f x

1212

(( ))

1ˆ ˆ( , , )nx x

ˆ( ) m ax ( )L L

ˆ

lnL( )

L( )lnL( )

1313

1 21

( , , , ; ) ( , )n

n ii

L x x x f x

1ln ( , )

n

ii

L ln f x

1414

ln 0, 1, 2, ,i

L i k

1 2ˆ ˆ ˆ, , , k

1515

1( ) , 0( )0, 0

xx e xf xx

1 2, , , nX X X

1616

1

1

(1) ( ) ( ) i

nx

ii

L x e

1 1ln (ln ln ) ( 1) ln

n n

i ii i

L n x x

1

1

ln( 2 ) 0

ˆ

n

ii

n

ii

d L n xd

n

x

1717

22

nnnn11 , n, n22 , n, n33 ((nn11+ n+ n22+ n+ n33 == nn))

2 2

1 2 3, 2 (1 ), (1 )p p p

1 2 3

2 1 2 3 2

2 2

2 2

( ) ( ) [2 (1 )] [(1 ) ]

2 (1 )

n n n

n n n n n

L

1 2 3 2 2ln ( ) (2 ) ln (2 ) ln(1 ) ln 2L n n n n n

1818

00

ˆ

1 2 3 22 20

1

n n n n

1 2 1 2

1 2 3

2 2ˆ2( ) 2

n n n nn n n n

21 2 3 2

2 2 2

ln ( ) 2 20

(1 )L n n n n

1919

33 NN(( ,, 22)) ==(( ,, 22))xx11, x, x22 ,, ……,, xxnn

22

21

2 / 2 22

1

2 2 22

1

( )1( , ) exp22

1(2 ) exp ( )2

1ln ( , ) ( ) ln ln(2 )2 2 2

ni

i

nn

ii

n

ii

xL

x

n nL x

2020

lnlnLL(( ,, 22))0,0,

2

21

ln ( , ) 1 ( ) 0n

ii

L x

22

2 4 21

ln ( , ) 1 ( ) 02 2

n

ii

L nx

2121

22

1

1ˆn

ii

x xn

2 22

1

1ˆ ( )n

ii

x x bn

2222

gg(( ))

ˆˆ( )g

232344 xx11 , x, x22 ,, ……,, xxnn NN(( ,, 22))22

::

22,X B

2ˆ B

3( 3)P X

2

3 XB

2424

5 X1, X2 ,…, Xn 1 2( , )U1 2,

2525

1 1 2

2 1 2

ˆ min( , , , )ˆ max( , , , )

n

n

X X X

X X X

2626

1ˆ ˆ( , , )nX X

ˆ( )E

ˆ

§§ .2.2

2727

BB2222

(1)(1) EE((BB22)) 22

BB2222

(2)(2) BB22

ss22

2 2

1

1 ( )1

n

ii

s x xn

2828

1 21 1

( ) ,

( , , , ) ( 1)n n

n i i ii i

E X

X X X a X a

2929

1 2ˆ ˆ,

1 2ˆ ˆD( ) D( ),

1̂ 2̂

3030

XX11, X, X22 ,, ……,, XXnn22

, ,, ,

nn>1>1

1 1ˆ X 2ˆ X

2 21 2ˆ ˆD( ) , D( ) / n

2ˆ 1ˆ

3131

RaoRao--CramerCramer

),;( xf

xfx }0);(|{ ˆ

)()ˆ( 0DD

20

)),(ln(

1)( XfnED

3232

nn>>00

1ˆ ˆ ( , , )n n nX X

ˆlim (| | ) 1nnP

n̂

)

3333

nn

,,

,,

n̂

n̂

3434

1

ˆ

,0)ˆ(lim,)ˆ( DEn

: 0)ˆ(}|ˆ{| 2nDP

1}|ˆ{|lim Pn

3535

2 pmXppmBX ˆ),,(~

3636

0,0

0,2);(

2

x

xxexfx

?

3737

§§7.37.3

(( ))

X1, X2 ,…, Xn

(0< <1)1 1 1( , , )nX X 2 2 1( , , )nX X

21̂ˆ( ) 1 ,P

3838

( )( )11-- ( )( )11-- ..

..

1 2,

1-100(1- )%

1 2,

1 2

3939(0<(0< <1)<1)

11-- (0<(0<<1)<1) ,,

11--

1

2

1( ) 1P

2( ) 1P

4040

:

UXXU n ),;,,()1( 1

2121 ,,1)()2( UUP

(3) (2))

4141

(( ))

(1)(1) ~ (0,1)XU Nn

2

2

( 2 ) 1 , ,

{ | | } 1

z

P U z

/ 2 /2(3)/

Xz zn

/2 /2X z X zn n

4242

1 ),,(~ 2NX

1.2 3.4 0.6 5.6

3 99%

: (1) 3 )1,0(~/

Nn

XU

/ 2

(2) 1 0.99, 0.01,2.575z

4343

/23(3) 2.7, ( ) (2.7 2.575 )4

( 1.1625, 6.5625)

x x zn

4444

22 99

0.10.1 0.950.95

11-- ==0.950.95 ==0.050.050.950.95

0.950.95(15.3347, 15.4653)(15.3347, 15.4653)

