Distributional results for thresholding estimators in high-dimensional Gaussian regression models

Munich Personal RePEc Archive

Distributional results for thresholding estimators in high-dimensional Gaussian regression models

Pötscher, Benedikt M. and Schneider, Ulrike

University of Vienna, University of Goettingen

June 2011

Online at https://mpra.ub.uni-muenchen.de/34706/

MPRA Paper No. 34706, posted 21 Nov 2011 17:15 UTC

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σ = (n−k)⁻ (Y −XˆθLS)^′(Y −XˆθLS)

; θ σ )

. ( n > k ˜θH . % *

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A

+ ( ) = max(,0) 1 () * % ˜θ_AS

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=





0 ˆθLS,i ≤σξˆ _i,nη_i,n ˆθLS,i−σ ξˆ _i,nη_i,n/ˆθLS,i ˆθLS,i >σξˆ _i,nη_i,n

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=





0 ˆθ_LS,i ≤σξ_i,nη_i,n ˆθLS,i−σ ξ_i,nη_i,n/ˆθLS,i ˆθLS,i > σξ_i,nη_i,n .

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% % * %

!

P_n,θ,σ ˆθ_i= 0 = 1−P_n,θ,σ ˆθ_i= 0 I ( % * (

P_n,θ,σ ˆθ_i= 0 = Φ n ^/ −θ_i/(σξ_i,n) +η_i,n −Φ n^/ −θ_i/(σξ_i,n)−η_i,n . <A?

@ % ) Pn,θ,σ ˆθi= 0 * θ ( % θi '

. ( % L * H.2 * H ( *

+

" # 0 < σ < ∞ i ≥ 1 i ≤ k = k(n)

n

! Pn,θ,σ ˆθi= 0 →0 n→ ∞ θ

θ_i= 0 θ_i n ξ_i,nη_i,n→0

! P_n,θ,σ ˆθ_i = 0 →1 n→ ∞ θ

θi= 0 n ^/ η_i,n→ ∞

! P_n,θ,σ ˆθ_i = 0 → c_i < 1 n → ∞

θ θi = 0 n ^/ η_i,n → ei" 0 ≤ ei < ∞ # ci ci =

Φ (ei)−Φ (−ei)

< ? % * * % ( I *

( , I + * ( % 1 < ? % (

I ( , I + * ( % 1

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e_i = 0? + % ) + ξ_i,n/n ^/ % ,

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% $ % & < ? * % < ? ξ_i,nη_i,n

n ^/ ξ⁻_i,n < % ?D n ^/ ξ⁻_i,n→ ∞) *

% < ? + exp −cnξ⁻_i,n / n ^/ ξ⁻_i,n c >0 *

θi/σ

< ? * % < ? exp −0.5nη_i,n / n ^/ η_i,n - < ? *

% % ( + n ^/ η_i,n ** e_i

E1 % + % : M-- $ " 1 <$=78? F

& 1 θ∈R^{k n} A_n(θ) ={i: 1≤i≤k(n), θ_i= 0} < ? % (i∈A_n(θ)

Pn,θ,σ ˆθi= 0 ≤Pn,θ,σ





j∈A θ

ˆθj= 0

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j∈A θ

Pn,θ,σ ˆθj = 0 .

** + θ + n< θ ( *

n?^$ ) % card(A_n(θ)) < * k(n)

?) * ( , I %

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i ∈A_n(θ) inf_{i∈A θ} |θ_i|>0 _{i∈A θ} exp −cnξ⁻_i,n / n ^/ ξ⁻_i,n →0 n→ ∞ c (inf_{i∈A θ} |θ_i|/σF

< ? 1 % (i /∈A_n(θ)+ % P_n,θ,σ ˆθ_i= 0 ≥ P_n,θ,σ





j /∈A θ

ˆθ_j= 0



= 1−P_n,θ,σ





j /∈A θ

ˆθ_j= 0





≥ 1−

j /∈A θ

1−P_n,θ,σ ˆθ_j= 0 .

** θ + n ) % card(A^c_n(θ))

< * k(n) ?) * ( (

( , * , % 0 n→ ∞ (

n ^/ η_i,n→ ∞ % (i∈A_n(θ) E- card(A^c_n(θ)) * ( % 0) ) _{i /∈A θ} exp −0.5nη_i,n / n ^/ η_i,n →0 n→ ∞F

< ? - X^′X ) % * P_{n,θ,σ i∈A θ} ˆθ_i= 0 +

Pn,θ,σ i /∈A θ ˆθi = 0 ( 2* * Pn,θ,σ ˆθi= 0

1−P_n,θ,σ ˆθ_i= 0 ) * A **

.2 * ( * + % *

% % * < : <"##7?)

