copula based spectral analysis - sfb649.wiwi.hu-berlin.de
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Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Copula based spectral analysis
Holger Dette, Ruhr-Universitat BochumMarc Hallin, Universite Libre de Bruxelles
Tobias Kley, Ruhr-Universitat BochumStefan Skowronek, Ruhr-Universitat Bochum
Stanislav Volgushev, Ruhr-Universitat Bochum
July , 2015
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Outline
1 Serial DependenceThe Traditional ApproachA Quantile-based Approach
2 Spectral Analysis of Time SeriesA Least Squares Interpretation of the PeriodogramA Quantile-based Approach
3 Asymptotic Properties
4 Data Example
5 Conclusions
6 ExtensionsDistributional properties of smoothed periodogramsLocal stationarity
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
The Traditional Approach
Traditional Time Series Models
Most traditional time series models are of the conditionallocation/scale type:
Xt = ψ(Xt−1,Xt−2, . . .) + σ(Xt−1,Xt−2, . . .)εt
where
the innovations (εt)t∈Z are white noise,
εt is independent of Xt−1,Xt−2, . . .,
Holger Dette Copula based spectral analysis 1 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
The Traditional Approach
Implications
The distribution of Xt conditional on Xt−1,Xt−2, . . . is thedistribution of εt rescaled and shifted (by conditionalparameters). Therefore
Xt is a linear function of εt
All standardized conditional distributions of Xt |Xt−1,Xt−2, . . .coincide with the distribution of εt ⇒all conditional quantiles (hence values at risk) follow fromthose of ε by a linear transformation
Interpretation of ψ and σ depends on the identificationconstraints on ε.
Interpretation as conditional mean and variance correspondsto (Gaussian) L2-legacy, which is widely used in time seriesanalysis.
Holger Dette Copula based spectral analysis 2 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
The Traditional Approach
Example: spectral density for 3 time series
Yt = Xt/(Var(Xt))1/2,where
Xt is i.i.d.Xt is ARCH(1)Xt = 0.1Φ−1(Ut) + 1.9(Ut − 0.5)Xt−1 is QAR (1)(Ut i.i.d. uniform, Φ cdf of the standard normal distribution
0.0 0.1 0.2 0.3 0.4 0.5
0.10
0.14
0.18
0.22
i.i.d.
0.0 0.1 0.2 0.3 0.4 0.5
0.10
0.14
0.18
0.22
QAR(1)
0.0 0.1 0.2 0.3 0.4 0.5
0.10
0.14
0.18
0.22
ARCH(1)
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Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
The Traditional Approach
Preceding analysis based on L2 methods and auto-covariances.
works well for Gaussian time series, not so for heavy tails
is good at capturing linear dynamics
is concerned with mean/variance (specific location-scale)effects
In this talk: robustness, richer view of dynamics, tails?
replace covariances joint distributions/copulas.
replace L2-loss by L1-based loss functions, quantile regression.
ranks.
Holger Dette Copula based spectral analysis 4 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
The Traditional Approach
The final result:
A family of copula spectral densities : ARCH(1)
0.0 0.1 0.2 0.3 0.4 0.5
0.01
00.
020
0.03
0
0.0 0.1 0.2 0.3 0.4 0.5
0.00
40.
008
0.01
2
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
15−0
.005
0.00
5
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
03−0
.001
0.00
10.
003
0.0 0.1 0.2 0.3 0.4 0.5
0.03
00.
040
0.05
0
0.0 0.1 0.2 0.3 0.4 0.5
0.00
40.
008
0.01
2
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
020.
000
0.00
2
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
030.
000
0.00
20.
004
0.0 0.1 0.2 0.3 0.4 0.50.
010
0.02
00.
030
τ 1=0
.1τ 1
=0.5
τ 1=0
.9
τ2=0.1 τ2=0.5 τ2=0.9
ω 2π
Holger Dette Copula based spectral analysis 5 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
The Traditional Approach
The final result:
A family of copula spectral densities : QAR
0.0 0.1 0.2 0.3 0.4 0.5
0.00
80.
012
0.01
6
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
050.
005
0.01
5
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
020.
000
0.00
20.
004
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
08−0
.004
0.00
0
0.0 0.1 0.2 0.3 0.4 0.5
0.03
00.
040
0.05
0
0.0 0.1 0.2 0.3 0.4 0.5
0.00
00.
010
0.02
0
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
06−0
.003
0.00
0
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
10−0
.006
−0.0
020.0 0.1 0.2 0.3 0.4 0.5
0.00
50.
015
0.02
5
τ 1=0
.1τ 1
=0.5
τ 1=0
.9
τ2=0.1 τ2=0.5 τ2=0.9
ω 2π
Holger Dette Copula based spectral analysis 6 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
The Traditional Approach
Being less restrictive
Modeling the entire distribution of the strictly stationary process(Xt)t∈Z?
Allow the conditional distribution of Xt |Xt−1,Xt−2, . . . to bearbitrary, but
make structural assumptions.
Example: (Xt)t∈Z a stationary, Markovian process of order one.
Then the distribution of (Xt)t∈Z is fully characterized by either
the joint distribution F1 of (Xt−1,Xt), or
the marginal distribution F of Xt andthe pair copula C1 of lag one, i. e. the joint distribution of(F (Xt−1),F (Xt)).
Holger Dette Copula based spectral analysis 7 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
The Traditional Approach
Being less restrictive
Modeling the entire distribution of the strictly stationary process(Xt)t∈Z?
Allow the conditional distribution of Xt |Xt−1,Xt−2, . . . to bearbitrary, but
make structural assumptions.
Example: (Xt)t∈Z a stationary, Markovian process of order one.
Then the distribution of (Xt)t∈Z is fully characterized by either
the joint distribution F1 of (Xt−1,Xt), or
the marginal distribution F of Xt andthe pair copula C1 of lag one, i. e. the joint distribution of(F (Xt−1),F (Xt)).
