1. No category

Asymptotic distributions of sample mean and ACVF
Felix Dietrich and Gundelinde Wiegel
University of Trento
29th of October 2013
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
1 / 24
Overview
1
Repetition
2
¯
Distribution of X
3
Distribution of γˆ
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
2 / 24
What we already know
Let {Xt }t∈Z a stationary process with E[Xt ] = µ and ACVF γ(·).
Definition (Estimator of µ)
n
X
¯n := 1
X
Xi
n
i=1
.
Definition (Estimator of ACVF)
n−h
γˆ (h) =
1X
¯n )(Xt+h − X
¯n )
(Xt − X
n
for
0≤h ≤n−1
i=1
Definition (Estimator of ACF)
ρˆ(h) =
Felix Dietrich and Gundelinde Wiegel (UTN)
γˆ (h)
γˆ (0)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
3 / 24
Mean Estimator
¯n =
For X
1
n
n
P
Xi we have already shown
i=1
¯n ] = µ
E[X
and
¯n ] =
Var [X
X
|h|
1−
γ(h) .
n
h<|n|
Hence
¯n ] → 0
Var [X
∞
X
¯n ] →
nVar [X
γ(h)
h=−∞
if
if
γ(h) → 0,
∞
X
|γ(h)| < ∞ .
h=−∞
¯n distributed?
But how is X
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
4 / 24
¯n
Distribution of X
Theorem
∞
P
Let {Xt } a stationary process, defined as Xt = µ +
ψj Zt−j with
j=−∞
P
¯n is
{Zt } ∼ IID(0, σ 2 ) for each t ∈ Z and ∞
6 0 < ∞. Then X
j=−∞ |ψj | =
asymptotically normal i.e.
¯n ∼ AN µ, ν
X
n
P∞
P
2
where ν = h=−∞ γ(h) = σ 2 ( ∞
j=−∞ ψj ) .
Remark: {Xn } is asymptotically normal distributed with µn and σn > 0 for
n large enough if for n → ∞
Xn ∼ AN(µn , σn2 ).
Felix Dietrich and Gundelinde Wiegel (UTN)
Xn −µn i.d.
−→
σn
Z where Z ∼ N(0, 1) . We write:
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
5 / 24
Tools
Proposition
Let {Xn }n∈N and {Ynj }j∈N,n∈N be random vectors such that
(i) Ynj −→ Yj for each j ∈ N
n→∞
(ii) Yj −→ Y
j→∞
(iii) lim lim supP(|Xn − Ynj | > ε) = 0 for every ε > 0. Then
j→∞ n→∞
i.d.
Xn −→ Y .
n→∞
Lemma (Chebyshevs inequality)
Let X be a random variable such that E [X ] = µ and Var[X ] = σ 2 < ∞
then the following inequality holds for any real number k > 0
P[|X − µ| ≥ k] ≤
Felix Dietrich and Gundelinde Wiegel (UTN)
σ2
k2
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
6 / 24
Tools II
Definition (m-Dependence)
A stationary process {Xt } is called m-dependent with 0 < m ∈ N if for
each t ∈ Z {Xj , j ≤ t} and {Xj , j ≥ t + m + 1} are independent.
Theorem (Central Limit Th. for Stationary m-Dependent Sequences)
Let {Xt } be a stationary and m-dependent
sequence of r.v. with mean
P
zero and ACVF γ(·). If νm = m
γ(j)
=
6
0, then
j=−m
¯n ] = νm
lim nVar [X
¯n ∼ AN 0, νm .
X
n
(1)
n→∞
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
(2)
29th of October 2013
7 / 24
Proof of the Theorem
Xtm := µ +
m
X
ψj Zt−j
j=−m
and
n
Ynm
X
¯nm := 1
Xtm .
:= X
n
t=1
{Ynm }n∈N is an 2m-dependent sequence.
With the central limit theorem for any m ∈ N:
⇒ Ynm −→ Xm as n → ∞ (in distribution)
P
where Xm ∼ N µ, n1 m
j=−m γ(j) .
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
8 / 24
Proof of the Theorem II
Hence we get
√
2
n(Ynm − µ) −→ Ym ∼ N 0, σ (
m
X
j=−m
2
ψj ) =
m
X
γ(j)
j=−m
P
P∞
2
2
2
Since σ 2 ( m
j=−m ψj ) → σ ( j=−∞ ψj ) > 0, we can show that
Ym −→ Y as m → ∞ (in distribuition)
P
2
where Y ∼ N 0, σ 2 ( ∞
j=−∞ ψj ) .Thus we showed property (i) and (ii) of
the first proposition. For (iii) we consider
Var
√
n X
X
X
¯n − Ynm ) = nVar 1
n(X
ψj Zt−j → σ 2 (
ψj )2 .
n
t=1 |j|>m
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
|j|>m
29th of October 2013
9 / 24
Proof of the Theorem III
Hence
n X
X
1 X
ψj Zt−j = lim σ 2 (
ψj )2
lim lim sup Var
m→∞
m→∞ n∈N
n
i=1 |j|>m
|j|>m
=0
Chebyshevs inequalitiy ⇒ (iii)
√
¯n − µ) −→ Y ∼ N 0, σ 2 (
n(X
∞
X
ψj )2
j=−∞
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
10 / 24
Distribution of γˆ
Now to the distribution of
n−h
γˆ (h) =
1X
¯n )(Xt+h − X
¯n )
(Xt − X
n
for
0≤h ≤n−1 .
i=1
Theorem (Distribution of γˆ )
P
Let {Xt } be a two-sided MA-process with Xt = ∞
j=−∞ ψj Zt−j ,
P∞
2
{Zt } ∼ IID(0, σ ), where j=−∞ |ψj | < ∞ and E[Zt4 ] = ησ 4 < ∞. Then
for any h ∈ N