: <"##=?? + H % * H + 2 ) ) + +

$ * () θ * k n .2 R^∞

C

θ + σ * * , n - * * + < ;

* ? + ( + ξ_i,nη_i,n →0 n→ ∞)

+ ˆθi % θi ; * A)

$6 + % ξ_i,nη_i,n→0) + +

n^/ η_i,n→e_i)0≤e_i<∞) n ^/ η_i,n→ ∞) + * A

+ H % % H H H) * % (^"

" ' $ i≥1 i≤k=k(n) n

ξ_i,nη_i,n→0 n ^/ η_i,n→ei 0≤ei≤ ∞

ei < ∞ $ θⁿ = (θ ,n, . . . , θk ,n) ∈ R^k σ_n∈(0,∞) n ^/ θ_i,n/(σ_nξ_i,n)→ν_i∈R #

n→∞lim P_{n,θ ,σ} ˆθi= 0 = Φ (−νi+ei)−Φ (−νi−ei).

e_i = ∞ $ θⁿ = (θ _,n, . . . , θ_{k ,n}) ∈ R^k σ_n∈(0,∞) θ_i,n/(σ_nξ_i,nη_i,n)→ζ_i∈R #

' |ζ_i|<1 lim_n→∞P_{n,θ ,σ} ˆθ_i= 0 = 1 ( |ζ_i|>1 limn→∞P_{n,θ ,σ} ˆθi= 0 = 0

) |ζ_i|= 1 r_i,n :=n ^/ η_i,n−ζ_iθ_i,n/(σ_nξ_i,n) →r_i" r_i∈R"

n→∞lim P_{n,θ ,σ} ˆθi = 0 = Φ(ri).

- .2 * ( * ( ) + * C *

θ_i,n≡θ_i σ_n ≡σ) * P_n,θ,σ ˆθ_i= 0 + ( 0 θ_i= 0)

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?D ( * * ( * . * % *

% * ( * ( ( * C * .

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( * ( + * ( $ ( % * *

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: <"##=?

( $ % * < ? * % < ? %

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exp −cnθ_i,n/(σ_nξ_i,n) / n ^/ θ_i,n/(σ_nξ_i,n)

" ( % n η <. . ? )

% ) ( % ; n η ) % ;

* 2

> () ( θ /σ ,

(c <1/2

< ? * % < $? + exp −cnη_i,n / n ^/ η_i,n + c

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< >? * % + * %

max exp −cnη_i,n / n ^/ η_i,n ,|ri,n−ri| (c <2 |r_i|<∞D |r_i|=∞ +

max exp −cnη_i,n / n^/ η_i,n ,exp −0.5r_i,n /|r_i,n| (c <2

* . >

: <"##=? > $ <"##=?

) !

- + % . * % L *

+ 5

P_n,θ,σ ˜θ_i= 0 =P_n,θ,σ ˆθ_LS,i ≤σξˆ _i,nη_i,n

=

∞

P_n,θ,σ ˆθ_LS,i ≤σξˆ _i,nη_i,n|ˆσ=sσ ρ_n−k(s)ds

=

∞

Pn,θ,σ ˆθi(sη_i,n) = 0 ρ_n−k(s)ds

=

∞

Φ n^/ −θi/(σξ_i,n) +sη_i,n

−Φ n ^/ −θ_i/(σξ_i,n)−sη_i,n ρ_n−k(s)ds

=T_{n−k,n θ / σξ} n ^/ η_i,n −T_{n−k,n θ / σξ} −n ^/ η_i,n . <7?

+ % <A?) * σˆ ˆθ_LS,i + * σˆ ( sσ

% ) : <"##>) * $$#? - % ρ_n−k (

(n−k)^{− /} ; ; % + n−k

- + % ρ_n−k(s) = 0 s <0) ρ_n−k

R

' + % + .2 * ( * % L

* + % * ( * *

+ % ) ) * A5

" * 0< σ < ∞ i ≥1 i ≤ k =k(n)

n

! P_n,θ,σ ˜θ_i= 0 →0 n→ ∞ θ

θ_i= 0 θ_i n ξ_i,nη_i,n→0

! Pn,θ,σ ˜θi = 0 →1 n→ ∞ θ

θ_i= 0 n ^/ η_i,n→ ∞

$#

! P_n,θ,σ ˜θ_i= 0 −c_i,n → 0 n → ∞ θ θi = 0 ci,n = Tn−k(ei)−Tn−k(−ei) lim sup_n→∞ci,n < 1 n ^/ η_i,n→ei"0≤ei<∞