Holger Dette Copula based spectral analysis 7 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Pair Copulae as a Measure for serial dependence
Notation:F denotes the distribution function of Xt
qτ = F−1(τ) is the τ -quantile of FFk denotes the joint distribution function of (Xt−k ,Xt)Ck denotes the copula of Fk , i.e the distribution of
(F (Xt−k),F (Xt))
(pair copula of lag k)
Note:In the previous example all the pair copulae Ck of lag k aredetermined by C1 (by Markov assumption).For other processes this is not necessarily the case; the paircopulae (Ck) vary freely.The copulae (Ck) are well suited to quantify serial dependence.Offer much richer information than the autocorrelations only.
Holger Dette Copula based spectral analysis 8 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Pair Copulae as a Measure for serial dependence
Notation:F denotes the distribution function of Xt
qτ = F−1(τ) is the τ -quantile of FFk denotes the joint distribution function of (Xt−k ,Xt)Ck denotes the copula of Fk , i.e the distribution of
(F (Xt−k),F (Xt))
(pair copula of lag k)
Note:In the previous example all the pair copulae Ck of lag k aredetermined by C1 (by Markov assumption).For other processes this is not necessarily the case; the paircopulae (Ck) vary freely.The copulae (Ck) are well suited to quantify serial dependence.Offer much richer information than the autocorrelations only.
Holger Dette Copula based spectral analysis 8 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Clipped Processes(1Xt ≤ 0.5)t∈N
t
Xt
-1
1
2
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Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Clipped Processes(1Xt ≤ 0.5)t∈N and (1Xt ≤ −1)t∈N
t
Xt
-1
1
2
t
Xt
-1
1
2
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Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Copula and Laplace Cross-covariance Kernels
Let (Xt)t∈Z be a real-valued, strictly stationary process (where Xt
has a positive density).
Definition
Laplace cross-covariance kernel of lag k ∈ Z
γk(q1, q2) = Cov(1Xt ≤ q1,1Xt+k ≤ q2)= Fk(q1, q2)− F (q1)F (q2)
= Ck(F (q1),F (q2))− F (q1)F (q2), q1, q2 ∈ R
Copula cross-covariance kernel of lag k ∈ Z
γk(τ1, τ2) = Cov(1Xt ≤ F−1(τ1),1Xt+k ≤ F−1(τ2))= γk(qτ1 , qτ2)
= Ck(τ1, τ2)− τ1τ2, τ1, τ2 ∈ (0, 1)
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Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Covariance and Cross-covariance
We use terminology of classical time series
But only covariances of indicators are considered
γk and γk always exist (no assumptions about moments)
Note:γk(q1, q2)| q1, q2 ∈ R
entirely characterizes the joint distribution of (Xt ,Xt+k)
γk(τ1, τ2)| τ1, τ2 ∈ (0, 1)
and F entirely characterize the joint distributionof (Xt ,Xt+k)
Holger Dette Copula based spectral analysis 12 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Example: AR(1) Process with independent Innovations
AR(1) process
Xt = ϑXt−1 + εt , t ∈ Z.
ϑ = −0.3
independent innovations with
(i) t1-distribution
(ii) standard normal distribution
Holger Dette Copula based spectral analysis 13 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Sample path: AR(1) Process, ϑ = −0.3; t1-distributed(left) and normal distributed innovations (right)
0 50 100 150 200
−80
−60
−40
−20
020
4060
t
Xt=
−0.
3Xt−
1+
ε t
0 50 100 150 200−
2−
10
12
34
t
Xt=
−0.
3Xt−
1+
ε t
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Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
γ1(τ1, τ2) = C1(τ1, τ2)− τ1τ2 = F1(qτ1, qτ2
)− τ1τ2
Holger Dette Copula based spectral analysis 15 / 58
γ1(τ
1,τ
2)
τ1
τ2
τ1
τ2
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
γ2(τ1, τ2) = C2(τ1, τ2)− τ1τ2 = F2(qτ1, qτ2
)− τ1τ2
Holger Dette Copula based spectral analysis 15 / 58
γ2(τ
1,τ
2)
τ1
τ2
τ1
τ2
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Properties
Laplace and Copula cross-covariance kernel
If EX 2t <∞, then (γk) and F determine the autocovariance
function of (Xt)t∈Z.
“Symmetry”
γk(q1, q2) = γ−k(q2, q1)
γk(τ1, τ2) = γ−k(τ2, τ1)
Invariance of γk under continuous monotone transformation
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Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
The Laplace Spectral Density Kernel
Assume that the Laplace cross-covariance kernels γk , k ∈ Z satisfy
∞∑k=−∞
|γk(q1, q2)| <∞ for all q1, q2 ∈ R
Laplace spectral density kernel
fq1,q2(ω) :=1
2π
∞∑k=−∞
γk(q1, q2)e−ikω
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Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Properties
fq1,q2(−ω) = fq2,q1(ω) = fq1,q2(ω)
If =fq1,q2 ≡ 0, then γk(q1, q2) = γ−k(q1, q2), ∀k
“Time reversibility”:
If =fq1,q2 ≡ 0, ∀q1, q2, then (Xt ,Xt+k)d= (Xt ,Xt−k).
Copula spectral density kernel
fqτ1 ,qτ2(ω) :=
1
2π
∞∑k=−∞
γk(τ1, τ2)e−ikω
:=1
2π
∞∑k=−∞
Ck(τ1, τ2)− τ1τ2e−ikω
Holger Dette Copula based spectral analysis 18 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Properties
fq1,q2(−ω) = fq2,q1(ω) = fq1,q2(ω)
If =fq1,q2 ≡ 0, then γk(q1, q2) = γ−k(q1, q2), ∀k
“Time reversibility”:
If =fq1,q2 ≡ 0, ∀q1, q2, then (Xt ,Xt+k)d= (Xt ,Xt−k).
Copula spectral density kernel
fqτ1 ,qτ2(ω) :=
1
2π
∞∑k=−∞
γk(τ1, τ2)e−ikω
:=1
2π
∞∑k=−∞
Ck(τ1, τ2)− τ1τ2e−ikω
Holger Dette Copula based spectral analysis 18 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Properties
fq1,q2(−ω) = fq2,q1(ω) = fq1,q2(ω)
If =fq1,q2 ≡ 0, then γk(q1, q2) = γ−k(q1, q2), ∀k
“Time reversibility”:
If =fq1,q2 ≡ 0, ∀q1, q2, then (Xt ,Xt+k)d= (Xt ,Xt−k).
