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63

* 3.4 Influencefunction

andasymptotic distribution

Theone-dimensional a

case

• h T

X ,...,

1

X i→

n

t

,consistency ∞

Functional

• h T

i F

. Appliedto

data:

T b F D

n

E

.

consistency:

T b F D

n

E

→ h T

i F

Central

•

limittheorem:

T b F D

n

E

≈∼N h

h T i F ,v i /n

= v E

h IF

; X T, i F

2

.

(2)

64

Modelwith

•

density

h f i x,θ

Maximum

− →

likelihood est.

Moregener

•

al:M-estimators

P ψ

i

D X ,

i

b θ E

=0

Functional:

ψ R h i x,θ

h dF i x

=0

MLE:

h ψ i x,θ

∂

= log

∂θ

h h f ii x,θ

sh =:

i x,θ

.

• IFhx;

i F

1

= ψ

c

hx,θ i

= c

∂

R ψ

∂θ

hx,θ i h f i x,θ

= dx R −

h ψ i x,θ

sh i x,θ

h f i x,θ

dx

(3)

65 3.4 Morethan b

onepar ameter

andpossib lymore

thanone

X

• h T

X ,...,

1

X i→

n

t

,consistency ∞

Functional

• h T

i F

. Appliedto

data:

T b F D

n

E

.

consistency:

T b F D

n

E

→ h T

i F

Central

•

limittheorem:

T b F D

n

E

≈∼N h

h T i F , /n V

i

= V E

h IF

; X T, i F

h IF

; X T, i F

T

.

(4)

66

Modelwith

•

density

h f i x,θ

Maximum

− →

likelihood est.

Moregener

•

al:M-estimators

P ψ

i

D X ,

i

b θ E

=0

Functional:

ψ R h i x,θ

h dF i x

=0

MLE:

h ψ i x,θ

∂

= log

∂θ

h h f ii x,θ

s =:

h i x,θ

.

• IFhx;

i F

=

−1

M hx,θ ψ

i

= M

∂

R ψ

∂θ

hx,θ i h f i x,θ

= dx R −

h ψ i x,θ

sh i x,θ

f

T

h i x,θ

dx

(5)

67

* 3.5 Robust

Estimators

MultidimensionalLocation. a

f x,µ

f =

0

x

− µ

f h

0

i z c =

· exp

D P −

(

j (

z

j

)

)

/

2

2 E

,

/c 1 π =(2

m/2

)

M-estimators b

P ψ

i j

x

−

i

µ

,

=0

Natural choice:

ψ

− x µ

w = h i u

− (x

,

µ) u

=

i

k x

−

i 2

µk

b µ

= X

w

i

h u i

i

(x

−

i

µ) .X w

i

h u i

i

= M Z w

k

− x µk

x (

−

i

)( µ x

−

i

) µ f

T 0

x

− µ dx

= Z h w

i u

(

uf

) u

h i u

· du

= I

1

I b

,

(6)

68

Sensitivität,Huber-Funktion. c

∗

γ h T, i F

x

=sup hk h IF

x T, ; ik F

i

Censorthe scores

− x

!

µ

− → h w

i u h =min

,c/u 1 i

M-Schätzungenfür d

|

Σ

.

|

b Σ

= X

w

i

|

Σ h u i

i

(x

−

i

µ)(x

−

i T

µ)

.X w

i

|

Σ h u i

i

u

=(

i

x

−

i

) µ b

T

|

Σ

− 1

x (

−

i

) µ

.

Breakdown e

Point.

Referenzen

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