* $# + ( % %

2* ( + % + %

1 ) c_i,n ** < ? % * * %

c_i = Φ(e_i)−Φ(−e_i) + n−k → ∞)

+ % / n−k ; m) () % ()

; c_i,n ; T_m(e_i)−T_m(−e_i) % ( ' . (

7 ** * $# %

1 + % + 2 % ( * %

% L * % * ( *

+ ' + n−k < % (? + n−k→ ∞

( + () ( *

N∪ {∞}+ + ( <* ( * ; ? n−k % N∪ {∞}

$ i≥1 i≤k=k(n) n

ξ_i,nη_i,n→0 n ^/ η_i,n→ei 0≤ei≤ ∞

e_i < ∞ $ θⁿ = (θ _,n, . . . , θ_{k ,n}) ∈ R^k σ_n∈(0,∞) n ^/ θ_i,n/(σ_nξ_i,n)→ν_i∈R

' + n−k , m" "

n→∞lim P_{n,θ ,σ} ˜θ_i= 0 =

∞

(Φ (−ν_i+se_i)−Φ (−ν_i−se_i))ρ_m(s)ds.

( + n−k→ ∞ "

n→∞lim P_{n,θ ,σ} ˜θi= 0 = Φ (−νi+ei)−Φ (−νi−ei).

e_i = ∞ $ θⁿ = (θ _,n, . . . , θ_{k ,n}) ∈ R^k σ_n∈(0,∞) θ_i,n/(σ_nξ_i,nη_i,n)→ζ_i∈R

' + n−k , m" "

n→∞lim P_{n,θ ,σ} ˜θ_i = 0 =

∞

|ζ|

ρ_m(s)ds= Pr(χ_m> mζ_i).

( + n−k→ ∞ "

' |ζ_i|<1 limn→∞P_{n,θ ,σ} ˜θi= 0 = 1 ( |ζ_i|>1 lim_n→∞P_{n,θ ,σ} ˜θ_i= 0 = 0

) |ζ_i|= 1 n ^/ η_i,n/(n−k) ^/ →0

n→∞lim P_{n,θ ,σ} ˜θi= 0 = Φ(ri) r_i,n:=n ^/ η_i,n−ζ_iθ_i,n/(σ_nξ_i,n) →r_i r_i∈R

& |ζ_i|= 1 n ^/ η_i,n/(n−k) ^/ →2^/ di 0< di<∞

n→∞lim P_{n,θ ,σ} ˜θi= 0 =

∞

−∞

Φ(dit+ri)φ(t)dt

ri,n →ri ri ∈R -. 1

ri=∞" 0 ri =−∞/

0 |ζ_i|= 1 n ^/ η_i,n/(n−k) ^/ → ∞

n→∞lim P_{n,θ ,σ} ˜θ_i= 0 = Φ(r_i^′) n ^/ η_i,n/(n−k)^{/ −} r_i,n→2^{− /} r^′_i r_i^′∈R

$$ + ) * ) + % %

% L * .2 * ( *

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+ % L * + %

+ % n−k . ( <

ˆ

σ/σ 1?D n−k % ( )

2 + % <+ σ * ( sσ?)

2 + * σ/σˆ E' +

e_i = 0 2 + %

F ' + 2* ) /

5 - n−k→ ∞ + %

|ζ_i| < 1 |ζ_i| > 1 ) ( 1 0) * % ( + % ) H (H

|ζ_i|= 1 + n−k % . ( % - %

n ^/ η_i,n/(n−k) ^/ →0) + %

) ( Φ(ri)) D n−k % . ( + () /

<+ ) ) A < "? ( % Φ(ri+ )+ *

? + n−k % (

% ( / + % % + *

ˆ

σ/σ * & % *

1 0) * % () ( ζ_i = 0 |ζ_i| =∞) * % () +

+ % < + % + n−k→ ∞?

< ? 1 + * C $$ + **

; ) (

< ? % * C % ; % % θ_i,n

σ_n ( % ( ; (θ_i,n, σ_n) + )

* R) ( ; n_j ; n_{j l}

; % ; n ^/ θ_i,n/(σ_nξ_i,n) < θ_i,n/(σ_nξ_i,nη_i,n)

ri,n? % R * C + ** ; 4 )

** * * ; n_{j l} ,

* * % L * +

%

< ? - ) % $$ <

n−k? ( ) * % , *

* % L * + %

$"

@ 4 ) % % + %

% * ( * + % + %

( ; * n−k→ ∞ e_i= 0<+ % % (

; % ( n ^/ η_i,n/(n−k) ^/ → 0? -

n−k→ ∞ n ^/ η_i,n/(n−k) ^/ →0

; + 5

" $ i ≥ 1 i ≤k = k(n) n

n ^/ η_i,n(n−k)^{− /} →0 n→ ∞ # sup

θ∈R , <σ<∞

P_n,θ,σ ˆθ_i= 0 −P_n,θ,σ ˜θ_i= 0 →0 n→ ∞.