γˆ (0)
γ(0)
 .. 
 .  V 
 .  ∼ AN  ..  , 
n
γˆ (h)
γ(h)
with
V = Cov (γ(p),P
γ(q))p,q=0,...,h =
(η − 3)γ(p)γ(q) + ∞
k=−∞ (γ(k)γ(k − p + q) + γ(k + q)γ(k − p)) p,q .
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
11 / 24
Tools
First of all we show some properties of
n
γ ∗ (h) =
1X
Xt Xt+h
n
with
h∈N .
t=1
Lemma (Lemma 1)
P
P∞
{Xt } MA-process Xt = ∞
j=−∞ ψj Zt−j , where
j=−∞ |ψj | < ∞ and
4
4
E[Zt ] = ησ < ∞. For p, q ∈ N we obtain
lim nCov [γ ∗ (p), γ ∗ (q)]
n→∞
= (η − 3)γ(p)γ(q) +
∞
X
(γ(k)γ(k − p + q) + γ(k + q)γ(k − p))
k=−∞
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
12 / 24
Proof of Lemma 1
As {Zt }t≥0 is i.i.d. with mean 0 and variance σ 2 plus the property
E [Zt4 ] = ησ 4 < ∞, we obtain

4

ησ if t = s = u = v
E [Zt Zs Zu Zv ] = σ 4
if s = t 6= u = v


0
else.
E [Xt Xt+p Xt+p+h Xt+p+h+q ]
XXXX
ψi ψj+p ψk+p+h ψl+h+p+q E [Zt−i Zt−j Zt−k Zt−l ]
=
i
j
= (η − 3)σ
k
l
4
X
ψi ψi+p ψi+p+h ψi+p+h+q + γ(p)γ(q)+
i
γ(h + p)γ(h + q) + γ(h + p + q)γ(h)
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
13 / 24
Proof of Lemma 1 II
Now back to γ ∗ . We know that
E [γ ∗ (h)] = γ(h)
Cov[γ ∗ (p), γ ∗ (q)] = E [γ ∗ (p)γ ∗ (q)] − E [γ ∗ (p)]E [γ ∗ (q)]
X
= n−1
(1 − n−1 |k|)Tk ,
k<|n|
where
Tk : = γ(k)γ(k − p + q) + γ(k + q)γ(k − p)
X
+ (η − 3)σ 4
ψi ψi+p ψi+k ψi+k+q
i
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
14 / 24
Proof of Lemma 1 III
Absolute summability of {ψj }j∈Z
⇒ {Tk }k∈Z is absolutely summable.
∗
∗
lim n Cov[γ (p), γ (q)] =
n→∞
∞
X
Tk
k=−∞
= (η − 3)γ(p)γ(q)
∞
X
+
γ(k)γ(k − p + q) + γ(k + q)γ(k − p) .
k=−∞
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
15 / 24
With this Lemma we show the convergence of the distribution of γ ∗ for a
truncated MA.
Lemma (Lemma 2)
{Xt } MA-process
Xt =
m
X
ψj Zt−j ,
j=−m
where
P∞
j=−∞ |ψj |
< ∞ and E[Zt4 ] = ησ 4 < ∞. Then for any h ∈ N




γ(0)
γ ∗ (0)
 .. 
 .  V 
 .  ∼ AN  ..  , 
n
γ ∗ (h)
γ(h)

