統計解析 19 02web.sfc.keio.ac.jp/~maunz/DSB19/DSB19_02.pdf"â ∑;+^ " ∑; ". n '= " =02 ( *

R( ) (

+ (

)-

* x rl rn Q

* : n l Q

. +/

( : ) : (

: )

X R Y

V

S)••

•• S X

Y

• S V• V S8 LLI : A B

••

•

•• X Y•••• !" #$% &$%• X Y• ' ("

•• V•

• g ), ++ = -. + -0)

nx

nXY

•-. -0

nX-. -0s Y•

V n

2 ..= - (- /

3AF :(2 -(1 ( (

s-0 -.

12

)• r o

r o v• s• o

r or o rl

• s r

• X Yn

n

•)0 )"• s+2 = -. + -0)02 + -")"2 + 12

r s

• + 3 )0, )", … , )5 r6n s 7 = 1,… , 6 Q

+0 = -. + -0)00 + ⋯+ -5)05 + 10+" = -. + -0)"0 + ⋯+ -5)"5 + 1"

⋮+2 = -. + -0)20 + ⋯+ -5)25 + 12

⋮+; = -. + -0);0 + ⋯+ -5);5 + 1;

• n

< =

+0+"⋮+2⋮+;

, = =

1 )00 ⋯1⋮1

)"0⋮)20

⋯⋱⋯

⋮1

⋮);0

⋱⋯

)05)"5⋮)25⋮);5

, ? =

-.-0-"⋮-5

, @ =

101"⋮12⋮1;

• n s

< = =? + @

-• S• S• S• S• S 375 475• S X Y• S• S6• S

• @ = < − =? BB = < − =? C < − =? = <C< − 2?C=C=< + ?C=C=?

•EBE?

= −2=C< + 2=C=? = 0

•

=C=? = =C<

• =C= G0 v H?

H? = =C= G0=C<

)• rn

• =r I Σ n=~L I, Σ s=? + @~L =? + @, ?Σ?C

• x @r 0 M" n@~L 0, M"$Q

• < =? M"$ n

< = =? + @~L =?, M"$

• ?R @R <H?R N<R O n

•H? = =C= G0=C<~L ?, M" =C= G0

•N< = =H? = = =C= G0=C< = P<~L =?, M"P

P = = =C= G0=C

•O = < − N< = $ − P <~L 0, M" $ − P

P

• n

H?~L ?, M" =C= G0

• n

H? − ?

M" =C= G0~L 0, 1

• V = V ? M"r o<r s n

• 7 )2 = 1, )20, … , )25 +2s n

Q +2|)2; ?, M" =1

2TM"U)Q −

+2 − )2? "

2M"

• p

•V +2|)2; ?, M" = )2?

•W +2|)2; ?, M" = M"

E yi | xi ;β,σ2( ) = xiβV yi | xi ;β,σ

2( ) =σ 2

• 7

Q +|=; ?, M" =X2Y0

;

Q +2|)2; ?, M"

=X2Y0

;1

2TM"U)Q −

+2 − )2? "

2M"

=1

2TM"

;

U)Q −∑2Y0; +2 − )2? "

2M"

•

ln Q <|=; ?, M" = ln X2Y0

;

Q +2|)2; ?, M"

= ln1

2TM"

;

U)Q −∑2Y0; +2 − )2? "

2M"

= −62ln2T −

62lnM" −

12M"

]2Y0

;+2 − )2? "

• V d

•

]2Y0

;+2 − )2? "

yi − xiβ( )2

i=1

n∑

-4 -2 0 2 4

-12

-10

-8-6

-4-2

Values of parameter

Log

likel

ihoo

d

山登り

対数尤度関数が最大となる点を点推定

• Vn

]2Y0

;+2 − )2? "

•H? =

∑2Y0; )2+2∑2Y0; )2

"

M" =1

6 − 3 + 1]2Y0

;

+2 − )2H?"

0 6 − (3 + 1)

• V = V ? M"r o+r s

n

Q <|=; ?, M" =1

2TM"U)Q −

+ − =? "

2M"

• p

•V <|=; ?, M" = =?

