• No results found

On the Second Order Behaviour of the Bootstrap of‎ L_1 Regression Estimators

N/A
N/A
Protected

Academic year: 2020

Share "On the Second Order Behaviour of the Bootstrap of‎ L_1 Regression Estimators"

Copied!
22
0
0

Loading.... (view fulltext now)

Full text

(1)

!

!

"

#$ % &

'

(

½

(2)

*

+

,

,,

,

+#

+

,

"#$

"$

-

"#$

"$+

. ")$

! "#$

" $

/

0 "#123$ ")$

"

"$+"-$$

.

'

- #

4

")$ 4

"'#$

"-$+#)

"-$+-.

"')$

#

+

+

(3)

6 "'#$ "')$

"

$

"

"7

$$

4 0 / "#123$

"

8 $

-

4

9 "#11#$:; 9 "#115$ < "#11=$ >

" $+

"

$ 6

"'#$ "')$

?

"$+ #

"

-$"

-$,

,

"$

"5$

" $

- > "5$

"

$ + #

)

"

-$"

-$,

+

,

"7$

:

!

¼  

"

!!

4

-"/#113

#111$

4

"7$

"5$.

(4)

0 / "0 #1==. / #1=2$

"

$

",$

" $+

",$

8 ' / "#112$ 0 /

#

"$ "@$

"

$

6 "'#$ "')$

<

"$+ "$" $

+" $" $

A"$" $" $" $

B+

" $

"=$

" "=$

$ 4 "@$

"

$

#

"$

"2$

'

"$ > #.

+ )--

0 /

' "#113$ /"#112$

4

4 C '

"#115$

4

" D $

4 "

$ "

$

"

$

(5)

> #? '

"$

+)--

"

$

"

$ '

!

'

")$ < "

$"

$

+

+#

(6)

+

"

$

"$ +

#

"

$

+ #

"

,

$

+ "

$ E

+

,

+#

"(

F

-.

- $ E

"$+

4 " $

"

$

"

$ > "

$

"

$

'

"$+

E

+ "

$ 6

"$+ #

"

-$"

-$,

,

"$

(7)

"$

" $

.

#

#

"

-$"

-$ G

"

$" #$

"

$ !

"

$ +

#

)

"

-$"

-$,

+

,

"3$

E

"7$

E

"$ 4

"'5$ >

"

$

"$+ " $"-$

"'7$ >

"')$

-"'@$ > Æ

#

Æ

(8)

"'=$ >

#

!"$

*"'5$"'7$"'@$"'=$

F >

F "

Æ

$ Æ-.

"'@$

+ "

$ "'@$ "'=$

"

$ H 8

" -$

"$"

"'5$ "'=$

#

"

$,"

$

"1$

"$

" $

"$

" $

+

"$" $" $" $

+ !" $.

! "

#

"7

$$

! % & ' $

"

$ %

(9)

>

",$

"$+)

", $

"$+ "

$" -$ > + "

$#"$+"$"-$

# "$ + "$ "-$ # "$ + # ",

$"

$

> -#

"$

#

"$ #"$

4 £ !#   £ ! £ $!! ! %   ! ! %   ! ! %   ! # £ !% !% !% ! E "$

" $

" $ "$ " $ "

F$ 0 "'5$

"'7$ "$ "$ " #$ < "$ + "$ " $ "$ " $

+!" $

(10)

8

<

"

$

4 8 ' / "#112$

4 # F F.

$+#"

#

"

$,"

$

$+#"

'

<

+

"

$,"

$E

"$

!"$

4G

E

#

"

$,"

$

+

G

"$

!"$

"$

"$ "

"7

$$ I

G

!"$

"$

4

" $

"5$

*

+ , , "+#$

(11)

+ - # )

F

"-$+- 4

"

$

"

"7

$$

+

# #)

#) #)

4 !"$+!"%

%

$ "'=$

!"$+ #

)

%

,%

,%

%

,%

%

,%

%

,%

!

0

" $

4

+ "-#$

' #-

> ) " + #$.

> 5

4

"

$ >

"$

#

"

$

"

$ '

"

$

! 4 #

"

$"

$

$"

$ 4

4 #

(12)

> )? ( #-

> 5? 4 #-

> )

(13)

" $+

&

"&

&

$ H &

,

,&

+.

"

$+ "

$, "

$

4

4 #

4

!

6 "7$

"

$"

$

+ "

$,"

$

,"

$,"

$

(

"

$ +

#

7

#

7

"#-$

H "

$

"

"$$

"

$

#

"$

+- "##$

(14)

" $ # "$ "$ "#)$

4 "##$"#)$

" Æ

$ Æ - !

A B '- ! " $+ , # 7 "#5$ # " $

+ "#7$

  £ £ !% £ ! £ £ !% £ !   "#@$ "

$ E "#5$ "#7$ "#@$

" $" $ # 7   £ £ ! £ % £ £ £ !   "#=$

"#-$ "##$ "#)$

" $" $ # 7

-4 "#=$

4

!(

(15)

(

" $

)

+ "(

(

$

#

)

"

(

$"

(

$

)

#

*

"+

$,*"+

$

*

* < A*

"$

*

"$

B + A*"$*"$

B + +

E #"7

$

>

"(

(

$

#

7

)

,*",$ )

)-")

,

$, "#2$

"#2$ E

# "

½

 ½

½

 ½

!! "

&

'&

%

!

# (

½

 ½

½

 ½

&

%!'&

%

!

#

"

4 4 ) ) : H "#133$

*

+

,

,

"+#$

+",#$

F

"-$+- 4

"

$

"

"7

$$

(16)

+

# #)

#) #5

C

.