Copula spectral density kernel
fqτ1 ,qτ2(ω) :=
1
2π
∞∑k=−∞
γk(τ1, τ2)e−ikω
:=1
2π
∞∑k=−∞
Ck(τ1, τ2)− τ1τ2e−ikω
Holger Dette Copula based spectral analysis 18 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Least Squares Interpretation of the Periodogram
Traditional Spectral Analysis
For the analysis of serial dependencies the autocovariance functionand its spectral representation are often considered.
Autocovariances measure linear serial dependencies.
The same is true for the corresponding spectral density.
Estimation of the spectral density is (often) based on theperiodogram
In(ωj) :=1
n
∣∣∣∣∣n∑
t=1
Xteitωj
∣∣∣∣∣2
where ωj := 2πjn ∈ (−π, π], j ∈ Z are the Fourier
frequencies.
Holger Dette Copula based spectral analysis 19 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Least Squares Interpretation of the Periodogram
A Least Squares Interpretation of the Periodogram
Note (representation as a quadratic form, i =√−1):
In(ωj) =n
4bn(ωj)
′(
1 −ii 1
)bn(ωj)
where bn(ωj) = (b1n(ωj), b2n(ωj))′,
(an(ωj), b1n(ωj), b2n(ωj)) = arg minb∈R3
n∑t=1
(Xt − ct(ωj)
′b)2
and ct(ωj) = (1, cos(tωj), sin(tωj))′.
Holger Dette Copula based spectral analysis 20 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
The Laplace Periodogram Kernel
Use weighted L1 instead of L2 projections:
(aτn(ω), bτ1n(ω), bτ2n(ω)) = arg minb∈R3
n∑t=1
ρτ(Xt − ct(ω)′b
)where ρτ (u) := u(τ − 1(−∞,0](u)) is the check function:
Use an inner product for two quantiles τ1, τ2
Holger Dette Copula based spectral analysis 21 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
The Laplace Periodogram Kernel
Main idea: Use weighted L1 projections for two differentvalues τ1, τ2
This gives two vectors:
bτn(ω) = (bτ1n(ω), bτ2n(ω))′ , τ = τ1, τ2.
Definition
The Laplace periodogram kernel is defined as
Lτ1,τ2n (ω) =
n
4bτ1n (ω)′
(1 −ii 1
)bτ2n (ω), τ1, τ2 ∈ (0, 1)
Holger Dette Copula based spectral analysis 22 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
The Laplace Periodogram Kernel
Main idea: Use weighted L1 projections for two differentvalues τ1, τ2
This gives two vectors:
bτn(ω) = (bτ1n(ω), bτ2n(ω))′ , τ = τ1, τ2.
Definition
The Laplace periodogram kernel is defined as
Lτ1,τ2n (ω) =
n
4bτ1n (ω)′
(1 −ii 1
)bτ2n (ω), τ1, τ2 ∈ (0, 1)
Holger Dette Copula based spectral analysis 22 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
A Quantile-based Approach
Remarks and questions:
Computation of Lτ1,τ2n (ω): Simplex algorithm
A special case (τ1 = τ2 = 12 ) of this L1 approach to spectral
analysis was suggested previously by Li (JASA 2008).
What object is this statistic “estimating”?
Consistency?
Holger Dette Copula based spectral analysis 23 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Technical assumptions
Assume that (Xt)t=1,...,n, n ∈ N is
strictly stationary
mn-decomposable (which includes linear processes) orβ-mixing or dependence concept of Wu and Shao (2004)
Let F be the absolutely continuous cdf of Xt ,
admitting the non-vanishing density f and
qτ := F−1(τ) is the quantile of F .
Assume that the Laplace cross-covariance kernels (γk) areabsolutely summable, such that the Laplace spectral densitykernel fq1,q2(ω) : q1, q2 ∈ R exists.
Holger Dette Copula based spectral analysis 24 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Theorem 1 (part 1/2)
For any ω1, . . . , ων ∈ (0, π) and any τ1, τ2 ∈ (0, 1) we have
(Lτ1,τ2n (ω1), . . . , Lτ1,τ2
n (ων)) (Lτ1,τ2(ω1), . . . , Lτ1,τ2(ων))
where Lτ1,τ2(ωj) are independent random variables such that
Lτ1,τ2(ωj) ∼1
2f τ1,τ2(ωj)χ
22 if τ1 = τ2,
and
f τ1,τ2(ω) := 2πfqτ1 ,qτ2
(ω)
f (qτ1)f (qτ2),
Holger Dette Copula based spectral analysis 25 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Theorem 1 (part 2/2)
and
Lτ1,τ2(ωj)d=
1
4
(Z11
Z12
)′(1 −ii 1
)(Z21
Z22
)if τ1 6= τ2
where (Z11,Z12,Z21,Z22) ∼ N4(0,Σ4(ωj)), with
Σ4(ωj) =1
2
f τ1,τ1(ωj) 0 <f τ1,τ2(ωj) −=f τ1,τ2(ωj)
0 f τ1,τ1(ωj) =f τ1,τ2(ωj) <f τ1,τ2(ωj)<f τ1,τ2(ωj) =f τ1,τ2(ωj) f τ2,τ2(ωj) 0−=f τ1,τ2(ωj) <f τ1,τ2(ωj) 0 f τ2,τ2(ωj)
..
Holger Dette Copula based spectral analysis 26 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
The bias?