•W <|=; ?, M" = M"

E y | X ;β,σ 2( ) = Xβ

• V

Q <|=; ?, M" =1

2TM"U)Q −

< − =? "

2M"

=1

2TM"U)Q −

< − =? C < − =?2M"

• V

ln Q <|=; ?, M" = ln1

2TM"U)Q −

< − =? C < − =?2M"

= −62ln2T −

62lnM" −

< − =? C < − =?2M"

• e r

H? = =C= G0=C<

M" =1a< − =H?

C< − =H?

a = 6 − 3 + 1c o

• b2 cde % cd ce• g ) = )0, … , ); + = +0, … , +;

), g+ Q s•

cde =∑2Y0; ) − )2 g+ − +2

6 − 1•

cd =∑2Y0; ) − )2 "

6 − 1

ce =∑2Y0; g+ − +2 "

6 − 1

b =cdecdce

=∑2Y0; ) − )2 g+ − +2

∑2Y0; ) − )2 " ∑2Y0

; g+ − +2 "

•

• −1 ≤ b ≤ 1 Q• b = 0 s• b = 1 s

• + cee cdd l• x

cee =]2Y2

;+2 − g+ "

cdd =]2Y2

;)2 − ) "

• + cee VcC

• ci = ⁄cde" cdd ckci n l r s

ck = cee − ci =]2Y2

;+2 − +2 "

ci = cee − ck =]2Y2

;+2 − g+ "

• ceeS + V• ckS s• ciS s

• ci cee !"r!" =

cicee

= 1 −ckcee

• X Y!" V• X Y

b" =cdecdce

"

=⁄cde" cddcee

=cicee

= !"

• Mk" ck Mk" =lm;G5

• !"!r

! = 1 −∑2G0; +2 − +2 "

∑2G0; +2 − g+ " = 1 −

ckcee

• o r)0 )" g X3 = 2Y l sr r)0

r)" r)0 )" gn i r o

rl• r3 s pv

25 l r•n XAdj. !"Y#$% &$% rl

Adj. !"

• X Y!" f• r n

Adj. !"r• X Y !" ck cee

Adj. !" = 1 −⁄ck 6 − 3 − 1⁄cee 6 − 1

= 1 −

∑2G0; +2 − +2 "

6 − 3 − 1∑2G0; +2 − g+ "

6 − 1

• 6 3 l

#$%

• #$% rlnrs 3 x

#$% = −2lnrs + 23

•

lnrs = −62ln 2TM" −

< − =H?C< − =H?

2M"• M" M" n

M" =< − =H?

C< − =H?

6

#$%

• r lnrs n s

lnrs = −62ln 2T

< − =H?C< − =H?

6−62

• #$% n x

#$% = −2lnrs + 23

= 6 ln 2T∑2Y2; +2 − +2 "

6+ 1 + 23

#$%

• #$% n X62T 6Yn v

6 t ln∑2Y2; +2 − +2 "

6+ 23

• #$% r n v #$%r xr x

L &$%• o

&$% = −2lnrs + 23 t ln 6

• r n n

&$% = 6 ln 2T∑2Y2; +2 − +2 "

6+ 1 + 23 t ln 6

• #$% &$%r x ro

• x U2 = +2 − +2XY r

• U2 X 7 Y M"nQU2~L 0, M"

• ck nck =]

2Y0

;+2 − +2 " =]

2Y0

;+2 − u-. − u-0)2 − ⋯− -5)25

"

• 6 − 3 + 1 − 1 = 6 − 3vk" n

vk" =∑2Y0; +2 − +2 "

6 − 3

• U2 = +2 − +2 g

•

]2Y0

;U2 =]

2Y0

;+2 − +2 = 0

•

]2Y0

;U2)2 =]

2Y0

;+2 − +2 )2 = 0

• 1r n 1~L 0, M"$ Un

U~L 0, M" $ − = =C= G0=C

= =

1 )00 ⋯1⋮1

)"0⋮)20

⋯⋱⋯

⋮1

⋮);0

⋱⋯

)05)"5⋮)25⋮);5

• P = = =C= G0=C

-

• P 7 X Y ℎ22 n•

16< ℎ22 < 1

• 3 r

]2Y0

;ℎ22 = 3 + 1

• X Y Vn r s

• U2

Wyb U2 = M" 1 − ℎ22

• X YU2z n

U2z =

U2M

• U2z ℎ22 V

n s

U2M 1 − ℎ22

• vk"

