Estymacja wspczynnika autokorelacji w szeregu czasowym AR(1) SLAJDY XXXIX Og\363lnpolska Konferencja Zastosowa\361 Matematyki Zakopane-Ko\234cielisko 7-14 wrze\234nia 2010 r

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Małgorzata Schroeder, IM Uniwersytet w Białymstoku

Ryszard Zieliński, IMPAN Warszawa

ESTYMACJA WSPÓŁCZYNNIKA AUTOKORELACJI

W SZEREGU CZASOWYM AR(1)

XXXIX Ogólnopolska Konferencja Zastosowań Matematyki

Zakopane-Kościelisko 7 - 14 września 2010 r.

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X

t

=

ρX

t−1

+

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

|ρ| < 1

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

i

.i.d.

N(0, σ),

σ = 1

Obserwacja

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X

t

=

ρX

t−1

+

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

|ρ| < 1

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

i

.i.d.

N(0, σ),

σ = 1

Obserwacja

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X

t

=

ρX

t−1

+

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

|ρ| < 1

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

i

.i.d.

N(0, σ),

σ = 1

Obserwacja

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X

t

=

ρX

t−1

+

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

|ρ| < 1

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

i

.i.d.

N(0, σ),

σ = 1

Obserwacja

X

T

, X

T +1

, . . . , X

T +n−1

,

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X

t

=

ρX

t−1

+

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

|ρ| < 1

ε

t

,

t =

. . . , −1, 0, 1, . . . ,

i

.i.d.

N(0, σ),

σ = 1

Obserwacja

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Maximum Likelihood Estimator:

ˆ

ρ

MLE

= arg max

_ρ

L(ρ; X

1

, X

2

, . . . , X

n

)

where

L(ρ; x

1

, x

2

, . . . , x

n

) = log(1 −

ρ

2

) − x

1

2

(1 −

ρ

2

) −

n

X

i =2

(x

i

− ρx

i −1

)

2

.

Observe that for every x

1

, x

2

, . . . , x

n

, function

L(ρ; x

1

, x

2

, . . . , x

n

)

, ρ ∈ (−1, 1) is concave (second derivative is

negative), L(

ρ; x

1

, x

2

, . . . , x

n

) → −∞ as

ρ → ±1 so that

ˆ

ρ

MLE

∈ (−1, 1) is uniquely defined. A disadvantage of the

estimator is that no simple explicit formula for computing ˆ

ρ

MLE

is

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Least Square Estimator:

ˆ

ρ

LSE

= arg min

_ρ

n

X

i =2

(X

i

− ρX

i −1

)

2

=

Pn

i =2

X

i

X

i −1

Pn

i =2

X

i −1

2

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Hurwicz estimator

ρ

HUR

= Med

_X

2

X

1

,

X

3

X

2

, . . . ,

X

n

X

n−1

where Med (

ξ

1

, ξ

2

, . . . , ξ

m

) denotes a median of

ξ

1

, ξ

2

, . . . , ξ

m

.

A nice property of the estimator is that it is median-unbiased

which means that

P

ρ

{ˆ

ρ

HUR

¬ ρ} = P

ρ

{ˆ

ρ

HUR

ρ} =

1

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Haddad estimator:

ρ

HAD

=

Med (X

t−1

X

t

)

Med (X

2 t

)

The estimator has been constructed as a robust counterpart of the

least square estimator.

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M-estimator with Huber loss function: ˇ

13 ρ

MHU

= arg min

_ρ

n−1

X

i =1

L(X

i +1

− ρX

i

)

with

L(x ) =







1

2

x

2

_{if |x | ¬ k}

_,

k|x | −

1 ₂

k

2

_{if |x |}

_{> k}

Following Lehmann we assume k = 3

/2

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Burg’s estimator:

The estimator has been constructed as that minimizing the forward

and backward prediction errors:

ρ

BUR

= arg min

_ρ

n

X

i =2

((X

i

− ρX

i −1

)

2

+ (X

i −1

− ρX

i

)

2

)

Then

ρ

BUR

=

2

Pn

_{i =2}

X

i

X

i −1

Pn

i =2

(X

i

2

+ X

i −1

2

)

It should be noted that the support of the estimator

ρ

BUR

is in the

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MLE

−1.0

−0.5

0.0

0.5

1.0

1.5

0 1000

BUR

−1.0

−0.5

0.0

0.5

1.0

1.5

0 1000

MHU

−1.0

−0.5

0.0

0.5

1.0

1.5

0 1000

LSE

−1.0

−0.5

0.0

0.5

1.0

1.5

0 1000

HAD

−1.0 −0.5

0.0

0.5

1.0

1.5

2.0

0

400 HUR

−1.0 −0.5

0.0

0.5

1.0

1.5

2.0

0

600 1200

Fig. 2. Histograms

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ENTROPY LOSS FUNCTION

-1

-0.5

0

0.5

1

5

10

15

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