ÿÿÿþþþ ‘‘‘laplace'''ÙÙÙ˙˙˙ııı‡‡‡555ÚÚÚoooíííäää

l

X Ú Æ ê Æ

J. Sys. Sci. & Math. Scis.

x(x) (20xx, x), 1–19

ÿÿÿþþþØØØLaplace©©©ÙÙÙ555ÚÚÚOOOíííäää∗

¤ïù

(ìÜÆêÆOÅÆÆ, 041004)

y¥(

(Department of Statistics, Kansas State University, Manhattan, KS, 66503)

Á p-ëêθÏLÝ^Em(X, θ) = 0½Â§XkLaplaceÿþØ§=·

U*ÿZ = X +U§©[7]Eθ üO. , TO=·^uU©

þÑlLaplace©ÙpÕá¹. ©·ò0«õLaplace©Ù§ ¿ò

©[7] í2äkù«õLaplace©ÙÿþØ.¥. ,§©[7]´Äu

Ã^Ï"'X Em(X, θ) = EH(Z, θ)§Ù¥H ,/ª®¼ê§ TéÚ

Oíä¯K¿Ø·^. ©·òE«Äu^Ï" E[m(X, θ)|Z] O. X

§·éùO5?1?Ø.

'c 5ÚOíä, ÿþØ, Laplace©Ù, .

MR(2000)ÌÌÌKKK©©©aaaÒÒÒ 62J02, 62F10

Nonlinear Statistical Inferences With Laplace Measurement Error

SHI Jianhong

(School of Mathematics and Computer Science, Shanxi Normal University, Linfen, 041004)

SONG Weixing

(Department of Statistics, Kansas State University, Manhattan, KS, 66503)

Abstract When a p-dimensional parameter θ is defined through the moment con-dition Em(X, θ) = 0, a simple estimation procedure of θ was proposed in [7] when X,

*II[g,ÆÄ7(NSF DMS 1205276)]ÏK.

ÏÕö: y¥(§[email protected]

ÂvFÏ: 200x-xx-xx, Â?UvFÏ: 200x-xx-xx.

?:

2 X Ú Æ ê Æ xò

a k-dimensional random vector, are contaminated with Laplace measurement errorU , that is, we can only observe Z = X +U . However, the estimation procedure wasdesigned particularly for the cases where the components of the measurement errorvector U are independent. In this paper, we first introduce a general multivariateLaplace distribution, then extend the methodology in [7] to this general multivari-ate scenario. Moreover, the moment estimation procedure in [7] is based on theunconditional expectation Em(X, θ) = EH(X, θ) for some function H. Exampleshows this techniques does not work in some cases. In this paper, we will proposean estimation procedure based on the condition expectation E(m(X, θ)|Z). Largesample properties of the proposed estimation procedure when X is one-dimensionalare discussed.

Keywords Nonlinear Statistical Inference, Measurement Error, Laplace Distribu-

tion, Bias Correction.

1 ÚÚÚ óóó

5ÚO.´ÚOÆ!²LOþÆ!)ÔÚOÆ!àÚOÆ¯õÆ¥A^

2a)ºCþm'XÚO.. u5.§5.3yêâ(!

[ÜCþ'X¡\(¹§ éuëê.5`§5.ëê kX~*

¢S)º§ÚOíä§S dêâû½§mk<0\. 3Nõ¹e§5.¥

ëêθdXoNÝO§û½µ

Em(X; θ) = 0 (1.1)

Ù¥Xk-þ§θ ∈ Θ ⊂ Rpëêþ§mdþ¼ê§Ï~kd ≥ p.,

3¢¥§UduÿþóäØ°(§Uduÿþ¤[B§³½´du<

Ï§·*ÿØX§*ÿ´Xk'CþZ. ~X§3IJwï

Ä¤?1xUþEïÄ(OPEN) ¥§*ÿI<Ø ¥x

\þ. , ùI*ÿØ¤[B§ §S¡. §Æ[À,

Cþ§=kx¹þZ§5X OCþ. 'uOPEN ïÄäN¹§ë

©[13]ó¶©[14]ïÄÖF´¾Ù>zþm'X§3TïÄ¥§A

CþY , Y = 1L«ÖkF´¾§Y = 0L«vk.XJ)ºCþ>zþX±

*ÿ§Logistic½öProbit £8Ï~¬^5éY X'X?1ï. , §3©[14]ï

Ä¥§ X*ÿØ§¤U*ÿ´Ö¤3[Ì¥Ó¹¿pz¹þZ.

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L*ÿêâ(Y, Z)5ïÄYX m'X=¤ÿþØ.ïÄSN. ©z¥

'uZXm'Xkéõ?Ø§k²(ÚO'X)\(Z = X + U§Berkson(

X = Z + U§£8(Z = α + γX + U§¦È(Z = XU §;Í[2]1!

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â(Instrumental Data/Validation Data)3ëêíä¥. ©·òæ^~\Ø

(: Z = X + U§Ù¥U L«ÿþØ§XUpÕá.

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xÏ ¤ïù§y¥(: 5ÿþØÚOíä 3

Ø½ü$ÿþØéÚO©ÛE¤K§ÒI\ré.b½öÂ8õ9Ïê

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.¥§cÙ´35½õª.¥~ÚÑlN(0,Σ). , 3Ù¦E,ëê

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§¿âÿþØA¼êéÙ?1©a. XJA¼êÜ±êÝP~0,

K¡TÿþØäk1w5(super smooth)¶XJA¼êÜ±õªÝªu0§

K¡TÿþØäk1w5(ordinary smooth).©Ù´äk1w5;.L§ ä

k1w5;.~f´Laplace ©Ù.Laplace©Ù3Ñ£OãØ ¡kX2

A^. 3ÚOÆ¥§duýé OÄu.Ê.d©Ù4q,OkXU,é

X§¤±Ù3è5ïÄ¥ÓkR/.

3bU©þpÕá§ÙgÑlLaplace©Ùê§©[7]JÑO.(1.1)

¥ëêθ«ü. 0B§PU = (U1, U2, . . . , Uk), Uj Ý¼êf(u) =

exp(−√

2|u|/σj)/(√

2σj), Uj , j = 1, 2, . . . , kpÕá. ,P

m(Z; θ, σ) = m(Z; θ) +k∑l=1

(−1

2

)l ∑j1<···<jl

σ2j1 · · ·σ

2jl

(∂2lm(Z; θ)

∂Z2jl· · · ∂Z2

jl

).