M" = vk" =∑2Y0; U2

"

6 − 3 − 1

• V L 0,1 n

• Wyb U2 = M" 1 − ℎ22 l ℎ22rf vU2 M" x

• s r V 7 VV r o r s o

• N< = P< l r

+2 = ℎ20+0 + ℎ2"+" + ⋯+ ℎ22+2 ⋯+ ℎ2;+;

• ℎ22r s 7 V +2r+2o r s

{•

H? = =C= G0=C<~L ?, M" =C= G0

•

u-5 = ~L -5, M" =C= 55G0

•

u-5 − -5M" =C= 55

G0 ~L 0, 1

{

• V l M" M" s o

u-5 − -5M" =C= 55

G0 ~{ 6 − 3 − 1

• r l nP.: u-5 = 0 { o

• r lr n

• P.: H? = 0 P.: u-0 = ⋯ = u-5 = 0 s

• ci vk"rM"

• ci x 3 (" n

ci =]2Y2

;+2 − g+ "

• +2 n+2~L =2?, M"P22

• g+2 = g+ l +2e}Gge~ �}}

n+2 − g+

M P22~L 0, 1

• e}Gge

~ �}}

"3 (" n

• 3 + 1 r o3 + 1 − 1 = 3

("

• =2r I M" n=2~L I, M" sx Ä2 =

Å}GÇ~

nÄ2~L 0,1

• sÄ2" (" (" 1 n r

Ä2" =

=2 − IM

"

~(" 1

• É = ∑2Y05 Ä2

" 3 (" (" 3 nÉ =]

2Y0

5Ä2" ~(" 3

0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

0.5

0.6

x

dchi

sq(x

, 1)

("("

(" (" 1

(" 3

(" 5(" 10

(" 20

("

• 3 (" (" 3 Ü ); 3 0 ≤ )r s ) < 0 s

Ü ); 3 =1

25"Γ

32

UGd")

5"G0

• Γ à

• 3 ("3p 23

'

• ci 3 vk"'

' =⁄ci 3

vk"

• ' n

' =⁄ci 3

vk"= â∑2Y2; +2 − g+ "

3

∑2Y0; U2

"

6 − 3 − 1

• 3 (" 6 − 3 − 1 ("

' ("

• É0r 30 (" (" 30 É"r 3" ("(" 3" É0 É"r l

• s É0 É"⁄äã 5ã⁄äå 5å

'30, 3" ' ' 30, 3" n

' =⁄É0 30⁄É" 3"

~' 30, 3"

'

• 30, 3" ' ' 30, 3" Ü ); 30, 3"r s

Ü ); 30, 3" =Γ30 + 3"

2 )5ãG""

Γ302 Γ

3"2 1 +

303")

5ãç5å"

303"

5ã"

' {

• L 0,1 nÄ 3 (" (" 3 nÉrl s { 3 { n

{ =Ä

⁄É 3• (

{" =Ä"

⁄É 3• Ä" (" (" 1 n {" 1, 3' ' 1, 3 n

• cC6 − 1 (" nX cC cee Y

cC =]2Y2

;+2 − g+ "

• ck n x 6 − 3 − 1 ("n

ck =]2Y2

;+2 − +2 "

• s cC ck ci x

cC = ck + ci

• n r s Q

]2Y2

;+2 − g+ " =]

2Y2

;+2 − +2 " +]

2Y2

;+2 − g+ "

6 − 1 6 − 3 − 1 3

• ci 3 vk" '3, 6 − 3 − 1 ' ' 3, 6 − 3 − 1 n

' =⁄ci 3

vk"= â∑2Y2; +2 − g+ "

3

∑2Y0; U2

"

6 − 3 − 1

• s P.: u-0 = ⋯ = u-5 = 0 l• + 3, 6 − 3 −1 ' /+ 'é• ' 3, 6 − 3 − 1 > 'é s• ' 3, 6 − 3 − 1 ≤ 'é s

• 'n

'

ci 3 ci ' =⁄ci 3

vk"

ck 6 − 3 − 1 vk" =ck

6 − 3 − 1

cC 6 − 1

(• ( - / g f

f• V V B L•• )02 )02 +2 • .