Corollary
For all τ1, τ2 ∈ (0, 1) and ω ∈ (0, π) we have
limn→∞
ELτ1,τ2n (ω) = f τ1,τ2(ω) = 2π
fqτ1 ,qτ2(ω)
f (qτ1)f (qτ2)
Note: The Laplace periodogram Lτ1,τ2(ωj) is not an asymptot-ically unbiased estimator of the copula spectral density fqτ1 ,qτ2
(ω),but of the quantity
2πfqτ1 ,qτ2
(ω)
f (qτ1)f (qτ2)
Holger Dette Copula based spectral analysis 27 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Finite Sample Properties
data from, AR(1) time series (ϑ = −0.3, n = 500)
t1- and N (0, 1) distributed innovations
Get an idea on the remaining bias of the estimate
ELτ1,τ2n (ω)
Corollary toTheorem 1−−−−−−−→
n→∞ELτ1,τ2(ω) = 2π
fqτ1 ,qτ2(ω)
f (qτ1)f (qτ2)
Holger Dette Copula based spectral analysis 28 / 58
Laplace Periodogram for AR(1) data (n = 500, εt ∼ t1)
0.0 0.1 0.2 0.3 0.4 0.5
010
0030
0050
00
τ 1=
0.05
0.0 0.1 0.2 0.3 0.4 0.5
−50
050
100
150
200
τ 1=
0.5
0.0 0.1 0.2 0.3 0.4 0.5
−20
000
2000
4000
τ2=0.05
τ 1=
0.95
0.0 0.1 0.2 0.3 0.4 0.5
−10
0−
500
5010
0
0.0 0.1 0.2 0.3 0.4 0.5
05
1015
2025
0.0 0.1 0.2 0.3 0.4 0.5
−50
050
100
150
200
τ2=0.5
0.0 0.1 0.2 0.3 0.4 0.5
−15
00−
500
050
015
00
0.0 0.1 0.2 0.3 0.4 0.5
−10
0−
500
5010
0
0.0 0.1 0.2 0.3 0.4 0.5
010
0030
0050
00τ2=0.95
ω
L n, Nτ 1, τ
2 (ω)
Laplace Periodogram for AR(1) data (n = 500,εt ∼ N (0, 1))
0.0 0.1 0.2 0.3 0.4 0.5
0.02
0.06
0.10
τ 1=
0.05
0.0 0.1 0.2 0.3 0.4 0.5
−0.
050.
050.
15
τ 1=
0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
040.
000.
04
τ2=0.05
τ 1=
0.95
0.0 0.1 0.2 0.3 0.4 0.5
−0.
100.
000.
050.
10
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
0.0 0.1 0.2 0.3 0.4 0.5
−0.
050.
050.
15
τ2=0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
040.
000.
020.
04
0.0 0.1 0.2 0.3 0.4 0.5
−0.
100.
000.
050.
10
0.0 0.1 0.2 0.3 0.4 0.5
0.02
0.06
0.10
τ2=0.95
ω
L n, Nτ 1, τ
2 (ω)
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Estimating the marginal distributions
Recall:
The Laplace periodogram Lτ1,τ2(ω) is not an asymptoticallyunbiased estimator for fqτ1 ,qτ2
(ω), but for
2πfqτ1 ,qτ2
(ω)
f (qτ1)f (qτ2)
If F ∼ U([0, 1]), then the Laplace periodogram Lτ1,τ2(ω) is anasymptotically unbiased estimator for 2πfqτ1 ,qτ2
(ω)
Idea: Use probability integral transformation Xt → F (Xt)!
Replace F (Xt) by Fn(Xt), where Fn is the empiricaldistribution function of X1, . . .Xn.
Note: nFn(Xt) gives the rank Rt(n) of Xt among X1, . . . ,Xn
Holger Dette Copula based spectral analysis 31 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Estimating the marginal distributions
Recall:
The Laplace periodogram Lτ1,τ2(ω) is not an asymptoticallyunbiased estimator for fqτ1 ,qτ2
(ω), but for
2πfqτ1 ,qτ2
(ω)
f (qτ1)f (qτ2)
If F ∼ U([0, 1]), then the Laplace periodogram Lτ1,τ2(ω) is anasymptotically unbiased estimator for 2πfqτ1 ,qτ2
(ω)
Idea: Use probability integral transformation Xt → F (Xt)!
Replace F (Xt) by Fn(Xt), where Fn is the empiricaldistribution function of X1, . . .Xn.
Note: nFn(Xt) gives the rank Rt(n) of Xt among X1, . . . ,Xn
Holger Dette Copula based spectral analysis 31 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Rank based Periodogram
The rank-based periodogram kernel
Lτ1,τ2
n,R (ω)
is the Laplace periodogram kernel calculated from
1
n + 1R1(n), . . . ,
1
n + 1Rn(n)
(instead of from X1, . . . ,Xn), where Rt(n) denotes the rank of Xt
among X1, . . . ,Xn.
(aτn(ω), bτ1n(ω), bτ2n(ω)) = arg minb∈R3
n∑t=1
ρτ
(1
n+1 Rt(n) − ct(ω)′b)
Holger Dette Copula based spectral analysis 32 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Theorem 2
For any ω1, . . . , ων ∈ (0, π) and any τ1, τ2 ∈ (0, 1) we have
(Lτ1,τ2
n,R (ω1), . . . , Lτ1,τ2
n,R (ων)) (Lτ1,τ2
R (ω1), . . . , Lτ1,τ2
R (ων))
where Lτ1,τ2
R (ωj) denote independent random variables distributedas Lτ1,τ2(ωj) , where
f τ1,τ2(ω) = 2πfqτ1 ,qτ2
(ω)
f (qτ1)f (qτ2)
has to be replaced by 2πfqτ1 ,qτ2(ω)
In particular the rank-based periodogram Lτ1,τ2
n,R (ω) is anasymptotically unbiased estimator for 2πfqτ1 ,qτ2
(ω).
Holger Dette Copula based spectral analysis 33 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Investigation of Finite Sample Properties
Consider the AR(1) time series with t1-distributed innovations(ϑ = −0.3) from the example
Get an idea on the remaining bias of the estimate
ELτ1,τ2n (ω) Theorem 2−−−−−−→
n→∞ELτ1,τ2(ω) = 2πfqτ1 ,qτ2
(ω)
Holger Dette Copula based spectral analysis 34 / 58
AR(1) Example: Rank-based Periodograms, n = 500,t1-distributed innovations
0.0 0.1 0.2 0.3 0.4 0.5
0.02
0.06
0.10
0.14
τ 1=
0.05
0.0 0.1 0.2 0.3 0.4 0.5
−0.