3é¼êm(x; θ) \1w5^§©[7]y²Xeúª

Em(X; θ) = Em(Z; θ, σ).

âþã(Ø§ÝÊm(X; θ) = 0=±dÝÊm(Z; θ, σ) = 05O. 5¿´§

¦+ù#Ý^6uZ§´%Ú\,Uëêσ. XJEm(Z; θ, σ) =

0±û½θ, σ§©[7]ÑÄu

(θ, σ) = argminθ,σ

(n∑i=1

m(Zi; θ, σ)

)′Wn

(n∑i=1

m(Zi; θ, σ)

)?ÝO. 3Kê§θ, σìC5y². XJEm(Z; θ, σ) = 0Øv±

û½θ, σ§|^σ2jØLZj¯¢§©[7]EθU8Ü.

¦+©[7]ó5Øó §, TkX²w". ÄkT·^

uU©þpÕá/§ Õá5éu9õCþÚOíä5`´~

b¶,Em(X; θ) = Em(Z; θ, σ)´k'Ã^Ï"ª. ÄuÃ^Ï"ÚOíä3

éõÿr<¿§cÙ´3£8.?Ø¥§=¦·Ä´X/. 'u

ö§·¬3¡?Ø¥^Poisson£8.~f5?Úã.

XJUÑl©Ù§aqÃ^Ï"ªéJ. k = 1, m(x; θ) = xp − θ§©[7]ÑXeª§

Em(X; θ) = EZp +

[p/2]∑l=1

(−1)l(σ2

2

)lp!

l!(p− 2l)!Zp−2l − θ.

XJUÑlõ©Ù§aqªéN´. ´éuõª5¼

êm(x, θ)§aq(Øvk.

©(SüXe§31!¥§·ò0«õLaplace©Ù½Â9ù

«©Ùü5§31n!¥§Äuù«õLaplace©Ù§·¬ãaq

4 X Ú Æ ê Æ xò

uEm(X; θ) = Em(X; θ, σ)Ã^Ï"ª§¿ïá²1u©[7]OO

5. Xd§·Òò3Õá5beïáÚOíä§Sí2õ¹¥. 3

1o!§·¬3X ©Ù¹e§ÑE(m(X; θ)|Z)Lª§,·¬|^

ù^Ï"Lª§5U?ÄuÃ^Ï"O. ¤knØ(Jy²Ñ31

Ê!.

2 õõõLaplace©©©ÙÙÙ

¯¢þ§î8§õLaplace©ÙE,vkÚ½Â. '~´d©[4,8]JÑ

ÄuA¼ê½Â. ¯¢þ©[4]¤½ÂõLaplace ©Ù´©[8]½ÂõLaplace©Ù

A~. 3©?Ø¥§·æ^©[4]½Â. âT½Â§k þX A¼ê

XJäk/ª

φ(t) =exp(iµ′t)

1 + t′Σt/2, t ∈ Rk,

K¡XÑlëê(µ,Σ)õLaplace©Ù§PX ∼ Lk(µ,Σ)§Ù¥µk-þ§Σ

k× k½Ý. éw,§Xd½ÂõLaplace ©Ù´ÄuLaplace ©ÙA¼ê

í2. ©ÙLk(µ,Σ)kXe5µ

(1). XJU ∼ Lk(µ,Σ)§@ol©Ù¿Âþ5`§U = µ +√V Σ1/2W§Ù¥VÑ

lþ1ê©Ù§WÑlõIO©Ù§V W pÕá;

(2). XJU ∼ Lk(µ,Σ)§@oÙÝ¼êäkXe/ª

ψ(u) =2

(2π)k/2B(d/2−1)(

√2q(u;µ,Σ))

(2q(u;µ,Σ))d/2−1, u ∈ Rk, (2.2)

Ù¥q(u;µ,Σ) = (u− µ)′Σ−1(u− µ)§Br(x)´r1a?Bessel¼ê¶

(3). XJU ∼ Lk(µ,Σ)§KEU = µ, Cov(U) = Σ.

5(1)`²õLaplace©Ù¢Sþ´õ©Ù«Ý·Ü§T5Ø=Jø«

)õLaplace©Ù§ Ñ|^T©Ù?1èÚOíä¦^EM U

5. ù¡óëw©[11,12]'uk = 1?Ø. 5(3)L²ëêΣ (¢xU¥©

þm'X.

duU'u0é¡´ÿþØ.©z¥~^^§¤±3e¡?Ø¥§ ·b

µ = 0.

3 ÃÃÃ^ÝÝÝOOO

±f, g©OPXZÝ¼ê§±φz(t), φx(t)φu(t)©OPZ,XUA¼ê. d

\(Z = X + U§±9XUpÕá§´φz(t) = φx(t)φu(t). Ïé¤kt ∈ Rk§φu(t) = (1 + t′Σt/2)−1 6= 0§¤±éN´φx(t) = (1 + t′Σt/2)φz(t).|^d¯¢§·±

?Úe¡(Ø.


ÚÚÚnnn 3.1 bφZ(t)²È§K

f(x) = g(x)− 1

2

k∑j,l=1

σjl∂2g(x)

∂xj∂xl,

Ù¥σjlΣ(j, l)§xjx1j©þ.

ÄuþãÚn§·±e¡Ã^Ï"ª.

½½½nnn 3.1 bëê¼êm(x, θ)Ý¼êg(z)÷ve¡^

(C1). ézθ ∈ Θ§ëê¼êm(x, θ)'ux§¿‖x‖ → ∞§

m(x, θ)g′(x)→ 0, m′(x, θ)g(x)→ 0.

(C2). ézθ ∈ Θ§

E‖m(Z, θ)‖ <∞, E

∥∥∥∥∂2m(Z, θ)

∂Zj∂Zl

∥∥∥∥ <∞,K·k

Em(X, θ) = Em(Z, θ)− 1

2

k∑j,l=1

σjlE∂2m(Z, θ)

∂Zj∂Zl. (3.3)

/ÏÚn3.1§½n3.1y²~ü. ¯¢þ§|^½n^¿?1©ÜÈ©§·

±y²§é¤kj, l = 1, 2, . . . , k,∫m(x, θ)

∂2g(x)

∂xj∂xldx = E

∂2m(Z, θ)

∂Zj∂Zl.