with
V = Cov (γ(p),P
γ(q))p,q=0,...,h =
(η − 3)γ(p)γ(q) + ∞
k=−∞ (γ(k)γ(k − p + q) + γ(k + q)γ(k − p)) p,q .
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
16 / 24
Proof of Lemma 2
Yt0 := (Xt Xt , Xt Xt+1 , . . . , Xt Xt+h ) .
{Yt } is again stationary and 2m + h-dependent and
n
1X
Yt =
n
t=1
n
n
1X
1X
Xt+0 Xt+0 , . . . ,
Xt Xt+h
n
n
t=1
!0
= (γ ∗ (0), . . . , γ ∗ (h))0 .
t=1
Central Limit Theorem: Let λ ∈ Rh+1 .
n
1X 0
λ Yt ] = lim nVar [λ0 (γ ∗ (0), . . . , γ ∗ (h))0 ]
lim nVar [
n→∞
n→∞
n
t=1
= lim nVar [λ0 γ ∗ (0) + . . . + λh γ ∗ (h)] = lim n
n→∞
Felix Dietrich and Gundelinde Wiegel (UTN)
n→∞
¯ and γ
Asymptotic distributions of X
ˆ
h
X
λi λj Cov [γ ∗ (i), γ ∗ (j)]
j,i=1
29th of October 2013
17 / 24
Proof of Lemma 2 II
=
Lemma 1
λ0 V λ > 0
with V =
P
(η − 3)γ(p)γ(q) + ∞
k=−∞ (γ(k)γ(k − p + q) + γ(k + q)γ(k − p)) p,q
Central Limit Theorem ⇒
 


γ(0)
n
1X 0
 
 1

λ Yt ∼ AN λ0  ...  , λ0 V λ
n
n
t=1
γ(h)
for every λ ∈ Rh+1 .
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
18 / 24
Lemma 3
Now back to our original two-sided MA.
Lemma (Lemma 3)
{Xt } MA-process
Xt =
∞
X
ψj Zt−j ,
j=−∞
where
P∞
j=−∞ |ψj |
< ∞ and E[Zt4 ] = ησ 4 < ∞. Then for any h ∈ N




γ(0)
γ ∗ (0)
 .. 
 .  V 
 .  ∼ AN  ..  , 
n
γ ∗ (h)
γ(h)

with
V = Cov (γ(p),P
γ(q))p,q=0,...,h =
(η − 3)γ(p)γ(q) + ∞
k=−∞ (γ(k)γ(k − p + q) + γ(k + q)γ(k − p)) p,q .
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
19 / 24
Proof of Lemma 3
The idea is to apply Lemma 2 on the truncated sequence
Xt,m :=
m
X
n
ψj Zt−j
and therefore
∗
γm
(p) :=
1X
Xt,m Xt+p,m
n
t=1
j=−m
Lemma 2 ⇒

∗ (0) − γ (0)
γm
m
√ 
 i.d.
..
n
 −→ Ym ∼ N(0, Vm ) .
.
∗
γm (h) − γm (h)

Besides Vm → V and Ym → Y ∼ N(0, V ). Also we can show, that
√ ∗
lim lim supP[ n|γm
(p) − γm (p) − (γ ∗ (p) − γ(p))| > ε] = 0
m→∞ n→∞
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
20 / 24
Proof of the Theorem
Left
γˆ
.
Hence
√
instead of
n(γ ∗ (p) − γˆ (p)) = op (1)
γ∗
for
n→∞




γ(0)
γˆ (0)
 .. 
 .  V 
 .  ∼ AN  ..  , 
n
γˆ (h)
γ(h)

Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
21 / 24
Some interesting corollaries
Corollary (Distribution of ρˆ)
P
Let {Xt } be the stationary process given by Xt = ∞
j=−∞ ψj Zt−j − µ,
P∞
4
4
where j=−∞ |ψj | < ∞ and E[Zt ] = ησ < ∞. Then for each h ∈ N
W
ρˆ(h) ∼ AN ρ(h),
n
where ρˆ(h)0 = [ˆ
ρ(1), ..., ρˆ(h)]0 and ρ(h)0 = [ρ(1), ..., ρ(h)]0 and W the
Covariance Matrix, whose elements are given by Bartlett’s formula:
wij =
∞
X
[ρ(k + i)ρ(k + j) + ρ(k − i)ρ(k + j) + 2ρ(i)ρ(j)ρ2 (k)
k=−∞
− 2ρ(i)ρ(k)ρ(k + j) − 2ρ(j)ρ(k)ρ(k + i)]
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
22 / 24
Some interesting corollaries
Corollary (Distribution of ρˆ II)
P
Let {Xt } be the stationary process given by Xt = ∞
j=−∞ ψj Zt−j − µ,
P∞
P∞
2
where j=−∞ |ψj | < ∞ and j=−∞ ψj |j| < ∞. Then for each h ∈ N
W
ρˆ(h) ∼ AN ρ(h),
n
where ρˆ(h)0 = [ˆ
ρ(1), ..., ρˆ(h)]0 and ρ(h)0 = [ρ(1), ..., ρ(h)]0 and W the
Covariance Matrix, whose elements are given by Bartlett’s formula:
wij =
∞
X
[ρ(k + i)ρ(k + j) + ρ(k − i)ρ(k + j) + 2ρ(i)ρ(j)ρ2 (k)
k=−∞
− 2ρ(i)ρ(k)ρ(k + j) − 2ρ(j)ρ(k)ρ(k + i)]
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
23 / 24
The End
Felix Dietrich and Gundelinde Wiegel (UTN)
¯ and γ
Asymptotic distributions of X
ˆ
29th of October 2013
24 / 24