050.
050.
150.
25
τ 1=
0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
050.
000.
050.
10
τ2=0.05
τ 1=
0.95
0.0 0.1 0.2 0.3 0.4 0.5
−0.
100.
000.
10
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
−0.
050.
050.
150.
25
τ2=0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
04−
0.02
0.00
0.02
0.04
0.0 0.1 0.2 0.3 0.4 0.5
−0.
15−
0.05
0.05
0.0 0.1 0.2 0.3 0.4 0.5
0.02
0.06
0.10
0.14
τ2=0.95
ω
L n, Nτ 1, τ
2 (ω)
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Smoothed Periodogram
Sequence of positive weights Wn = Wn(j) : |j | ≤ NnWn(k) = Wn(−k) for all k∑|k|≤Nn
Wn(k) = 1
Definition
smoothed rank-based periodogram kernel:
f τ1,τ2
n,R (ωj) :=∑|k|≤Nn
Wn(k)Lτ1,τ2
n,R (ωj+k)
Holger Dette Copula based spectral analysis 36 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Asymptotics of the Laplace Periodogram Kernels
Theorem 3
For any ω ∈ (0, π) and any τ1, τ2 ∈ (0, 1) we have
f τ1,τ2
n,R (gn(ω)) = 2πfqτ1 ,qτ2(ω) + oP(1),
where gn(ω) denotes the Fourier frequency closest to ω.
Holger Dette Copula based spectral analysis 37 / 58
AR(1) Example: Means of the Smoothed Rank-basedPeriodograms, n = 500, t1− distributed innovations
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.05
0.10
0.15
τ 1=
0.05
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.1
0.2
0.3
τ 1=
0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
050.
000.
050.
10
τ2=0.05
τ 1=
0.95
0.0 0.1 0.2 0.3 0.4 0.5
−0.
100.
000.
10
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.1
0.2
0.3
τ2=0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
04−
0.02
0.00
0.02
0.04
0.0 0.1 0.2 0.3 0.4 0.5
−0.
15−
0.05
0.05
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.05
0.10
0.15
τ2=0.95
ω
f nτ 1, τ
2 (ω)
AR(1) Example: Means of the Smoothed Rank-basedPeriodograms, n = 500, N (0, 1)-distributed innovations
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.04
0.08
0.12
τ 1=
0.05
0.0 0.1 0.2 0.3 0.4 0.5
−0.
050.
050.
15
τ 1=
0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
040.
000.
040.
08
τ2=0.05
τ 1=
0.95
0.0 0.1 0.2 0.3 0.4 0.5
−0.
100.
000.
050.
10
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.1 0.2 0.3 0.4 0.5
−0.
050.
050.
150.
25
τ2=0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
040.
000.
020.
04
0.0 0.1 0.2 0.3 0.4 0.5
−0.
100.
000.
050.
10
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.04
0.08
0.12
τ2=0.95
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Standard & Poor’s 500: Daily (log-)Returns, 1963–2009
Year
S&
P 5
00 R
etur
n
1970 1980 1990 2000 2010
−0.
050.
000.
05
Holger Dette Copula based spectral analysis 40 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Traditional Spectral Analysis (Returns)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
0.0 0.1 0.2 0.3 0.4 0.5
0.00
007
0.00
008
0.00
009
0.00
011
frequency
spec
trum
bandwidth = 0.0142
Holger Dette Copula based spectral analysis 41 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Traditional Spectral Analysis (Quadratic (log-)Returns)
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
Lag
AC
F
0.0 0.1 0.2 0.3 0.4 0.5
1e−
072e
−07
3e−
075e
−07
frequency
spec
trum
bandwidth = 0.0142
Holger Dette Copula based spectral analysis 42 / 58
Smoothed Rank-based Periodograms
0.0 0.1 0.2 0.3 0.4 0.5
−0.
10.
00.
10.
20.
3
τ 1=
0.05
0.0 0.1 0.2 0.3 0.4 0.5
−0.
10.
00.
10.
20.
3
τ 1=
0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
10.
00.
10.
20.
3
τ2=0.05
τ 1=
0.95
0.0 0.1 0.2 0.3 0.4 0.5
−0.
10.
00.
10.
20.
3
0.0 0.1 0.2 0.3 0.4 0.5
−0.
10.
00.
10.
20.
3
0.0 0.1 0.2 0.3 0.4 0.5
−0.
10.
00.
10.
20.
3
τ2=0.5
0.0 0.1 0.2 0.3 0.4 0.5
−0.
10.
00.
10.
20.
3
0.0 0.1 0.2 0.3 0.4 0.5
−0.
10.
00.
10.
20.
30.0 0.1 0.2 0.3 0.4 0.5
−0.
10.
00.
10.
20.
3
τ2=0.95
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Comments
The classical approach does not detect any serial structure
... the Laplace periodogram does ...
For extreme quantiles the smoothed periodograms peak at lowfrequencies
This indicates long-range dependence in the tails (ornon-stationarity)
Imaginary parts have smaller (absolute) values than the realparts, which might indicate “time reversibility”
Holger Dette Copula based spectral analysis 44 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Summary: “model-free” and “nonlinear” Spectral Analysis
The rank-based periodogram kernel seems to inherit much ofthe properties of the ordinary periodogram
Robustness can be expected due to the L1-nature of the toolsinvolved
Analysis of conditional distributions, not simply conditionalmeans and variances
No linearity, distribution, nor even moment assumptions arerequired
Separation of serial dependencies and marginal features
Invariance under monotone transformations of theobservations
Holger Dette Copula based spectral analysis 45 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Much work remains on the research agenda
Distributional properties of smoothed periodograms
Locally stationary processes
Prediction
Testing time reversibility
Extreme quantiles - tail dependence - tail copulas
Integrated spectra, higher order spectra
Graphical models
Holger Dette Copula based spectral analysis 46 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Distributional properties of smoothed periodograms
Distributional properties of smoothed periodograms:
In(ωj) :=1
n
∣∣∣ n∑t=1
Xte−itωj
∣∣∣2So far we have used generalizations of L2 projections
In(ωj) =n
4bn(ωj)
′(
1 −ii 1
)bn(ωj)
where bn(ωj) = (b1n(ωj), b2n(ωj))′ with
(an(ωj), b1n(ωj), b2n(ωj)) = arg minb∈R3
n∑t=1
(Xt − ct(ωj)
′b)2
and ct(ωj) = (1, cos(tωj), sin(tωj))′.