Ú©[7]'uÕáLaplaceÿþØ©þ(J'§½n3.1(Øl/ªþw~

ü§Ã^Ï"ªmà9ëê¼êm(z, θ)'uz . Äu½n3.1§·

±X²1u©[7](J. B?Ø§P

m(Z; θ,Σ) = m(Z, θ)− 1

2

k∑j,l=1

σjl∂2m(Z, θ)

∂Zj∂Zl.

3.1 £££OOO¹¹¹eee:::OOO

bÝ^Em(Z; θ,Σ) = 0±(½θ, Σ. é?¿é¡½ÝWn, ½Â

(θ, Σ) = argminθ,Σ

(n∑i=1

m(Zi; θ,Σ)

)′Wn

(n∑i=1

m(Zi; θ,Σ)

). (3.4)

Ó§·b

(C3). Em(Z; θ,Σ) = 0 =θ = θ0, Σ = Σ0 > 0, Ù¥θ0,Σ0ý¢ëê¶

(C4). é,½é¡Ý§WnVÇÂñuW ;

(C5). ±σPþ(σjl, j ≥ l), α = (θ′, σ′)′.E∂m(Z; θ,Σ)/∂α 3§÷§¿3α0,

S§∂m(Z; θ,Σ)/∂α÷vLipschitz^.E‖m(Z; θ0,Σ0)‖2 <∞.

6 X Ú Æ ê Æ xò

3þã^e§·ke¡½n.

½½½nnn 3.2 b^(C1)-(C5)¤á§K(θ, σ)VÇªu(θ0, σ0)§¿

√n

(θ − θ0

σ − σ0

)=⇒ N

(0, (A′WA)−1(A′WΩWA)(A′WA)−1

),

Ù¥

A =

∂m(Z; θ)

∂θ− 1

2

k∑j,l=1

σjl∂m3(Z; θ)

∂Zj∂Zl∂θ,

(− 1

2δ(j,l)∂2m(Z; θ)

∂Zj∂Zl, j ≥ l

) ,Ω = Em(Z; θ,Σ)m′(Z; θ,Σ); XJj = l, δ(j, l) = 1, ÄKδ(j, l) = 0.

dθ, ΣÜ5§·ØJe¡(Ø.

½½½nnn 3.3 b^(C1)-(C5)¤á§Wn = n(∑ni=1 m(Zi; θ, Σ)m′(Zi; θ, Σ))−1, ,

(θ, Σ) = argminθ,σ

(n∑i=1

m(Zi; θ, σ)

)′Wn

(n∑i=1

m(Zi; θ, σ)

)(θ,Σ)O§KWnVÇªuΩ−1§(θ, Σ)E(θ0,Σ0)ÜO§¿

√n

(θ − θ0

σ − σ0

)=⇒ N

(0, (A′ΩA)−1

).

Ù¥σ = (σjl, j ≥ l), σjlΣ(j, l)©þ.

ÌIO'uÝOÜ5IOØy§·±½n3.23.3y².

'å§©·Ñ[y²L§.

3.2 ØØØ£££OOO¹¹¹eeeUUUOOO

XJÝ§Em(Z; θ,Σ) = 0Øv±û½θ,Σ§Ò´`§^(C3)Ø¤á§@oUì

©[7]Jøg´§·±E'uθU8Ü. ½Â

η0(θ) = Em(Z, θ)− 1

2

k∑j,l=1

√τ2j τ

2l

[E∂2m(Z, θ)

∂Zj∂Zl

]−.

η1(θ) = Em(Z, θ) +1

2

k∑j,l=1

√τ2j τ

2l

[E∂2m(Z, θ)

∂Zj∂Zl

]+

.

Ù¥τ2jZ1j©þ§j = 1, 2, . . . , k. þªmàÑy3ê þ+, −L«A¼

êÜKÜ. dué?¿j, l = 1, 2, . . . , k§Ñk|σjl| ≤√σ2jσ

2l ≤

√τ2j τ

2l §¤±·k

η0(θ0) ≤ Em(Z; θ0,Σ0) = 0 ≤ η1(θ0),

dØª¿XëêθU8ÜÀ

Θ0 = θ|θ ∈ Θ, η0(θ) ≤ 0 ≤ θ1(θ) .

ÚOíäO§?½ÝW§9ªuW?U6êâ½ÝSWn§½Â

Q(θ) = [η0(θ)I(η0(θ) > 0)]′W [η0(θ)I(η0(θ) > 0)] + [η1(θ)I(η1(θ) < 0)]′W [η1(θ)I(η1(θ) < 0)],


Qn(θ) = [ηn0(θ)I(ηn0(θ) > 0)]′Wn[ηn0(θ)I(ηn0(θ) > 0)]

+[ηn1(θ)I(ηn1(θ) < 0)]′Wn[ηn1(θ)I(ηn1(θ) < 0)],

Ù¥

ηn0(θ) =1

n

n∑i=1

m(Zi, θ)−1

2

k∑j,l=1

√τ2j τ

2l

[1

n

n∑i=1

∂2m(Zi, θ)

∂Zij∂Zil

]−.

ηn1(θ) =1

n

n∑i=1

m(Zi, θ) +1

2

k∑j,l=1

√τ2j τ

2l

[1

n

n∑i=1

∂2m(Zi, θ)

∂Zij∂Zil

]+

,

ùpτ2jZij , i = 1, 2, . . . , n. 3e¡8Üål½Âe

ρ(A,B) = supa∈A

infb∈B|a− b|,

|a− b|a, bü:mîªål§·ke¡(Ø.

½½½nnn 3.4 b^(C1)-(C3)§(C5)¤á. éuªu0êSγn§½Â8Ü

Θn = θ|θ ∈ Θ, Qn(θ) ≤ argminb∈ΘQn(b) + γn,

Kn→∞, ρ(Θn,Θ0)→ 0, a.s..?Ú§XJ·ksupθ∈Θ

Qn(θ)−Q(θ)

/γn → 0, a.s.,

@oρ(Θ0,Θn)→ 0, a.s..

