Estimation in Spectral Analysis - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 90

Let us study the estimation of the parameter vector a = a 1 a 2 … a p ] T . As shown in
Chapter 6, the natural estimator of a may be written as:

−
= aRr

1

ˆ ˆ

[3.26]

p

() ( ) ( )

ˆ ˆ ˆ

⎛ c 0 c − 1 c − + ⎞ p 1

xx xx xx

⎜ ⎟

() () ( )

ˆ ˆ ˆ

c 1 c 0 c −+ p 2

⎜ ⎟

xx xx xx

⎜ ⎟

R = ⎜

p

⎟

⎜ ⎟

⎜ ⎟

( ) ( ) ( )

ˆ ˆ ˆ

cp − 1 cp − 2 c 0

⎝ ⎠

xx xx xx

()
()

⎛ c

c

ˆ 1 ⎞

xx
xx

⎜ ⎟

ˆ 2

⎜ ⎟

ˆ

r = ⎜

⎟

⎜ ⎟

⎜ ⎟

()

cp ˆ

⎝ ⎠

xx

Thus, from results 3.8 and 3.2, it directly follows that:

T

⎛ ⎞

as

∂∂ aa

c

( )

N ˆ aa ∼ ,

⎜ 0 Σ ⎟

⎜ ⎟

T

∂

∂ c

⎝ ⎠

The matrix of derivatives ∂ a / ∂ c T can easily be calculated by using the Yule-
Walker equations [POR 94]. The previous approach directly used result 3.8, which
is very general. Nevertheless, the approach may be followed more specifically for
each case. For the AR estimation, we can write:

− 1

( )

( )

N aa ˆ −=− N R r ˆ + Ra

p p

{ }

− 1

( )

)

⎡

(

⎤

=− R Nrr ˆ − + R − R a

p p p

⎣ ⎦

[3.27]

{ } ()

( )

− 1

)

⎡

(

⎤

=− R Nrr ˆ − + R − R a + o 1

p p p p

⎣ ⎦

− 1

()

=− R No ε + 1

p p

The vector N ε is asymptotically distributed according to a normal law of 0
mean and covariance matrix S whose element ( )

k is:

pp

= ∑∑

( )

S k , a a σ −−

ijki , j

i == 00 j