Holger Dette Copula based spectral analysis 47 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Distributional properties of smoothed periodograms
Alternative representation of periodogram (DFT of ACF):
Fourier transform of empirical auto-covariance
In(ωj) =1
n
∣∣∣ n∑t=1
Xte−itωj
∣∣∣2 =∑|k|<n
e−ikωjn − k
nγk
for Fourier frequencies ωj = 2πjn ∈ (0, π)
γk is empirical auto-covariance at lag k, i.e.
γk :=1
n − |k |
n−|k|∑t=1
(Xt − X )(Xt+|k| − X ).
Holger Dette Copula based spectral analysis 48 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Distributional properties of smoothed periodograms
Alternative estimator:
Discrete Fourier transform: dτn,R(ω) :=∑n
t=1 I 1n+1 Rt(n) ≤ τe−iωt
Copula periodogram
I τ1,τ2
n,R (ω) :=1
ndτ1
n,R(ω)dτ2
n,R(−ω), ω ∈ (0, π), (τ1, τ2) ∈ [0, 1]2,
For ω = 2πj/n with j = 1, ..., (n − 1)
I τ1,τ2
n,R (ω) =∑|k|<n
n − k
ne iωk Γτ1,τ2
k
where (k > 0)
Γτ1,τ2
k :=1
n − k
n−k∑t=1
I 1n+1 Rt(n) ≤ τ1I 1
n+1 Rt+k(n) ≤ τ2
is an estimator of the pair copula at lag k (but not the empiricalcopula!).
Holger Dette Copula based spectral analysis 49 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Distributional properties of smoothed periodograms
Alternative smoothed estimator:
Definition
The smoothed copula periodogram kernel:
fτ1,τ2
n,R (ω) :=2π
n
n−1∑s=1
Wn
(ω − 2πs/n
)I τ1,τ2
n,R (2πs/n)
where
Wn(u) :=∞∑
j=−∞b−1n W (b−1
n [u + 2πj ])
for a kernel W and bandwidth bn.
Holger Dette Copula based spectral analysis 50 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Distributional properties of smoothed periodograms
Theorem 4
Under suitable assumptions, for any fixed ω ∈ (0, π)√nbn
(fτ1,τ2
n,R (ω)− fqτ1,qτ2
(ω)− B(k)n (τ1, τ2;ω)
)τ1,τ2∈[0,1]
H(·, ·;ω)
in `∞([0, 1]2) where
B(k)n (τ1, τ2;ω) :=
k∑j=1
bjn
j!
∫v jW (v)dv
dj
dωjfqτ1
,qτ2(ω),
and H(·, ·;ω) is a centered Gaussian process with
Cov(H(x1, y1;ω
),H(x2, y2, ω)) = fqx1
,qy1(ω)fqx2
,qy2 (ω)
∫W 2(u)du.
Weak convergence above holds jointly for any finite collections offrequencies (asymptotic independence).
Conjecture: similar result holds for rank based periodogram
Holger Dette Copula based spectral analysis 51 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Distributional properties of smoothed periodograms
Distributional properties of smoothed periodograms:
Estimates of copula spectra and point wise confidence intervals forsimulated ARCH-data
0.0 0.1 0.2 0.3 0.4 0.5
0.01
00.
020
0.03
0
0.0 0.1 0.2 0.3 0.4 0.5
0.00
40.
008
0.01
2
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
100.
000
0.00
5
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
03−0
.001
0.00
10.
003
0.0 0.1 0.2 0.3 0.4 0.5
0.03
00.
035
0.04
00.
045
0.05
0
0.0 0.1 0.2 0.3 0.4 0.5
0.00
40.
008
0.01
2
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
020.
000
0.00
2
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
020.
000
0.00
20.
004
0.0 0.1 0.2 0.3 0.4 0.5
0.01
00.
020
0.03
0
τ 1=0
.1τ 1
=0.5
τ 1=0
.9
τ2=0.1 τ2=0.5 τ2=0.9
ω 2πHolger Dette Copula based spectral analysis 52 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Distributional properties of smoothed periodograms
Distributional properties of smoothed periodograms:
Estimates of copula spectra and point wise confidence intervals forsimulated e QAR-data
0.0 0.1 0.2 0.3 0.4 0.5
0.00
80.
012
0.01
6
0.0 0.1 0.2 0.3 0.4 0.5
0.00
00.
005
0.01
00.
015
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
020.
000
0.00
20.
004
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
08−0
.004
0.00
0
0.0 0.1 0.2 0.3 0.4 0.5
0.03
50.
040
0.04
50.
050
0.0 0.1 0.2 0.3 0.4 0.5
0.00
00.
010
0.02
0
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
05−0
.003
−0.0
010.
001
0.0 0.1 0.2 0.3 0.4 0.5
−0.0
10−0
.006
−0.0
02
0.0 0.1 0.2 0.3 0.4 0.5
0.01
00.
015
0.02
00.
025
τ 1=0
.1τ 1
=0.5
τ 1=0
.9
τ2=0.1 τ2=0.5 τ2=0.9
ω 2πHolger Dette Copula based spectral analysis 53 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Local stationarity
Local stationarity:
Definition
A triangular array (Xt,T )t∈ZT∈N of processes is called locally strictlystationary (of order two) if there exists a constant L > 0 and, forevery ϑ ∈ (0, 1), a strictly stationary process Xϑ
t , t ∈ Z such that, forevery 1 ≤ r , s ≤ T ,∥∥Fr ,s;T (·, ·)− Gϑ
r−s(·, ·)∥∥∞ ≤ L
(max(|r/T − ϑ|, |s/T − ϑ|) + 1/T
),
where
‖ · ‖∞ is the supremum norm
Fr ,s;T (·, ·) is the joint distribution functions of (Xr ,T ,Xs,T )
Gϑk (·, ·) is the joint distribution functions of (Xϑ
0 ,Xϑ−k)
Holger Dette Copula based spectral analysis 54 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Local stationarity
Local smoothed copula periodograms:
Data: dailylog-returns of S&P500 since the sixties,13000 observations.