½n3.4y²aqu©[9]¥·K35y²§dÑ.

4 ^ÝÝÝOOO

·±kÿþØPoisson£8~f5m©!?Ø. 3½ÅCþX

^e§bYÑlþexp(Xθ)Poisson©Ù.=

P (Y = y|X) =exp(yXθ) exp(− exp(Xθ))

y!, y = 0, 1, . . . .

XJ±(X,Y )|*ÿ(Xi, Yi), i = 1, 2, . . . , n§@oθÐOAT´4q,

O§Ò´¦

Ln(θ) =1

n

n∑i=1

[− exp(Xiθ) + YiXiθ − log Yi!]

θ. 3XØ*ÿÑlÿþØ.Z = X + U§·U´Ä±

^¦

Ln(θ) =1

n

n∑i=1

[− exp(Ziθ) + YiZiθ − log Yi!]

θ5θO. ùÒ´¤¢NaiveO. 5¿n→∞ §

Ln(θ)→ E[− exp(Xθ) + Y Xθ − log Y !] + E[− exp(Zθ) + exp(Xθ)],

8 X Ú Æ ê Æ xò

Ò´`§XJ·½Â

L0n(θ) =1

n

n∑i=1

[YiZiθ − log Yi!]− E exp(Xθ)

KL0n(θ)Ln(θ)VÇÂñuE[− exp(Xθ) + Y Xθ − log Y !]. Äuù?q,¼

êL(θ)§©[6]Ñ§XJE exp(Xθ)®§KL0n(θ):±θO. ©[6]¡ù

O©O.3½Kê§©OÜ5ìC5

y².

´§E exp(Xθ)®^ ¿X·IéX©Ùk¤)§ 3ÿþØ

©z¥§ïÄ<ééX©ÙÑb§¤±XJA^©[6]§ 1

É exp(Xθ) O. ù5§3ÿþØÑlLaplace©Ù¹e§©[7]ó§½

ö·31n!(J§BJøE exp(Xθ)«UO. ¯¢þ§·k

E exp(Xθ) =

[1− 1

2σ2θ2

]E exp(Zθ).

Xd§XJ?ÚbÿþØUσ2®§·=±ÏL4z

L1n(θ) =1

n

n∑i=1

[YiZiθ − log Yi!−

(1− 1

2σ2θ2

)exp(Ziθ)

]5¦θO. , c[©Û±w§éu|½(Zi, Yi), i = 1, 2, . . . , n§θ¼ê§

θ → ±∞§L1n(θ) → +∞. Ò´`§L1n(θ):¿Ø3. ù¿X3þã

¹¥§©[7]ó§½ö·31n!(J¿Ø·^.

4.1. ^ÏÏÏ"""úúúªªª

3ÿþØ.ÚOíä¥§£8(Regression Calibration)´«Uk~$

O . ¦^TÄkE(X|Z)O§,3AÚOíä§S¥§

±E(X|Z)OX. 'uT[¹§±ëw;Í[2]14Ù§½ö©[3]¥?

Ø. ¯¢þ§£8¿ØUØÿþØ5K§¦Ù-<÷¿J§

Øk°(E(X|Z) O§Im(x, θ)x¹1w¼ê, ½ökXÿþØ

. 3e¡?Ø¥¬w§ÿþØ.äkLaplace©Ù§·Ø=UE(X|Z)

°(Lª§ é1w¼êm(x, θ)§·UE[m(X, θ)|Z]°(Lª. ¦

^ö ±kO.

·Äkã·ûõX(J. éuëê¼êm(x; θ)§XJE(m(X; θ)) = 0§@

o¿XE(E[m(X; θ)|Z]) = 0. ¤±XJ·UE[m(X; θ)|Z] Lª(w,ùL

ª6uΣ)§K33.1 !¥m(Z; θ,Σ)§Ìaqg´±Eθ,ΣO§¿*

þ5`§d^Ï"E[m(X; θ|Z)]ÑuEO'âÃ^Ï"'X¥m(Z; θ,Σ) E

OÐ§.¾3¤kZ ÿ¼ê¥§^Ï"E[m(X; θ|Z)]3þØ¿Âe

Cum(X; θ).

bm(x)ÿ¼ê. e¡½nÑ^Ï"E(m(X)|Z) Lª.

½½½nnn 4.1 bÿþØUÑld(2.2)½Âþ0õLaplace©Ù§ZA¼


êφZ(t)²È§K

E[m(X)|Z] =1

g(Z)

∫ m(x)ψ(Z − x)g(x)dx− 1

2

n∑j,l=1

σjl

∫m(x)ψ(Z − x)

∂2g(x)

∂xj∂xldx

. (4.5)

Ù¥ψd(2.2)½ÂUÝ¼ê.

XJk = 1§K·ke¡íØ

íííØØØ 4.1 b¼êm§g§¿éb(x) = m(x)g(x), m′(x)g(x)½m(x)g′(x)5`

limx→±∞

b(x) exp(−|x|)→ 0.

K·k

E[m(X)|Z] = m(Z) + l(Z), (4.6)

Ù¥

l(z) =ez/b

g(z)

∫ ∞z

[m′(x)− bm′′(x)

2

]g(x)e−x/bdx− e−z/b

g(z)

∫ z

−∞

[m′(x) +

bm′′(x)

2

]g(x)ex/bdx,

b = σ/√

2.

5551. UÑl©Ù§ÃØX©ÙÛ§·okE(X|Z) = Z + σ2g′(Z)/g(Z).

©[10]Jdúª´Maurice Tweedie3T©öh<Ï&¥JÑ§¤±Túª¡

Tweedieúª. duÙ²BayesnØ'X§Tweedieúª~2A^§

[¹ë©[1]ó. ½n4.1 ¿Â3uÿþØÑlLaplace ©Ù§·

aqTweedie úª(4.5)§Ù¥m(x) Ø==ux 5¼ê.

5552. ØJy²§é(4.5)¥l(Z)Ï"=

El(Z) = −σ2

2E

[∂2m(Z)

∂Z2

],

dd=(3.3)k = 1/.

