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" " "4 0 " "0 " % %" & " "4 " & % 1 & : % % " X % " & " " & 5 &
& 1 / & 6 " Z % " & "0 & & " 0 % -
Y = m(X, Z, ξ)
5.6
% ξ " " 5 6 % " 5.=+*6 "" 3 %1 % " " "
" % "4 & " % &
" & % & " % " & " " % " % & "" C Y = m(X, Z) + ξ
5)6
" % m(., .) 0 " " & & " Y
X " Z & 0 & / & & & % & " 1 & " " " % " & " " 3 " & " & "& " & " " % " 5.=F)6 1 " 0 & % & " " " " " "" & 7 " & " " " & " % 0 & "
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B
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5.6 % %" % ξ
" ? " H 5),,-6 % 1 % Y = m(X, Z) + ξ "" " " " " " C " & X " Z & & " " %1 &
& % 0 & 5;6 / 5;6 " " & " " % & " 0 " " " Y = β X + m(Z) + ξ
57 8 #$$. " * ! ) " !
+
m(z)
' 0
m(Z) = γ Z
0
m(Z) = m(Z1 , · · · , ZL )
"" 0
m(Z) =
L l=1
gl (Zl )
& "/ 0
m(Z) = G(γ Z)
E " "
→ Y = β X + γ Z + ξ
56
' " " 56
→ Y = β X + m(Z1 , · · · , ZL) + ξ
→
' " "" " L Y = β X + l=1 gl (Zl ) + ξ 56 ' " & "/ " 56
→ Y = β X + G(γ Z) + ξ
" "4 " & % m(z) "0 " & " 5!.6 & & " 0 5!)6 5!;6 " 5!*6 " " % %
0
" 3 " " I β " m(z) " % 1 % & Y − βX " "
& β " % % & " 0 & % " 0 0 " 5.=FF6 % m(.) 0" " & % & 0 m(.) "0 " 0 " & 0 5 6 "" " "
F
0 & " & β " " " 5.=FF6 " & % " 5;6 " / z % C Y − E(Y |Z = z) = β X − E(X|Z = z) + ξ
5*6
" " " % C . &
yi " xi zi i ≡ xi − E(X|Z = zi ) Yi ≡ yi − E(Y |Z = zi ) " X
"
) GE " & β 5*6 5.=FF6 %" " & " " √ " n " β β " / & / E(Y |Z = zi ) 5 E(X|Z = zi )6 % θ0 % % & 2 C min
(θ0 ,...,θL )
n
[Yk − θ0 −
L
θj (Zkj − zij )]2 Kh (Zk − zi )
j=1
k=1
% Kh (.) " 1 " " & "%" " " %&" 3 & & θˆ = (Z ΥZ)−1 Z ΥY
% Y
= (Y1 , . . . , Yn )
"
" * " ") ! ' 6 0 9 ,--2 !
=
⎡ ⎢ ⎢ ⎢ Z=⎢ ⎢ ⎢ ⎣
1 Z11 − zi1 1 Z21 − zi1 . . . . . . 1 Zn1 − zi1
. . . Z1L − ziL . . . Z2L − ziL ... . ... . ... . . . . ZnL − ziL
⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
" Υ = diag(Kh(Zk − zi )) E(Y |Z = zi) 0 θˆ E(Y |Z = zi ) ≡ θ0 % E(X|Z = zi ) & "%" h E(Y |Z = zi ) " E(X|Z = zi ) "%" & " " " : % % & " " 5 1 .==. & " D J .==B " H " " E ),,;6 2 n 1 i )2 CV (h) = (Yi − β X n i=1
5-6
% "%" h 5 Yi Xi " β h6
" " G β " % W = Y − βX " " " m(z) " & " @ % " " " " % % & "
" " 1 & " " " .,
m(z) " & " & 7 " " & Y − β X Z " & m(·) m(z) %
"" " C m(Z) =
L
5B6
gl (Zl )
l=1
% gl(.)Ll=1 L 1 % & " 0 "
5+6
E(gl (Zl )) = 0
l = 1, . . . , L "" " " & : " 7 5.==,6 10 & & & " % C .
C L L 0 gl (.)l=1 " " gl (.)l=1
G gki (.) & w − Ll=1,l=k gli−1(Zl ) Zk ;
C & ""
)
C
@ & & 3 " " & & % "" " "" " " & & " E "
5.==-6 % " " " / "" ?K
,--/ 4*
#$$$ 5 )6 #$
%$
..