15.4x

/2( ) (15.4 1.96 0.1 9) (15.4 0.0653)x z n

/2 1.96z

4545

33 NN(( ,1,1))0.950.95 1.21.2

0.950.95

nnnn (2/1.2)(2/1.2)22zz22

/2/2.. 11--= 0.95= 0.95 zz /2/2=1.96=1.96 nn (5/3)(5/3)22 1.961.9622 ==

10.6710.67 1111 11110.950.95 1.21.2

/ 22 z n

/ 22 1.2z n

/2 /2( / , / )x z n x z n

4646

22

t1-

2

( ) ~ ( 1)/

XT t ns n

/2 /2( ( 1) , ( 1) )S SX t n X t nn n

2 21 ( )1 is x x

n

4747

4),(~ 2NX

20

(510(1832 Sx

95%

4848

: )1(~/

)1( ntnS

XT

/2 0.025

(2) 1 0.95, 0.05.( 1) (19) 2.093t n t

/ 2(3) 95% ( 1) )

510(1832 2.093 ) (1593.3, 2070.7)20

Sx t nn

4949

55

1212

4.68 4.85 4.32 4.85 4.61 5.024.68 4.85 4.32 4.85 4.61 5.025.20 4.60 4.58 4.72 4.38 4.705.20 4.60 4.58 4.72 4.38 4.70:(1):(1)

)1(~/

ntnS

XT

)05.0(

5050

0.95

(4.7092 2.2010 0.0615 / 12) (4.5516,4.8668)

0 .0 2 5

( 2 )1 0 .9 5 , 0 .0 5 ,/ 2 ( 1) (1 1) 2 .2 0 1 0t n t

0615.0,7092.4)3( 2sx

5151

11--

tt 0.050.05(11)=(11)=1.79591.79590.950.95 4.58064.5806

( )( ( 1) ) 1/

xP t ns n

( 1)x t n s n

5252

:, X

5353

21

221

222

211

211

,),,,),,(~

,),,,),,(~

2

1

YX

YSYYNY

XSXXNX

n

n

5454

1 12

22

2 12 , 2

2

2 2 2 21 2 1 2

/2 /21 2 1 2

,X Y z X Y zn n n n

(1)

2 2 2 21 2 1 2 1 2

2 21 2

/ 21 2

( , 50), , , ,

( )

n n S S

S SX Y zn n

5555

/2 1 2 /2 1 21 2 1 2

1 1 1 12 , 2w wX Y S t n n X Y S t n nn n n n

222

21)2(

2)()1(

21

222

2112

nnSnSnS w

5656

11 1818

881010

//

628 583 510 554 612 523 530628 583 510 554 612 523 530615615

535 433 398 470 567 480 498535 433 398 470 567 480 498560560

503 426503 426

5757

XX11 ,, ……, X, X88 YY11 ,, ……, Y, Y1010

=569.3750=569.3750 SSxx22 =2140.5536=2140.5536 nn11=8=8

=487.0000=487.0000 SSyy22=3256.2222=3256.2222 nn22 =10=10

xy

2 21 2

1 2

1 1 7 2110.5536 9 3256.2222 52.48802 16

x yw

n s n ss

n n

)2(~11

)()()1( 21

21

21 nnt

nnS

YXT

w

5858

/ 2 1 2 0.025(2) 0.05, 2 16 2.1199t n n t

1 21 2

1 1 1 1(3) 2 2.1199 52.4880 52.77978 10wt n n s

n n

1 2 (29.5953, 135.1547)

595922

22 2

2

( 1)(1) ~ 1n s n

2 2/ 2 1 / 2

22 2

1 / 2 / 22

( 1) , ( 1) ,

1

( 2 )

1

n n

n sP

1 - ,

2

2 2 2 22 1 2

(3)

1 1 , 1 1n s n n s n

1-

6060

22 NN(( ,, 22))99

45.3 45.4 45.1 45.3 45.5 45.7 45.4 45.3 45.645.3 45.4 45.1 45.3 45.5 45.7 45.4 45.3 45.60.950.95

)1(~)1()1( 22

22 nSn

180.2)8(,535.17)8(05.0,95.01)2(

2975.0

20.025

)3454.0,1218.0()1193.0,0148.0(%95,0325.0)3( 22S

6161

1122// 22

22

2 2 2 21 1 1 2

1 22 2 2 22 2 1 2

/ ~ 1, 1/

s s sF F n ns

22

21

221

222

211

211

,),,,),,(~

,),,,),,(~

2

1

YX

YSYYNY

XSXXNX

n

n

)1(~)1(),1(~)1(2

222

222

12

21

211 nSnnSn

6262

2 21 12 22 2 1 2 2 1 2 1 2

1 1,1, 1 1, 1

s ss F n n s F n n

2 21 2

1 2 1 2 2 1 22 22 1

1, 1 1, 1 1sP F n n F n ns

),1,1(),1,1(,1

212/1

212/

nnFnnF

22

21

6363

33

55 66

5.06 5.08 5.03 5.00 5.075.06 5.08 5.03 5.00 5.074.98 5.03 4.97 4.99 5.024.98 5.03 4.97 4.99 5.02

4.954.9522// 22

0.950.95

6464

:0.975

0.025

0.05,1 14,5 0.1068

5,4 9.36F

F

0.025 4, 5 7.39F

00092.0,00099.0,99.4,048.5

22

21 SS

yx

)072.10,1463.0(

%9522

21F