ãB'å§·e5Ä/. ùÏkü:§´3õ

¹e§d(4.7)§gêÑy3^Ï"Lª¥§^gØÝO

êgéU¬éOnØ&?5æ. ´3½^e§·,±

ò(4.7) zaqu(4.8) /ª§Ò´`3E(m(X)|Z)Lª¥Ñy´g§ Ø

´gê§´3õ¹?n¥§ØPÒ,±§õênØØyL§

¹ÓÉ.

4.2 ^ÝÝÝOOO

duXÝ¼ê§l ZÝ¼êÃl¡.XJ±Z1, Z2, . . . , ZnL«Z

|§@o·Ò±¦^gØÝO.±KPÀ½Ø¼ê§hI°§KgØÝ

O½Â

gh(z) =1

nh

n∑i=1

K

(Zi − zh

).

10 X Ú Æ ê Æ xò

BO§P

µl(x; θ, σ) =

[m′(x; θ) + (−1)l

σ

2√

2m′′(x; θ)

]exp

((−1)l

x√

2

σ

),

l = 1, 2.âíØ4.1, é½θ, σ§E(m(X, θ)|Z)±êªO

E(m(X, θ)|Z) = m(Z; θ) +eZ√

2/σ

gh(Z)

∫ ∞Z

µ1(x; θ, σ)gh(x)dx− e−Z√

2/σ

gh(Z)

∫ Z

−∞µ2(x; θ, σ)gh(x)dx.

I5¿´§©1ÑyØO§3nØØy· yØOk"e.. ´

ù¦é| ¢ê8Ü©Ù¿ØÜn. ©z¥®²k5?nù¯K. 3

©¥§·æ^´òë$Zi, i = 1, 2, . . . , n3,4«mS§ù

Ib'¼êëY5§=£;K©1C"5nØ(J. d§v

êa, -H(Z; θ, σ) = I[−a,a](Z)E(m(X, θ)|Z)§¿^

H(Z; θ, σ) = I[−a,a](Z)E(m(X, θ)|Z)

H(Z; θ, σ)O§ùpI[−a,a](·)4«m[−a, a]«5¼ê. BO§3e¡?Ø¥

·Îa(·)5L«[−a, a]«5¼ê.

X3.1!¥?Ø§XJÝÊH(Z; θ, σ) = 0±(½θ, σ§é?¿6uêâ

½ÝWn§·±^


(n∑i=1

H(Zi; θ, σ)

)′Wn

(n∑i=1

H(Zi; θ, σ)

). (4.7)

Ó§·b

(S1). θ, σëêm©ORp, R+¥;f8¶

(S2). EH(Z; θ, σ) = 0 =θ = θ0, σ = σ0 > 0, Ù¥θ0, σ0ý¢ëê¶

(S3). é,½é¡Ý§WnVÇÂñuW ;

(S4). ±Pα = (θ′, σ)′. EH(Z; θ, σ)/∂α3§÷§¿3α0,S§∂H(Z; θ, σ)/∂α

÷vLipschitz^. E‖H(Z; θ0, σ0)‖2 <∞.

(S5). Ø¼êK÷v

K(x) ≥ 0,

∫K(x)dx = 1,

∫xK(x)dx = 0, 0 <

∫x2K(x)dx <∞.

(S6). I°h÷vh→ 0, nh2 →∞, nh4 → 0.

3þãê§·ke¡½n.

½½½nnn 4.2 b^(S1)-(S6)¤á§K(θ, σ)VÇªu(θ0, σ0)§¿

√n

(θ − θ0

σ − σ0

)=⇒ N

(0, (A′WA)−1(A′WΩWA)(A′WA)−1

),

Ù¥A = E∂H(Z; θ, σ)/∂α, Ω = Cov(S11(Z)− S12(Z) + S21(Z)− S22(Z)),Ù¥

S11(Z) = µ1(Z; θ0, σ0)

∫ a

−aI(u < Z) exp(u

√2/σ0)du;


S12(Z) =exp(Z

√2/σ)

g(Z)Ia(Z)

∫ ∞Z

µ1(x; θ0, σ0)g(x)dx;

S21(Z) = µ2(Z; θ0, σ0)

∫ a

−aI(u > Z) exp(−u

√2/σ0)du;

S22(Z) =exp(−Z

√2/σ)

g(Z)Ia(Z)

∫ Z

−∞µ2(x; θ0, σ0)g(x)dx.

dθ, σÜ5§·ØJe¡(Ø.

½½½nnn 4.3 b^(S1)-(S6)¤á§Wn = n(∑ni=1 H(Zi; θ, σ)H ′(Zi; θ, σ))−1, ,


(n∑i=1

H(Zi; θ, σ)

)′Wn

(n∑i=1

h(Zi; θ, σ)

)(β, σ)O§KWnVÇªuΩ−1§(θ, σ)E(θ0, σ0)ÜO§¿

√n

(θ − θ0

σ − σ0

)=⇒ N

(0, (A′ΩA)−1

).

5 ÌÌÌ(((JJJyyy²²²

·Äk5y²Ún3.1.

yyy²²²µµµ ±φZ(t), φu(t)©OL«ZUA¼ê. dA¼êüúª§XÝ¼ê

f(x) =1

(2π)k

∫exp(−it′x)φZ(t)φ−1

u (t)dt =1

(2π)k

∫exp(−it′x)φZ(t)

[1 +

1

2t′Σt

]dt

=1

(2π)k

∫exp(−it′x)φZ(t)dt+

1

2(2π)k

k∑j,l=1

σjl

∫exp(−it′x)tjtlφZ(t)φ−1

u (t)dt

= g(x) +1

2(−i)2

k∑j,l=1

σjl∂2

∂xj∂xl

[1

(2π)k

∫exp(−it′x)φZ(t)dt

]

= g(x)− 1

2

k∑j,l=1

σjl∂2g(x)

∂xj∂xl.

e5·y²íØ4.1.

yyy²²²µµµ Bå§·òm(x; θ)m(x), b = σ/√

2. |^^Ï"½Â§·k

E[m(X)|Z] =

∫m(x)f(x|z)dx =

∫m(x) 1

2be−|x−z|

b [g(x)− b2g′′(x)]dx

g(z).