Compute spectrafrom local windowsand plot as heatplots.
Colours: indicatedeviations from whitenoise
Holger Dette Copula based spectral analysis 55 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Local stationarity
Local smoothed copula periodograms:
Holger Dette Copula based spectral analysis 56 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Local stationarity
Local smoothed copula periodograms:
Holger Dette Copula based spectral analysis 57 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Local stationarity
Some references
- Davis, R. A., Mikosch, T. and Zhao, Y. (2013). Measures of serial extremal dependence and theirestimation. Stochastic Processes and their Applications 123 2575-2602.
- Dette, H., Hallin, M., Kley, T., Volgushev, S. (2011,2014). Of copulas, quantiles, ranks and spectra: AnL1-approach to spectral analysis. Available on Arxiv. To appear in: Bernoulli
- Hagemann, A. (2011). Robust Spectral Analysis (arXiv:1111.1965v1).
- Li, T. H. (2008). Laplace periodogram for time series analysis. Journal of the American StatisticalAssociation 103, 757- 768.
- Li, T.-H. (2012). Quantile periodograms. Journal of the American Statistical Association 107, 765-776.
- Kley, T. Volgushev, S. , Dette, H., Hallin, M., (2013). Quantile spectral processes - asymptotic analysisand inference. Available on Arxiv. To appear in: Bernoulli
- Skowronek, S., Volgushev, S., Kley, T., Dette, H., Hallin, M., (2013). Quantile Spectral Analysis forLocally Stationary Time Series,. Available on Arxiv.
- Kley, T. (2014). quantspec: Quantile-based Spectral Analysis Functions. R package version 0.1.Available on http://cran.r-project.org/web/packages/quantspec/index.html.
Holger Dette Copula based spectral analysis 58 / 58
Serial Dependence Spectral Analysis Asymptotic Properties Data Example Conclusions Extensions
Copula based spectral analysis
Holger Dette, Ruhr-Universitat BochumMarc Hallin, Universite Libre de Bruxelles
Tobias Kley, Ruhr-Universitat BochumStefan Skowronek, Ruhr-Universitat Bochum
Stanislav Volgushev, Ruhr-Universitat Bochum
July , 2015
Proof of Theorem 1 Proof of Theorem 3
Backup Slides
7 Proof of Theorem 1
8 Proof of Theorem 3
Proof of Theorem 1 Proof of Theorem 3
Proof of Theorem 1 (main idea)
We show for Ω := ω1, . . . , ων ⊂ Fn ⊂ (0, π) and allT := τ1, . . . , τp ⊂ (0, 1)
√n(
bτn(ω))τ∈T , ω∈Ω
L−−−→n→∞
(Nτ (ω)
)τ∈T , ω∈Ω
Here (Nτ (ω))τ∈T , ω∈Ω is a vector of centered (bivariate) normaldistributed random variables with Cov(Nτ1(ω1),Nτ2(ω2)) = 0 ifω1 6= ω2 and
Cov(Nτ1(ω1),Nτ2(ω1)) = 2
(<f τ1,τ2(ω1) =f τ1,τ2(ω1)−=f τ1,τ2(ω1) <f τ1,τ2(ω1)
)otherwise
Holger Dette Copula based spectral analysis BU 32 / 36
Proof of Theorem 1 Proof of Theorem 3
Most important step: uniform linearization
Define
Zn,τ,ω(δ) :=∑n
t=1
(ρτ (Xt − qτ − n−1/2c′t(ω)δ)− ρτ (Xt − qτ )
)ZX ′n,τ,ω(δ) = −δ′ζX ′
n,τ,ω + 12δ′QX ′
n,τ,ωδ
where
ζX ′
n,τ,ω = n−1/2∑n
t=1 ct(ω)(τ − IX ′t,n ≤ qn,τ)QX ′
n,τ,ω = fn,X ′(qn,τ ) n−1∑n
t=1 ct(ω)c′t(ω)fn,X ′ is the is the density of X ′t,n
then
supω∈Fn
sup‖δ−δX ′
n,τ,ω‖≤ε|Zn,τ,ω(δ)−ZX ′
n,τ,ω(δ)| = OP
((n−1/4∨(n−1/2mn))(log n)2
)where δX
′n,τ,ω = (QX ′
n,τ,ω)−1ζX′
n,τ,ω
Holger Dette Copula based spectral analysis BU 33 / 36
Proof of Theorem 1 Proof of Theorem 3
Most important step: uniform linearization
Define
Zn,τ,ω(δ) :=∑n
t=1
(ρτ (Xt − qτ − n−1/2c′t(ω)δ)− ρτ (Xt − qτ )
)ZX ′n,τ,ω(δ) = −δ′ζX ′
n,τ,ω + 12δ′QX ′
n,τ,ωδ
where
ζX ′
n,τ,ω = n−1/2∑n
t=1 ct(ω)(τ − IX ′t,n ≤ qn,τ)QX ′
n,τ,ω = fn,X ′(qn,τ ) n−1∑n
t=1 ct(ω)c′t(ω)fn,X ′ is the is the density of X ′t,n
then
supω∈Fn
sup‖δ−δX ′
n,τ,ω‖≤ε|Zn,τ,ω(δ)−ZX ′
n,τ,ω(δ)| = OP
((n−1/4∨(n−1/2mn))(log n)2
)where δX
′n,τ,ω = (QX ′
n,τ,ω)−1ζX′
n,τ,ω
Holger Dette Copula based spectral analysis BU 33 / 36
Proof of Theorem 1 Proof of Theorem 3
Consequences:
If
δn,τ,ω = arg minδ
Zn,τ,ω(δ)
δX′
n,τ,ω = arg minδ
ZX ′n,τ,ω(δ) = (QX ′
n,τ,ω)−1ζX′
n,τ,ω
then
supω∈Fn
‖δn,τ,ω − δX′
n,τ,ω‖ = OP
((n−1/8 ∨ (n−1/4m
1/2n )) log n
).