# ** ξ_i,nη_i,n →0 n→ ∞)

- n ^/ η_i,n(n−k)^{− /} % , n → ∞) +

* C $$ % * <

** * < ? ; (θⁿ , σ_n )? + % + %

+ n ^/ η_i,n(n−k)^{− /} →0 % * *

+ < ξ_i,nη_i,n→0 ?

# ! +, ) - ! +, . !

/ 0

1 * * * + + + % * * ) + (

+ % ˆθ_LS,i N(θ_i, σ ξ_i,n/n)

" $ i ≥ 1 i ≤ k = k(n) n

ξ_i,n/n ^/ →0 ! ˆθLS,i θi"

ξ_i,n/n ^/

$ ξ_i,n/n^/ →0 # ˆθLS,i θi

ε >0

n→∞lim sup

θ∈R

sup

<σ<∞

P_n,θ,σ ˆθ_LS,i−θ_i > σε = 0.

+ "ˆθ_LS,i n ^/ /ξ_i,n θ_i ε >0

M >0 sup

n∈N

sup

θ∈R

sup

<σ<∞Pn,θ,σ n ^/ /ξ_i,n ˆθLS,i−θi > σM < ε.

-. θ σ + "

, 2Φ(−εn ^/ /ξ_i,n) 2Φ(−M)" /

* ˜θ_H,i)˜θ_S,i) ˜θ_AS,i

* ˆθ_H,i)ˆθ_S,i) ˆθ_AS,i + +

% ˜θ_i ˜θ_H,i" ˜θ_S,i" ˜θ_AS,i # i≥1

i≤k=k(n) n

˜θ_i θ_i ξ_i,nη_i,n→0 ξ_i,n/n ^/ →0

$ ξ_i,nη_i,n → 0 ξ_i,n/n ^/ →0 # ˜θ_i

ε >0

n→∞lim sup

θ∈R

sup

<σ<∞Pn,θ,σ ˜θi−θi > σε = 0.

"˜θ_i a_i,n a_i,n= min n ^/ /ξ_i,n,(ξ_i,nη_i,n)⁻

ε >0 M >0

sup

n∈N

sup

θ∈R

sup

<σ<∞Pn,θ,σ ai,n ˜θi−θi > σM < ε.

$ ξ_i,nη_i,n → 0 ξ_i,n/n ^/ →0 b_i,n ≥0 + ε >0 M >0

lim sup

n→∞ sup

θ∈R

sup

<σ<∞P_n,θ,σ b_i,n ˜θ_i−θ_i > σM < ε <6?

" b_i,n=O(a_i,n)

ˆθi ˆθH,i"ˆθS,i" ˆθAS,i #

ˆθi

* + ˜θ_H,i)˜θ_S,i) ˜θ_AS,i < +

% ? (a_i,n *

* % - * ) % % (n ^/ /ξ_i,n

) + 2* .

+ % ) * ( ) * +

( ( (ξ_i,nη_i,n)⁻ ) ) % + 2

% ; < 2 ? < % % (

? 1 * + %

6 A

& - n^/ η_i,n→ei= 0) ˜θi ( * ( ; % ˆθLS,i

% (ε >0

n→∞lim sup

θ∈R

sup

<σ<∞

P_n,θ,σ n ^/ /ξ_i,n |˜θ_i−ˆθ_LS,i|> σε = 0.

@ ˆθ_i 1 ˜θ_i + ( <"8? C

( * ρ_n−k D ˆθ_i +

ˆθ_i−ˆθ_LS,i ≤σξ_i,nη_i,n

' < ? @ % * $6 +

(

n→∞lim sup

θ∈R

sup

<σ<∞Pn,θ,σ ai,n ˆθi−θi > σM = 0

% (M >1)

n→∞lim sup

θ∈R

sup

<σ<∞

P_n,θ,σ a_i,n ˜θ_i−θ_i > σM = 0

$A

% (M >1* % n−k→ ∞

< ? - * * + $6< ?

* % R^k * ( * % ( 0

σ .2 ( * % %

< ? - σε σM * (ε M) * % () * ( * $7

$6 + $8) * %

* % 0< σ <∞ * ( * % 0< σ≤c) + c >0 (

$ 1 " 2

$ !

' 2 * . * - +

% % % ) + α_i,n

* % ) + ( .