þª¥©f±?Úe¡üªÚµ

S1n(z) =1

2bez/b

∫ ∞z

m(x)e−x/b[g(x)− b2g′′(x)]dx,

S2n(z) =1

2be−z/b

∫ z

−∞m(x)ex/b[g(x)− b2g′′(x)]dx.


|^½n^¿?1©ÜÈ©§±∫ ∞z

m(x)e−x/bg′′(x)dx =

∫ ∞z

m(x)e−x/bdg′(x)

= −m(z)e−z/bg′(z)−∫ ∞z

g′(x)

[m′(x)e−x − 1

bm(x)e−x/b

]dx

= −m(z)e−z/bg′(z)−∫ ∞z

g′(x)m′(x)e−x/bdx+1

b

∫ ∞z

g′(x)m(x)e−x/bdx

= −m(z)e−z/bg′(z) + g(z)m′(z)e−z/b +

∫ ∞z

g(x)m′′(x)e−x/bdx− 1

bg(z)m(z)e−z/b

−2

b

∫ ∞z

g(x)m′(x)e−x/bdx+1

b2

∫ ∞z

g(x)m(x)e−xbdx.

aq± ∫ z

−∞m(x)ex/bg′′(x)dx =

∫ z

−∞m(x)ex/bdg′(x)

= m(z)ez/bg′(z)− g(z)m′(z)ez/b +

∫ z

−∞g(x)m′′(x)ex/bdx

−1

bg(z)m(z)ez/b +

2

b

∫ z

−∞g(x)m′(x)ex/bdx+

1

b2

∫ z

−∞g(x)m(x)ex/bdx.

òþãüªf\Sn1(z), Sn2(z)Lª¥§·±

Sn1(z) = − b2ez/b

∫ ∞z

g(x)m′′(x)e−x/bdx+ ez/b∫ ∞z

g(x)m′(x)e−x/bdx

+b

2m(z)g′(z)− b

2g(z)m′(z) +

1

2g(z)m(z),

Sn2(z) = − b2e−z/b

∫ z

−∞g(x)m′′(x)ex/bdx− e−z/b

∫ z

−∞g(x)m′(x)ex/bdx

− b2m(z)g′(z) +

b

2g(z)m′(z) +

1

2g(z)m(z).

òþãüªf\E[m(X)|Z]Lª¥, \n=íØ4.1(Ø.

y²½n4.2§·ÄkÑüÚn. bX1, X2, . . . , XnÑlÝ¼êf(x)Õá

Ó©ÙÅCþ§fn(x)f(x)ØÝO§Ù¥Ø¼êK§I°h.

ÚÚÚnnn 5.1 bÝ¼êf(x)| R§÷vf(x)§Ùêk..KK ÷v

^(S5)§Ké?¿êa,

supx∈[−a,a]

|fn(x)− f(x)| = O

(√log n

nh

)+O(h2), a.s..

Ún5.1ã´ØÝO¥2<(J§¤±·ã Øy.

ÚÚÚnnn 5.2 bη(x)ëY¼ê§Ý¼êf(x)êk.. ·k

1√n

n∑i=1

η(Xi)Ia(Xi)[fn(Xi)−f(Xi)] =1√n

n∑i=1

[η(Xi)f(Xi)Ia(Xi)−Eη(X)f(X)Ia(X)]+op (1) .


yyy²²²µµµ \n§·k

1√n

n∑i=1

η(Xi)Ia(Xi)fn(Xi) =1√

n(n− 1)h

∑i 6=j

η(Xi)Ia(Xi)K

(Xi −Xj

h

)+

K(0)√nnh

n∑i=1

η(Xi)Ia(Xi)

− 1√nn(n− 1)h

∑i 6=j

η(Xi)Ia(Xi)K

(Xi −Xj

h

),

duE|η(X)Ia(X)| <∞§E|η(X)f(X)Ia(X)| <∞e§üþop(1).P

h(Xi, Xj) =1

2[η(Xi)Ia(Xi) + η(Xj)Ia(Xj)]

1

hK

(Xi −Xj

h

),

K·k

1

n(n− 1)

∑i 6=j

η(Xi)Ia(Xi)1

hK

(Xi −Xj

h

)=

2

n(n− 1)

∑i<j

h(Xi, Xj).

PþªUn§KØJwÑUnU -ÚOþ§

1√n

n∑i=1

η(Xi)Ia(Xi)fn(Xi) =√nUn + op(1).

½Âθ = Eh(X1, X2). ·5ÄUn3z:þÝK§E(Un|Xi), i = 1, 2, . . . , n. ü

O±uy§ézi = 1, 2, . . . , n,

E(Un|Xi) =2

nE[h(X,Xi)|Xi] +

n− 2

nθ,

þªmà1¥XXiÕáÓ©Ù.- Un =∑ni=1E(Un|Xi)− (n− 1)θ, ´

Un − θ =2

n

n∑i=1

(E[h(X,Xi)|Xi]− θ) .

Ph(Xi) = E[h(X,Xi)|Xi]§H(Xi, Xj) = h(Xi, Xj)− h(Xi)− h(Xj)− θ.K

Un − Un =2

n(n− 1)

∑i<j

[h(Xi, Xj)− h(Xi)− h(Xj)− θ] =2

n(n− 1)

∑i<j

H(Xi, Xj).

5¿i, j, k(= 1, 2, . . . , n)üüØ§·kEH(Xi, Xj) = 0, E[H(Xi, Xj)|Xi] = 0§

ö¿XEH(Xi, Xj)H(Xj , Xk) = 0§l ±íÑ

E(Un − Un)2 =2

n(n− 1)EH2(X1, X2).

üO±uyEH2(X1, X2) = O(1/h)§Xd·=

E(Un − Un)2 = O

(1

n2h

), or Un = Un +Op

(1

n√h

).

dunh→∞§¤±ù¯¢¿X√n(Un − θ) =

√n(Un − θ) + op(1). ,§·k

θ = E

(η(X1)Ia(X1)

1

hK

(X1 −X2

h

))=

∫∫ a

−aη(u)f(v)f(u)

1

hK

(u− vh

)dudv

=

∫ a

−aη(u)f(u)

[∫K(v)f(u+ vh)dv

]du = Ef(X)η(X)Ia(X) +O(h2).