"/0$ +

/,0

.

"/0$ +

/,0

.

"/0$ +

/,0

4 !"$+!"%

%

$ "'=$

!"$+

.

"%

%

$ .

"%

%

$

.

"%

%

$ .

"%

%

$

4

"

$

5

&

½

 ½

½

 ½

 

!

! "#3$

A"$" $"$" $

B + !" $ "$

"

"7

$$ + "-#$

4

"#3$

1

+ )

!"$,!" $!" $

"

$"

$"$" $

+

)#-)

"#1$

H*

"1@J )#-)--#$

4

"

$

(17)

¾

Ò

4 #? H * 1

")-$ .

1

F

#= " +)-$ 7- " +#----$

E .

"

$

)-- > #---

1

+

"

$

5

")-$

+")-$

4 1

"#1$ 3)1 3 ' 4 # 1

CK L"#11)$ "#$

Æ

,"#Æ$

"

$

")#$

Æ A-#B. Æ

'

F - M "

$+

1

"'#$ "')$ Æ A-#B

"Æ$ ")#$

"

"Æ$$ +

#

)Æ,)"#Æ$

Æ"

$,)"#Æ$

,

"Æ$

(18)

"2"Æ$ $ "Æ$+ " $ 2"Æ$ + Æ

,7"#Æ$

1

,7Æ"#Æ$"

$

)Æ,)"#Æ$ "))$ 6 "Æ$ Æ

#,Æ"#$

""Æ$$

< "=$

"Æ$"2"Æ$

$. "Æ$ <

"Æ $"Æ $ + Æ Æ

,7"#Æ

$"#Æ

$1

,)A

B Æ

"#Æ

$,Æ

"#Æ

$

,)"#Æ

$)Æ

,)"#Æ

$ CÆ

ÆA-#B 2"Æ$

"))$. Æ Æ Æ " " Æ $$ "2"Æ $ $ "I "Æ$ "Æ$ F " Æ

$ "Æ

$ Æ

Æ Æ

$ 4 C

K L "#11)$ Æ + Æ , " $ Æ + Æ , " $ ' Æ M

"Æ$ Æ A-#B H

" "Æ$ "Æ$$" "Æ$ "Æ$$ 2"Æ$ Æ

!Æ" # ½ ¼ " "Æ## "Æ# Ì ! "Æ# Ì " "Æ## ½ ¼

+3"Æ$

(19)

"Æ$ "2"Æ$

$ " $

"Æ$ 4 3"Æ$ E

Æ Æ # ! Æ

"-#$

Æ

"$ " Æ$

A

"Æ$

"Æ$

B

Æ

"$Æ

4"$

E

Æ

E

Æ

A-#B :

Æ

4

"Æ$ ")#$

Æ A-#B.

Æ

! K

8 ")--)$ "%&$

"

Æ$. Æ

"' *

: "#117$ $ 4 Æ

Æ

Æ

.

Æ

,

"

$

!

!

" $

(20)

4 "

"

$

"

$

% & $

+ -.

+-

4

%&

"0 #11=$ %&"0N O)--)$

4

"$ "$

%&

!

.

" "7$$

"

$+

,

"7$

>

M "

$M "

$

+

"$ "@$

:

"#!

4 E (

D P * *

(21)

' H ' "#113$ (

'!

(H 32##2

0 P P "#1==$' '

H ( @22@3-

0 < / P "#123$ '

K ' (

' =#3=))

0 8 "#11=$0 H8 #)5

#7-

0N 9 O 0 ")--)$ ' '

( 1)21=#

*O H :Q "#117$'

( 9 8 #52#7)

C O K LR K "#11)$ '

!

(' PHDCO

5#7@

D 0 "#121$ 0 ?

' ( #)=

C ' C : 9 O < ' "#115$ '

K ' ( '

#5#-#5#=

< * K "#11=$ I

6

(22)

: 9 H H ' "#133$D

9 4 P

> )=#)33

:; E 8 9 C "#115$ '

(P P 6

I

K C 8 C ")--)$ *

? %& 6

/ I "#12=$PH 0 ? 'M

/ K "#1=2$ I 0 F

' H ( #5)5#57)

// "#112$ '

6

// "#113$ 8

'(!2@@22-

// "#111$ '

* K (

712@-2

/ P 0 < "#123$ P D

! 55@-

9 C "#11#$ '

D 4 #3=#11

References

Related documents

These individual patient-based cost estimates for both nonsevere and severe CL-CIE were then multiplied by the estimated annual incidence of both nonsevere and severe CL-CIEs

During his evidence he told the court that Mr Knight did not inform him that he had been receiving chiropractic treatment since 2005, that he had a history of neck pain for 15

Finally, the articulation of Bahrom’s hierarchies of trade in Tajikistan emphasises this family’s Uzbek background and their historical pedigree as traders in contrast to

more than four additional runs were required, they were needed for the 2 7-3 design, which is intuitive as this design has one more factor than the 2 6-2 design

Measuring Academic Achievement of University of Kentucky Measuring Academic Achievement of University of Kentucky Student Athletes: Helping the Center for Academic and Tutorial

The threshold into the stadium is through a series of layers which delaminate from the geometry of the field to the geometry of the city and creates zones of separation,

The level of accuracy was determined by comparing the spatial analysis results with the fuzzy inference system and the fieldwork analysis to assess the suitability of

There are infinitely many principles of justice (conclusion). 24 “These, Socrates, said Parmenides, are a few, and only a few of the difficulties in which we are involved if