(•

rl n•

•rl n

•

INC1

20 30 40 50

50100

150

200

2030

4050

CONS1

50 100 150 200 0.5 1.0 1.5 2.0

0.5

1.0

1.5

2.0

WORK

•n

/.+ ./.+ . /. . /

• n {+ = 12.04 + 0.187)0 + 2.98)"

• u-0 = 0.19 > 0 u-0 { 2 ) . +• u-" = 2.98 > 0 u-" { 2( + +• !" = 0.977

• u-0 u-" s or s

.( ) .!" = 0.977

( +

• n+ = 12.04 + 0.187)0 + 2.98)"

• u-0 u-" s or s

• )0rf o r .- 2 .-☓ Y o• )"rf f o r(/. 2( /.☓ Y o• r

.( ) .!" = 0.977

( + {

• fS + = -. + -0)0• gS + = -. + -0)0 + -")"• S + = -. + -")"• #$% &$%

n

Adj. !" #$% &$%

f / / -+ -. )g /-- - ) - /

-() / -

• Vr n s Q

r

•• 9 9• 8• X Y

Q• +2 U2

• r2M or nV n

n s Vl rl

• 7 = 1, 4, 13r rl

20 30 40 50

−3−2

−10

12

Fitted values

Resid

uals

lm(CONS1 ~ INC1 + WORK)

Residuals vs Fitted

13

1

4

Q• V

9 9• 9 9 V

n s +aV r rlQ r v+a x

• rg or

rl• 7 = 1, 13, 18r rl

−2 −1 0 1 2

−2−1

01

Theoretical QuantilesSt

anda

rdize

d re

sidua

ls


Normal Q−Q

1318

1

Q• V

• r 2 o nV r l

r

• 7 = 1, 13, 18r rl 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Fitted values

Stan

dard

ized

resid

uals


Scale−Location13

181

• ℎ22r ⁄2 3 + 1 6 xvr

• î2r 0.2 <î2 ≤ 0.5 0.5 < î2

Vr

• 7 = 18nr 7 = 1, 13

V s 0.0 0.2 0.4 0.6

−2−1

01

LeverageSt

anda

rdize

d re

sidua

ls


Cook's distance

1

0.5

0.5

1

Residuals vs Leverage

181

13

• f V x Vx

î2 =∑ïY0; +ï − +ï 2

"

Q t vk"=∑ïY0; +ï − +ï 2

"

3 + 1 t vk"

• +ï V +ï 2 V7 Q X

sQ = 3 + 1Y vk" l Q

• r rX Sj Y

• RV

• ) V ñd nñd =

) − )vó )

•n

• o +2 V

• (/ ++2 − 29.1110.05V ñer

• V ñe R

. / (( () ( ( (+ (- ) ) ) (/ )( ) + ++

+ - + + ( / / / (/ / . +. ( +.

•

n

• Vr

−1 0 1 2 3

−10

12

z.kakei[, 4]z.

kake

i[, 5

]

X Y

(• ñd0 ñd"p ñe

n r

òñe = 0.00 + 0.853ñd0 + 0.162ñd"

•+ l• r r so nr s• t

V p rl Q

) .!" = 0.977

( + {

• l

l•

n

INC1

20 30 40 50

50100

150

200

2030

4050

CONS1

50 100 150 200 0.5 1.0 1.5 2.0

0.5

1.0

1.5

2.0

WORK/.+. . /

• X rYX Yr

on

• r

•s

/.+

.

. /

• rl n rrl

• { r x X Y• X Yr s X r Y• r s• r s

• B C I C I 0 <76

• )0 )" Md0 Md"p Md0d"br s

b =Md0d"Md0Md"

• b b" <76 nx

W$' =1

1 − b"

• <76r s r x X( YQ<76r s

nQ

• )0 )" W$' =0

0G..ö0õå≈ 3.04

• g rl o

統計解析 19 02web.sfc.keio.ac.jp/~maunz/DSB19/DSB19_02.pdf"â ∑;+^ " ∑; ". n '= " =02 ( *

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