, α_i,n

1 ) % . * * θ (

θ_i

" ( # H_H,n,θ,σⁱ :=H_H,ηⁱ _,n,θ,σ σ⁻ α_i,n(ˆθ_H,i−θ_i)

H_H,n,θ,σⁱ (x) = Φ n ^/ x/(α_i,nξ_i,n) α⁻_i,nx+θ_i/σ > ξ_i,nη_i,n

+Φ n ^/ −θ_i/(σξ_i,n) +η_i,n 0≤α⁻_i,nx+θ_i/σ≤ξ_i,nη_i,n

+Φ n ^/ −θi/(σξ_i,n)−η_i,n −ξ_i,nη_i,n≤α⁻_i,nx+θi/σ <0 , <8?

" , "

dH_H,n,θ,σⁱ (x) = Φ n ^/ −θi/(σξ_i,n) +η_i,n −Φ n ^/ −θi/(σξ_i,n)−η_i,n dδ_{−α θ /σ}(x) + n ^/ /(αi,nξ_i,n) φ n ^/ x/(αi,nξ_i,n) α⁻_i,nx+θi/σ > ξ_i,nη_i,n dx <C?

δ_z z

" * # H_S,n,θ,σⁱ :=H_S,ηⁱ _,n,θ,σ σ⁻ α_i,n(ˆθ_S,i−θ_i)

H_S,n,θ,σⁱ (x) = Φ n ^/ x/(α_i,nξ_i,n) +n ^/ η_i,n α⁻_i,nx+θ_i/σ≥0

+Φ n ^/ x/(α_i,nξ_i,n)−n ^/ η_i,n α⁻_i,nx+θ_i/σ <0 , <=?

" , "

dH_S,n,θ,σⁱ (x) = Φ n ^/ −θ_i/(σξ_i,n) +η_i,n −Φ n ^/ −θ_i/(σξ_i,n)−η_i,n dδ_{−α θ /σ}(x) + n ^/ /(α_i,nξ_i,n) φ n ^/ x/(α_i,nξ_i,n) +n ^/ η_i,n α⁻_i,nx+θ_i/σ >0 <$#?

+φ n ^/ x/(αi,nξ_i,n)−n ^/ η_i,n α⁻_i,nx+θi/σ <0 dx.

" # H_AS,n,θ,σⁱ :=H_AS,ηⁱ _,n,θ,σ σ⁻ α_i,n(ˆθ_AS,i−θ_i)

H_AS,n,θ,σⁱ (x) = Φ z_n,θ,σ(x, η_i,n) α⁻_i,nx+θ_i/σ≥0 +Φ z_n,θ,σ(x, η_i,n) α⁻_i,nx+θ_i/σ <0 ,

<$$?

z_n,θ,σ(x, y)≤z_n,θ,σ(x, y)

0.5n ^/ ξ⁻_i,n(α⁻_i,nx−θ_i/σ)±n ^/ 0.5ξ⁻_i,n(α⁻_i,nx+θ_i/σ) +y .

" , "

dH_AS,n,θ,σⁱ (x) = Φ n ^/ −θ_i/(σξ_i,n) +η_i,n −Φ n ^/ −θ_i/(σξ_i,n)−η_i,n dδ_{−α θ /σ}(x) +(0.5n ^/ /(α_i,nξ_i,n)) φ z_n,θ,σ(x, η_i,n) (1 +t_n,θ,σ(x, η_i,n)) α⁻_i,nx+θ_i/σ >0 +φ z_n,θ,σ(x, η_i,n) (1−t_n,θ,σ(x, η_i,n)) α⁻_i,nx+θ_i/σ <0 ,

tn,θ,σ(x, y) = 0.5ξ⁻_i,n α⁻_i,nx+θi/σ / (0.5ξ⁻_i,n α⁻_i,nx+θi/σ ) +y ^/ .

. * ˆθH,i)ˆθS,i) ˆθAS,i (

* + * ) * * −α_i,nθ_i/σ

( + ( ( 1

* * : <"##=?

<"##=?

- + X^′X ) * θ_i θ_j

i =j * % ( +

. * % ˆθ_H) ˆθ_S) ˆθ_AS - * ) * %

. * : ˆθL * % : ˆθAS

< $ "?

$ ) !

. * ˜θ_H,i)˜θ_S,i) ˜θ_AS,i 2

* % **

" # H_H,n,θ,σⁱ :=H_H,ηⁱ _,n,θ,σ σ⁻ αi,n(˜θH,i−θi)

H_H,n,θ,σⁱ (x) = Φ n ^/ x/(αi,nξ_i,n)

∞

α⁻_i,nx+θi/σ > ξ_i,nsη_i,n ρ_n−k(s)ds <$"?