5¿E[h(Xi, Xj)|Xi]±L«

1

2E

[η(Xi)Ia(Xi)

1

hK

(Xi −Xj

h

) ∣∣∣Xi

]+

1

2E

[η(Xj)Ia(Xj)

1

hK

(Xi −Xj

h

) ∣∣∣Xi

],


üO±¦

E

[η(Xi)Ia(Xi)

1

hK

(Xi −Xj

h

) ∣∣∣Xi

]= η(Xi)f(Xi)Ia(Xi) +Op(h

2)

e¡5y²eª¤á

E

[η(Xj)Ia(Xj)

1

hK

(Xi −Xj

h

) ∣∣∣Xi

]= η(Xi)f(Xi)Ia(Xi) +Op(h

2).

'å§PþªàQh(Xi).¯¢þ§·k

n−1/2n∑i=1

[Qh(Xi)− η(Xi)f(Xi)Ia(Xi)]

=1√n

n∑i=1

[Qh(Xi)− θ]−1√n

n∑i=1

[η(Xi)f(Xi)Ia(Xi)− Eη(X)f(X)Ia(X)] + op(1).

Var

(1√n

n∑i=1

[Qh(Xi)− θ]−1√n

n∑i=1

[η(Xi)f(Xi)Ia(Xi)− Eη(X)f(X)Ia(X)]

)= E ([Qh(X1)− θ]− [η(X1)f(X1)Ia(X1)− Eη(X)f(X)Ia(X)])

2

= E[Qh(X1)− θ]2 + E[η(X1)f(X1)Ia(X1)− Eη(X)f(X)Ia(X)]2

−2E[Qh(X1)− θ][η(X1)f(X1)Ia(X1)− Eη(X)f(X)Ia(X)].

±y²n→∞§þªmànÏ"ÑÂñuVar(η(X)f(X)Ia(X)). ù¿X

1√n

n∑i=1

E

[η(Xj)Ia(Xj)

1

hK

(Xi −Xj

h

) ∣∣∣Xi

]=

1√n

n∑i=1

η(Xi)f(Xi)Ia(Xi)] + op(1).

nþ¤ã§·k

√n(Un − θ) =

2√n

n∑i=1

[η(Xi)f(Xi)Ia(Xi)− Eη(X)f(X)Ia(X)] +Op(√nh2).

XJ^nh4 → 0¤á§K·ª

1√n

n∑i=1

η(Xi)[fn(Xi)− f(Xi)] =√nUn −

1√n

n∑i=1

η(Xi)f(Xi)Ia(Xi) + op(1)

=√n(Un − θ)−

1√n

n∑i=1

[η(Xi)f(Xi)Ia(Xi)− θ] + op(1).

|^cã¤§éþªmà\n=Ún(Ø.

ÚÚÚnnn 5.3 bµ(x), η(x)ëY¼ê§∫ a

−aη2(u)

[∫ ∞u

f(x)µ2(x)dx

]f(u)du <∞,

K·k

1√n

n∑i=1

η(Xi)Ia(Xi)

∫ ∞Xi

[fn(x)− f(x)]µ(x)dx =1√n

n∑i=1

[P (Xi)− EP (X)] + op(1).

Ù¥

P (Xi) = µ(Xi)

∫ a

−aη(u)I(u < Xi)f(u)du.


yyy²²²µµµ duÚny²aquÚn5.2y²§¤±·=QãØÓ?. Äk§

1√n

n∑i=1

η(Xi)Ia(Xi)

∫ ∞Xi

fn(x)µ(x)dx =1

nh√n

n∑i=1

η(Xi)Ia(Xi)

∫ ∞Xi

K

(Xi − xh

)µ(x)dx

− 1

n(n− 1)h√n

∑i 6=j

η(Xi)Ia(Xi)

∫ ∞Xi

K

(Xj − xh

)µ(x)dx

+1

(n− 1)h√n

∑i 6=j

η(Xi)Ia(Xi)

∫ ∞Xi

K

(Xj − xh

)µ(x)dx.

du

E

∣∣∣∣∣ 1

nh

n∑i=1

η(Xi)Ia(Xi)

∫ ∞Xi

K

(Xi − xh

)µ(x)dx

∣∣∣∣∣ ≤∫ a

−a|η(u)|f(u)

[∫ ∞0

K(v)µ|(v + uh)|dv]du,

þªmàO(1), ¤±

1

nh√n

n∑i=1

η(Xi)Ia(Xi)

∫ ∞Xi

K

(Xi − xh

)µ(x)dx = op(1).

·íÑ

E

∣∣∣∣∣∣ 1

n(n− 1)h

∑i6=j

η(Xi)Ia(Xi)

∫ ∞Xi

K

(Xj − xh

)µ(x)dx

∣∣∣∣∣∣≤

∫ a

−a|η(u)|f(u)

[∫ ∞u

f(x)|µ(x)|dx]du+O(h2) = O(1),

¤±1

n(n− 1)h√n

∑i 6=j

η(Xi)Ia(Xi)

∫ ∞Xi

K

(Xj − xh

)µ(x)dx = op(1).

Xd§·Ò±

1√n

n∑i=1

η(Xi)

∫ ∞Xi

fn(x)µ(x)dx =√nUn + op(1),

Ù¥Un = 2n−1(n− 1)−1∑ni=1 h(Xi, Xj)§

h(Xi, Xj) =1

2

[η(Xi)

∫ ∞Xi

1

hK

(Xj − xh

)µ(x)dx+ η(Xj)

∫ ∞Xj

1

hK

(Xi − xh

)µ(x)dx

].

½Âθ = Eh(X1, X2)§¿ÄUn3z:þÝK§·±ÙGLª

Un − θ =

n∑i=1

E[Un|Xi]− (n− 1)θ =2

n

n∑i=1

(E[h(X,Xi)|Xi]− θ) ,

Un − Un =2

n(n− 1)

∑i<j

[h(Xi, Xj)− h(Xi)− h(Xj)− θ] =2

n(n− 1)

∑i<j

H(Xi, Xj).

¿·k

E[Un − Un]2 =2

n(n− 1)EH2(X1, X2).

3Ún^e§±y²EH2(X1, X2) = O(1). dd=√n(Un − θ) =

√n(Un − θ) + op(1).


eP

Rh(Xi) = E

[η(Xi)Ia(Xi)

∫ ∞Xi

1

hK

(X − xh

)µ(x)dx

∣∣∣Xi

]

Ph(Xi) = E

[η(X)Ia(X)

∫ ∞X

1

hK

(Xi − xh

)µ(x)dx

∣∣∣Xi

]´y

Rh(Xi) = η(Xi)Ia(Xi)

∫ ∞Xi

f(u)µ(u)du+Op(h2).