Therefore the asymptotic properties of√
nbn,τ (ωj) can be obtainedfrom those of the random variables δX
′n,τ,ω
Holger Dette Copula based spectral analysis BU 34 / 36
Proof of Theorem 1 Proof of Theorem 3
Consequences:
If
δn,τ,ω = arg minδ
Zn,τ,ω(δ)
δX′
n,τ,ω = arg minδ
ZX ′n,τ,ω(δ) = (QX ′
n,τ,ω)−1ζX′
n,τ,ω
then
supω∈Fn
‖δn,τ,ω − δX′
n,τ,ω‖ = OP
((n−1/8 ∨ (n−1/4m
1/2n )) log n
).
Therefore the asymptotic properties of√
nbn,τ (ωj) can be obtainedfrom those of the random variables δX
′n,τ,ω
Holger Dette Copula based spectral analysis BU 34 / 36
Proof of Theorem 1 Proof of Theorem 3
Consequences:
If
δn,τ,ω = arg minδ
Zn,τ,ω(δ)
δX′
n,τ,ω = arg minδ
ZX ′n,τ,ω(δ) = (QX ′
n,τ,ω)−1ζX′
n,τ,ω
then
supω∈Fn
‖δn,τ,ω − δX′
n,τ,ω‖ = OP
((n−1/8 ∨ (n−1/4m
1/2n )) log n
).
Therefore the asymptotic properties of√
nbn,τ (ωj) can be obtainedfrom those of the random variables δX
′n,τ,ω
Holger Dette Copula based spectral analysis BU 34 / 36
Proof of Theorem 1 Proof of Theorem 3
Proof of Theorem 3 (main idea)
Let Fn denote the empirical distribution function of X1, . . . ,Xn,We have to minimize
n∑t=1
(ρτ (Fn(Xt)− τ − n−1/2c′t(ω)δ)− ρτ (Fn(Xt)− τ)
)Replace Fn(Xt) by Fn(X ′t,n), where Xt = X ′t,n + X ′,′t,n refers to mn
decomposibility
n∑t=1
(ρτ (Fn(X ′t,n)− τ − n−1/2c′t(ω)δ)− ρτ (Fn(X ′t,n)− τ))
Replace Fn(Xt,n) by Ut,n = Fn,X (X ′t,n), where Fn,X denotes theempirical distribution function of X ′1,n, . . . ,X
′n,n
Holger Dette Copula based spectral analysis BU 35 / 36
Proof of Theorem 1 Proof of Theorem 3
Proof of Theorem 3 (main idea)
Let Fn denote the empirical distribution function of X1, . . . ,Xn,We have to minimize
n∑t=1
(ρτ (Fn(Xt)− τ − n−1/2c′t(ω)δ)− ρτ (Fn(Xt)− τ)
)Replace Fn(Xt) by Fn(X ′t,n), where Xt = X ′t,n + X ′,′t,n refers to mn
decomposibility
n∑t=1
(ρτ (Fn(X ′t,n)− τ − n−1/2c′t(ω)δ)− ρτ (Fn(X ′t,n)− τ))
Replace Fn(Xt,n) by Ut,n = Fn,X (X ′t,n), where Fn,X denotes theempirical distribution function of X ′1,n, . . . ,X
′n,n
Holger Dette Copula based spectral analysis BU 35 / 36
Proof of Theorem 1 Proof of Theorem 3
Proof of Theorem 3 (main idea)
Let Fn denote the empirical distribution function of X1, . . . ,Xn,We have to minimize
n∑t=1
(ρτ (Fn(Xt)− τ − n−1/2c′t(ω)δ)− ρτ (Fn(Xt)− τ)
)Replace Fn(Xt) by Fn(X ′t,n), where Xt = X ′t,n + X ′,′t,n refers to mn
decomposibility
n∑t=1
(ρτ (Fn(X ′t,n)− τ − n−1/2c′t(ω)δ)− ρτ (Fn(X ′t,n)− τ))
Replace Fn(Xt,n) by Ut,n = Fn,X (X ′t,n), where Fn,X denotes theempirical distribution function of X ′1,n, . . . ,X
′n,n
Holger Dette Copula based spectral analysis BU 35 / 36
Proof of Theorem 1 Proof of Theorem 3
Proof of Theorem 3 (main idea)
n∑t=1
(ρτ (Ut,n−τ−n−1/2c′t(ω)δ)−ρτ (Ut,n−τ)
)−δ1
√n(Fn,X (F−1
n (τ))−τ)
where δ = (δ1, δ2, δ3)′.
Linearization (e1 = (1, 0, 0)′)
−δ′(ζUn,τ,ω + e1
√n(Fn,X (F−1
n (τ))− τ))
+1
2δ′QU
n,ωδ
where
QUn,ω =
1
n
n∑t=1
ct(ω)c′t(ω)
ζUn,τ,ω = n−1/2n∑
t=1
ct(ω)(τ − IUt,n ≤ τ
)Holger Dette Copula based spectral analysis BU 36 / 36
Proof of Theorem 1 Proof of Theorem 3
Proof of Theorem 3 (main idea)
n∑t=1
(ρτ (Ut,n−τ−n−1/2c′t(ω)δ)−ρτ (Ut,n−τ)
)−δ1
√n(Fn,X (F−1
n (τ))−τ)
where δ = (δ1, δ2, δ3)′.
Linearization (e1 = (1, 0, 0)′)
−δ′(ζUn,τ,ω + e1
√n(Fn,X (F−1
n (τ))− τ))
+1
2δ′QU
n,ωδ
where
QUn,ω =
1
n
n∑t=1
ct(ω)c′t(ω)
ζUn,τ,ω = n−1/2n∑
t=1
ct(ω)(τ − IUt,n ≤ τ
)Holger Dette Copula based spectral analysis BU 36 / 36