+

∞

Φ n ^/ −θi/(σξ_i,n) +sη_i,n 0≤α⁻_i,nx+θi/σ≤ξ_i,nsη_i,n ρ_n−k(s)ds +

∞

Φ n ^/ −θ_i/(σξ_i,n)−sη_i,n −ξ_i,nsη_i,n≤α⁻_i,nx+θ_i/σ <0 ρ_n−k(s)ds.

" , "

dH_H,n,θ,σⁱ (x) =

∞

Φ n ^/ −θ_i/(σξ_i,n) +sη_i,n <$>?

−Φ n ^/ −θi/(σξ_i,n)−sη_i,n ρ_n−k(s)dsdδ_{−α θ /σ}(x) +n^/ α⁻_i,nξ⁻_i,n

×φ n ^/ x/(αi,nξ_i,n)

∞

α⁻_i,nx+θi/σ > ξ_i,nsη_i,n ρ_n−k(s)dsdx.

$6

" # # H_S,n,θ,σⁱ :=H_S,ηⁱ _,n,θ,σ σ⁻ α_i,n(˜θ_S,i−θ_i) H_S,n,θ,σⁱ (x) =

∞

Φ n ^/ x/(α_i,nξ_i,n) +n ^/ sη_i,n ρ_n−k(s)ds α⁻_i,nx+θ_i/σ≥0 +

∞

Φ n ^/ x/(α_i,nξ_i,n)−n ^/ sη_i,n ρ_n−k(s)ds α⁻_i,nx+θ_i/σ <0

= T_{n−k,−n x/ α ξ} n ^/ η_i,n α⁻_i,nx+θi/σ≥0

+T_{n−k,−n x/ α ξ} −n ^/ η_i,n α⁻_i,nx+θi/σ <0 . <$A?

" , "

dH_S,n,θ,σⁱ (x) =

∞

Φ n ^/ −θi/(σξ_i,n) +sη_i,n <$7?

−Φ n ^/ −θ_i/(σξ_i,n)−sη_i,n ρ_n−k(s)dsdδ_{−α θ /σ}(x) +n ^/ α⁻_i,nξ⁻_i,n

×

∞

φ n ^/ x/(αi,nξ_i,n) +n ^/ sη_i,n ρ_n−k(s)ds α⁻_i,nx+θi/σ >0 +

∞

φ n ^/ x/(αi,nξ_i,n)−n ^/ sη_i,n ρ_n−k(s)ds α⁻_i,nx+θi/σ <0 dx.

" $ # H_AS,n,θ,σⁱ :=H_AS,ηⁱ _,n,θ,σ σ⁻ α_i,n(˜θ_AS,i−θ_i)

H_AS,n,θ,σⁱ (x) =

∞

Φ z_n,θ,σ(x, sη_i,n) ρ_n−k(s)ds α⁻_i,nx+θ_i/σ≥0 +

∞

Φ z_n,θ,σ(x, sη_i,n) ρ_n−k(s)ds α⁻_i,nx+θ_i/σ <0 . <$6?

" , "

dH_AS,n,θ,σⁱ (x) =

∞

Φ n^/ −θi/(σξ_i,n) +sη_i,n <$8?

−Φ n ^/ −θi/(σξ_i,n)−sη_i,n ρ_n−k(s)dsdδ_{−α θ /σ}(x) + (0.5n ^/ /(αi,nξ_i,n))

×

∞

φ z_n,θ,σ(x, sη_i,n) (1 +t_n,θ,σ(x, sη_i,n))ρ_n−k(s)ds α⁻_i,nx+θ_i/σ >0 +

∞

φ z_n,θ,σ(x, sη_i,n) (1−tn,θ,σ(x, sη_i,n))ρ_n−k(s)ds α⁻_i,nx+θi/σ <0 dx.

@ + % % 2 *

( * - ) % + *

(ρ_n−k (

( * <+ + % ( ( *

+ ) 1 $ : <"##=??

* % ) + % + * ( ρ_n−k

/ % % D * ; % (

+ % <1 " : <"##=? 1 $

<"##=??

% - + X^′X ) . *

% ˜θ_H)˜θ_S) ˜θ_AS ˆθ_H)ˆθ_S) ˆθ_AS < ""?

( ˆσ=sσ + * ρ_n−k(s) - * ) * %

. * : ˜θ_L * % : ˜θ_AS

< $ "?

% 3 0 " 2

' 2 % ( * %

* < ( .2 * ? + + + ( *

( .2 * +

* < 6 A?

% !