·±

1√n

n∑i=1

Ph(Xi) =1√n

n∑i=1

P (Xi) + op(1),

¯¢þ§·IyE[Ph(Xi)− P (Xi)]2 = o(1)=.

,§·éN´wE[h(X,Xi)|Xi] = [Ph(Xi)+Rh(Xi)]/2§EPh(X) = ERh(X)§θ =

[EPh(X) + ERh(X)]/2. ¤±k

1√n

n∑i=1

η(Xi)Ia(Xi)

∫ ∞Xi

[fn(x)−f(x)]µ(x)dx =√nUn−

1√n

n∑i=1

η(Xi)Ia(Xi)

∫ ∞Xi

f(x)µ(x)dx+op(1),

òc¡¤¯¢\þªmà§=Ún(Ø.

íííØØØ 5.2 bµ(x), η(x)ëY¼ê§∫ a

−aη2(u)

[∫ u

−∞f(x)µ2(x)dx

]f(u)du <∞,

1√n

n∑i=1

η(Xi)Ia(Xi)

∫ Xi

−∞[fn(x)− f(x)]µ(x)dx =

1√n

n∑i=1

[Q(Xi)− EQ(X)] + op(1).

Ù¥

Q(Xi) = µ(Xi)

∫ a

−aη(u)I(u > Xi)f(u)du.

íØ5.2y²L§aquÚn5.3y²§ã Øy.

e¡·5¤½n4.2y².

½½½nnn4.2yyy²²²µµµ duëêm;5±9'¼ê'uëêëY5§Ü5y²'ü,

dÑ. ·=Iy²OìC5.

dõ¼êVÐm§θ, σ÷veã§

0 =

(1

n

n∑i=1

˙H(Zi; θ0, σ0)

)′Wn

(1√n

n∑i=1

H(Zi; θ0, σ0)

)

+

(1

n

n∑i=1

¨H(Zi; θ, σ)

)′Wn

(1

n

n∑i=1

H(Zi; θ, σ)

)⊗ I(p+1)×(p+1)

(α− α0)

+

(1

n

n∑i=1

˙H(Zi; θ, σ)

)′Wn

(1

n

n∑i=1

˙H(Zi; θ, σ)

)√n(α− α0),


Ù¥˙H(Z; θ, σ),

¨H(Z; θ, σ)©OL«H(Z; θ, σ)é(θ, σ) §θ0uθθ0m§σ0

uσσ0m. dθ, σÜ5§9^(S2)§·Iy²n−1/2(∑n

i=1 H(Zi; θ0, σ0))ä

k¤IìC5.

âH(Zi; θ0, σ0)½Â§·±ò n−1/2(∑n

i=1 H(Zi; θ0, σ0))

1√n

n∑i=1

H(Zi; θ0, σ0) =1√n

n∑i=1

H(Zi; θ0, σ0) +

6∑j=1

Tjn,

Ù¥§

T1n =1√n

n∑i=1

[1

gn(Zi)− 1

g(Zi)

]exp(Zi

√2/σ)Ia(Zi)

∫ ∞Zi

µ1(x; θ0, σ0)[gn(x)− g(x)]dx

T2n =1√n

n∑i=1

g−1(Zi) exp(Zi√

2/σ)Ia(Zi)

∫ ∞Zi

µ1(x; θ0, σ0)[gn(x)− g(x)]dx

T3n =1√n

n∑i=1

[1

gn(Zi)− 1

g(Zi)

]exp(Zi

√2/σ)Ia(Zi)

∫ ∞Zi

µ1(x; θ0, σ0)g(x)dx

T4n =1√n

n∑i=1

[1

gn(Zi)− 1

g(Zi)

]exp(−Zi

√2/σ)Ia(Zi)

∫ Zi

−∞µ2(x; θ0, σ0)[gn(x)− g(x)]dx

T5n =1√n

n∑i=1

g−1(Zi) exp(−Zi√

2/σ)Ia(Zi)

∫ Zi

−∞µ2(x; θ0, σ0)[gn(x)− g(x)]dx

T6n =1√n

n∑i=1

[1

gn(Zi)− 1

g(Zi)

]exp(−Zi

√2/σ)Ia(Zi)

∫ Zi

−∞µ2(x; θ0, σ0)g(x)dx.

dÚn5.1±íT1n = op(1), T4n = op(1). 3Ún5.3¥§η(Z) = g−1(Z) exp(Z√

2/σ0),

µ(x) = µ1(x; θ0, σ0), |^½n¥PÒS11(Z), Kk

T2n =1√n

n∑i=1

[S11(Zi)− ES11(Z)] + op(1),

dÚn5.1

T3n =1√n

n∑i=1

[[g(Zi)− gn(Zi)]

g2(Zi)

]exp(Zi

√2/σ)Ia(Zi)

∫ ∞Zi

µ1(x; θ0, σ0)g(x)dx+ op(1).

3Ún5.2¥

η(Z) = g−2(Zi) exp(Zi√

2/σ)Ia(Zi)

∫ ∞Zi

µ1(x; θ0, σ0)g(x)dx,

Kk

T2n = − 1√n

n∑i=1

[S12(Zi)− ES12(Z)] + op(1),

|^Ún5.2§±9íØ5.2§aq

T5n =1√n

n∑i=1

[S21(Zi)− ES21(Z)] + op(1),

T6n = − 1√n

n∑i=1

[S22(Zi)− ES22(Z)] + op(1).


dd=½n(Ø.

ëëë ©©© zzz

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[7] Hong, H.and Tamer, E.A. Simple Estimator for Nonlinear Error in Variable Models. Journal ofEconometrics, 2003, 117(1): 1-19.

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[11] Shi, J., Chen, K., Song, W. Robust errors-in-variables linear regression via Laplace distribution.Statistics & Probability Letters, 2014, 84: 113-120.

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[14] Tosteson T.D., Stefanski L.A., Schafer D.W. A measurement-error model for binary and ordinalregression. Stat Med., 1989 8(9):1139-1147.

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