" " 5.==-6 " 5.==B6 & " " " 3 &% "C
m(Z) m(Z) ≡ E(Y |Z = z) = Ll=1 gl (zl )
" %
gj (zl ) =
...
m(z) ϕ−l (z−l ) dz−l
5F6
% z−l z l− " ϕ−l (z−l ) " ? " 5L − 16" : gl " 0" m(z) & & " D " (Wi, Zi) 5F6 1 gl (zl ) = m(Zi1 , . . . , Zi,l−1, zl , Zi,l+1, . . . , ZiL ) n i=1 n
5=6
gl (zl ) " & 1 %
m(Zi1 , . . . , Zi,l−1, zl , Zi,l+1, . . . , ZiL) %
& "/ " " " " 2" & "/ γ Z % γ 1 % I 1" "& & 1 % 1 G(.) C m(z) = G(γ Z)
5.,6
" " & " " " & " & & "/ & " " 0 ∗ δ " ν 3 5.,6 3 m(z) = G ν + δ(γ Z) " 2 " 2 "" .)
G % 1 % γ %" " & % & 3 C γ = arg min γ
n
Wi − G(γ Zi )
2
5..6
i=1
i )i=1,...,n % (Wi )i=1,...,n " 0 " (Yi − βX G(·) 1 % " 3 " "I
" Z # & # " 5#@#6 " γ % & 2 5 '% .=F=6 " & γ γ∝E
" C
∂G(γ Z) . ∂Z
2 yi · ϕ(Z i) n i=1 n
γDW ADE = −
% ϕ(·) 1 % " ϕ(·) Z
" &
& %1 " 0 % 0 m(z.) " " " 89 " 1 " 0 89
" 0 " " 5!)6 % & 5!.6 5!;6 " 5!*6 " 0 !
" 0 & " & 5 & E & .;
" L & .==B :M" " ! .==;6 :%2 " 1 5),,.6 "" % " @ H0 m(Z) & / γ ∈ Γ m(Z) = M(Z, γ) & H1 γ " " Sk (γ) % 1
m(Z) " 1 " & M(Z, γ) Sh (γ) =
n
h (Zi, γ) m h (Zi ) − M
2
5.)6
i=1
n h (Zi , γ) = % M j=1 Wh (Zi , Zj )M(Zj , γ) 1 " % 1 %& Wh (·, ·) T
" % " " "%" " "%" h ∈ Hn " " " " 2" C
T = maxh∈Hn
h Sh (γ) − N h V
5.;6
@ T " *
D2 " E 5),,.6 & "" & H0 & %1 "" " % & " % " " 1 % 1 L(·) & & : % % %1 % 1 % 1 5LN" 6 "H0 " γ ∈ Γ m(Z) = M(Z; γ) = γ0 + Dd=1 gd(Zd ) " " : 1 τ0 " % 5 " 6 " " " & H0 τ1 5 τ36 " E & " L &1 τ2 C 2 1 (Zi ; γ) π(Zi) m(Z i) − M n i=1 n
τ0 =
.*
1 (Zi ; γ) π(Zi ) i) − M = ui m(Z n i=1 1 Kij ui ui π(Zi )π(Zj ) = n2 hdn i=j n
τ1 τ2
1 = (ui 2 − ui)2 π(Zi) n i=1 n
τ3
(Zi , γ) % ui = Yi − m(Z i ) " " ui = Yi − M "" 5 "6 " π(·) %& & " Kij =
K((Zi − Zj )/h)(Zi = Zj )
& "/ 0 & m(z) % " " " E 5.==B6 " H0 m(z) = G(γ z) γ ∈ Rd " 1 % G(·) : R → R & & Ho "
3 ν = Y − G(γZ) " H0 E[ν|Z] = 0 E[νE[ν|Z]] = E[E[ν|Z]2 ] 0 " 3 " " H0 ! T c " " ν C Tc = with
Inc
=
nhd/2 Inc (2)σc
n 1 f( γ f( γ Z ) ( ν Z )) K ν i i j j ij n(n − 1)hd i=1 j=i
Zi ) & "/ 5 "6 " % νi = Yi − G(γ f( γ Zi ) 1 " f (γ Z) γ Z " Kij = K((γ Zi − γ Zj )/h) % K(·) 1 " H0 " T c N(0, 1)
.-
" " & " &C " " " & % " A" A # % " ! 6 " " " " " " " & " " 2 " / " 1" % "
" " " " " C & ! ""# " 2 $"% " 0 & . " & / " & % @ " % % % " " " " % 5 " 6 " & 5 & 6 ! " % & " " & % 1 % & 5" " "' %( 6 " " & % & & " & & 1 % " C " & & % " "' %( & . 7 % & A" A & " & %" 3 " " 0" 7 % " " 1 7 " & 3
4' 6 " 6 " "!