' . %

" & $ i ≥ 1 i ≤k = k(n) n

ξ_i,nη_i,n→0 n ^/ η_i,n→e_i 0≤e_i≤ ∞

e_i<∞ $ α_i,n=n ^/ /ξ_i,n $

θⁿ = (θ ,n, . . . , θk ,n) ∈ R^k σn ∈ (0,∞) n ^/ θi,n/(σnξ_i,n) → νi ∈ R #

H_H,n,θⁱ _,σ 1

Φ (x) (|x+ν_i|> e_i) + Φ (−ν_i+e_i) (0≤x+ν_i≤e_i) + Φ (−ν_i−e_i) (−e_i≤x+ν_i<0),

{Φ (−νi+ei)−Φ (−νi−ei)}dδ−ν (x) +φ(x) (|x+νi|> ei)dx. <$C?

-# |νi|=∞ ei= 0/

e_i = ∞ $ α_i,n = ξ_i,nη_i,n ⁻ $

θⁿ = (θ ,n, . . . , θk ,n)∈R^k σn∈(0,∞) θi,n/(σnξ_i,nη_i,n)→ζ_i∈R ' + |ζ_i|<1" H_H,n,θⁱ _,σ 1 δ_−ζ

( + |ζ_i|>1" H_H,n,θⁱ _,σ 1 δ

) + |ζ_i|= 1 n^/ η_i,n−ζ_iθ_i,n/(σ_nξ_i,n) →r_i" r_i ∈R" H_H,n,θⁱ _,σ

1

Φ(r_i)δ_−ζ + (1−Φ(r_i))δ .

" ' $ i ≥ 1 i ≤k = k(n) n

ξ_i,nη_i,n→0 n ^/ η_i,n→e_i 0≤e_i≤ ∞

e_i<∞ $ α_i,n=n ^/ /ξ_i,n $

θⁿ = (θ _,n, . . . , θ_{k ,n}) ∈ R^k σ_n ∈ (0,∞) n ^/ θ_i,n/(σ_nξ_i,n) → ν_i ∈ R #

H_S,n,θⁱ _,σ 1

Φ (x+ei) (x+νi≥0) + Φ (x−ei) (x+νi <0),

{Φ (−ν_i+e_i)−Φ (−ν_i−e_i)}dδ_−ν (x)+{φ(x+e_i) (x+ν_i>0) +φ(x−e_i) (x+ν_i <0)}dx.

<$=?

$C

-# N(−sign(ν_i)e_i,1) |ν_i|=∞ e_i= 0/

e_i = ∞ $ α_i,n = ξ_i,nη_i,n ⁻ $

θⁿ = (θ _,n, . . . , θ_{k ,n})∈ R^k σ_n ∈ (0,∞) θ_i,n/(σ_nξ_i,nη_i,n)→ ζ_i ∈ R

# H_S,n,θⁱ _,σ 1 δ_{− ζ ,|ζ|}

" ( $ i ≥ 1 i ≤k = k(n) n

ξ_i,nη_i,n→0 n ^/ η_i,n→e_i 0≤e_i≤ ∞

ei<∞ $ αi,n=n ^/ /ξ_i,n $

θⁿ = (θ _,n, . . . , θ_{k ,n}) ∈ R^k σ_n ∈ (0,∞) n ^/ θ_i,n/(σ_nξ_i,n) → ν_i ∈ R #

H_AS,n,θⁱ _,σ 1

Φ 0.5(x−ν_i) + (0.5(x+ν_i)) +e_i (x+ν_i≥0)

+Φ 0.5(x−ν_i)− (0.5(x+ν_i)) +e_i (x+ν_i<0) <"#?

|νi|<∞"

{Φ (−νi+ei)−Φ (−νi−ei)}dδ−ν (x)

+0.5 φ 0.5(x−ν_i) + (0.5(x+ν_i)) +e_i (1 +t(x)) (x+ν_i>0) +φ 0.5(x−ν_i)− (0.5(x+ν_i)) +e_i (1−t(x)) (x+ν_i <0) dx,

t(x) = (x+ν_i)/ (x+ν_i) + 4e_i + |ν_i| =∞" H_AS,n,θⁱ _,σ

1 Φ" " -+ ei = 0

/

ei = ∞ $ αi,n = ξ_i,nη_i,n ⁻ $

θⁿ = (θ _,n, . . . , θ_{k ,n})∈R^k σ_n∈(0,∞) θ_i,n/(σ_nξ_i,nη_i,n)→ζ_i∈R ' + |ζ_i|<1" H_AS,n,θⁱ _,σ 1 δ−ζ

( + 1≤ |ζ_i|<∞" H_AS,n,θⁱ _,σ 1 δ_{− /ζ}

) + |ζ_i|=∞" H_AS,n,θⁱ _,σ 1 δ

0 % αi,n % * * 2 (

ai,n % % +

* 2 % - % %

% ( . * )

% * ( * + * . * %

( + ( - ) .2 * ( * + )

+ * θi,n ≡θi σn ≡σ % * * ) *

. * * * + % θ_i = 0 ) .2

* G * %

G + ( N(0,1)) , θ_i 1 + %

* ( + . * .2 * θ_i= 0+

% (N(−sign(θ_i)e_i,1) - * ) % * * (

% . * .2 * +

- + % * ) (

+ * * < + ( , N?)