.B
" 1 & " D & " % &&& " % % "" ) * + , - C • 0 5 % " 6 "
& 1 % % " " • " " "" 5% ) 6
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" " % " % " & % % " " / % " " " 1 "
% " " % "
* * * ' ) 8 ! ' ' ,-/2 !
.+
.C L " # &' & ++ .+/ +,/) & &,/ + & , / & , / +
! &( * ,"- * - . 0 "1 2"( "1 " & ( - /- 1 - 5 , (
" ) = 1 * ( # # = 1 *
# 3 ###4 " !(67
## # # # # # ### # # # # # # ###
$ %$ $$# # ### % $ # % $
% $ # $ % % # $ # #$ % # %
" " % "0 " %" C •
' 0 5!.6C ln P ricei = α + β1 AGEi + β2 REP AIRi + β3 ROOMSi + β4 LOTi + β5 COUNT Yi + β6 V ACANTi + β7 P OPi + β8 AV INCi + γ1 T MEADi + γ2 NIT ROi + ξi
•
0 5!)6C ln P ricei = α + β1 AGEi + β2 REP AIRi + β3 ROOMSi + β4 LOTi + β5 COUNT Yi + β6 V ACANTi + β7 P OPi + β8 AV INCi + m(T MEADi , NIT ROi ) + ξi
•
"" 0 5!;6C ln P ricei = α + β1 AGEi + β2 REP AIRi + β3 ROOMSi + β4 LOTi + β5 COUNT Yi + β6 V ACANTi + β7 P OPi + β8 AV INCi + g1 (T MEADi ) + g2 (NIT ROi ) + ξi
.F
% # $ #
# %%
# % % $
−4
x 10
5 250 4.5
4 200
Density peak
3.5
NITRO
Mean value
3
150
2.5 100
2
1.5 50 1
0.5 0
5
10
15
20
25
30
35
40
45
50
TMEAD
& .C . •
/ (z1 , z2 )
& "/ 0 5!*6C ln P ricei = α + β1 AGEi + β2 REP AIRi + β3 ROOMSi + β4 LOTi + β5 COUNT Yi + β6 V ACANTi + β7 P OPi + β8 AV INCi + G(γ1 T MEADi + γ2 NIT ROi ) + ξi
% & % 0 " " @ 0 " " " 50 &6 @ )#& 0 m(z) 5 " &6 %" 0 5" &6 ! & % & " " " " 0 "
! "
β " 0 " ) "%" "
.=
)C 0 " " & "/ "
&( - . "1 &
/- ,
8# ## # % # # # #% # #% 8# # # # # ##
# $
%$9 9
!
!
8# ##
!
8# ###
8# ##
! "#$ $$% & "'" & () * #
γT mead
& #+
"$ ) & #
" / 3 * " & H " " E7 5),,;6 " ! " "%" " &" " & "/ % " " " & "/ 0 m(z) @ %" &" & :M" " 1 5.==;6 " " :%2 " :M" 5.==B6 " "%" & "/ % "%" hDW ADE #@# C 2
hDW ADE = h0 n− 2p+k+2
5.*6
% k " Z " " " " Z " p + 1 / " % p = 1 " k = 2 hDW ADE = h0 n−1/3 @ " % &" h0 "%" ),
$ " & & & "% & 0 " / " & " & " / & 0 O
% " 2" " & , )P " 1 & ? " " ;-P 0 & && 2 & L "0 & % % " & " / : " % " 5L 6 / & = .P " " % % & & %" / % % 4 %" 1 % " " % & & E1 & . % " / " O " I & 0 " & " " , ;P " " " O 5 16 & & & " % I 1 3 0 & & 0 % %
" & @' " " " " % ) "/ " / " % I & &% " "" " / G(.)
! ' : * 7 ,--/ ! ;