% 2 + * +

( ( * (

* H * H *

H % (H ^A - .2 * + +

% δ % ( % θi * %

@ . + % ;

+ n) * 2 %

' * 6 A /

+ .2 * δ_{− θ} ) + δ (

θ_i = 0D O + + * (

* * %

% ) - ! - 2 )

!

' 2 + . * ˜θH,i)˜θS,i) ˜θAS,i

* ˆθH,i)ˆθS,i) ˆθAS,i) * % () ( <+ * * ?

% < * ? * %

n−k % . ( @* + )

+ ; ' >#

+ ( α_i,n

* $ i≥1 i≤k=k(n) n

n ^/ η_i,n(n−k)^{− /} →0 n→ ∞ # sup

θ∈R , <σ<∞

H_H,n,θ,σⁱ −HH,n,θ,σ T Vⁱ →0 n→ ∞, sup

θ∈R , <σ<∞

H_S,n,θ,σⁱ −HS,n,θ,σ T Vⁱ →0 n→ ∞,

sup

θ∈R , <σ<∞

H_AS,n,θ,σⁱ −H_AS,n,θ,σⁱ _∞→0 n→ ∞

7

- % % ) n ^/ η_i,n(n−k)^{− /} → 0 + (

. n−k → ∞ E- ; % n−k → ∞ e_i = 0F -

n−k → ∞ ( + n ^/ η_i,n(n−k)^{− /} → 0

+ % ) ) I n ^/ η_i,n(n−k)^{− /} → 0 η_i,n → 0

lim sup_n→∞k/n <1

A1 % % ζ ) + +

% * ( θ ( + % %

+

) + θ

7 * % * % %

) ( * ) * ( 2*

* ' *

"#

** ξ_i,nη_i,n→0 n→ ∞ - n ^/ η_i,n(n−k)^{− /} %

, n→ ∞) $A + % >#

E % * + *

* % F + n ^/ η_i,n(n−k)^{− /} →0

% + < ξ_i,nη_i,n→0 ?

% ) !

% ! / / 0

' 2 ˜θ_H,i)˜θ_S,i) ˜θ_AS,i % * +

% %

2 $ i ≥ 1

i ≤ k = k(n) n ξ_i,nη_i,n → 0 n ^/ η_i,n → ei

0 ≤ e_i < ∞ $ α_i,n = n ^/ /ξ_i,n $

θⁿ = (θ ,n, . . . , θk ,n)∈R^k σn ∈(0,∞) n ^/ θi,n/(σnξ_i,n)→νi ∈R

+ n−k , m" " H_H,n,θⁱ _,σ 1

∞

{Φ (x) (|x+ν_i|> se_i) + Φ (−ν_i+se_i) (0≤x+ν_i≤se_i) + Φ (−ν_i−se_i) (−se_i≤x+ν_i<0)}ρ_m(s)ds,

∞

{Φ (−ν_i+se_i)−Φ (−ν_i−se_i)}ρ_m(s)dsdδ_−ν (x) +φ(x)

∞

(|x+ν_i|> se_i)ρ_m(s)dsdx.

<"$?

-# |νi|=∞ ei = 0/

+ n−k → ∞ " H_H,n,θⁱ _,σ 1

% (3

# $ $ i≥1

i≤k=k(n) n ξ_i,nη_i,n →0 n ^/ η_i,n→e_i 0≤e_i<∞ $ αi,n =n ^/ /ξ_i,n $ θⁿ = (θ ,n, . . . , θk ,n)∈ R^k σ_n∈(0,∞) n ^/ θ_i,n/(σ_nξ_i,n)→ν_i∈R

+ n−k , m" " H_S,n,θⁱ _,σ 1

∞

{Φ (x+sei) (x+νi≥0) + Φ (x−sei) (x+νi<0)}ρ_m(s)ds,

∞

{Φ (−ν_i+se_i)−Φ (−ν_i−se_i)}ρ_m(s)dsdδ_−ν (x) +

∞

{φ(x+se_i) (x+ν_i>0) +φ(x−se_i) (x+ν_i<0)}ρ_m(s)dsdx. <""?