Robust scale and autocovariance estimation
Transcript of Robust scale and autocovariance estimation
![Page 1: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/1.jpg)
Robust scale and autocovariance estimation
Garth Tarr, Samuel Muller and Neville Weber
School of Mathematics and Statistics
THE UNIVERSITY OF SYDNEY
EMS 2013
![Page 2: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/2.jpg)
Scale Autocovariance Conclusion
Outline
Robust Scale Estimator, Pn
Covariance
Autocovariance
Long RangeDependence
Short RangeDependence
Inverse CovarianceMatrix Estimation
Covariance Matrix
![Page 3: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/3.jpg)
Scale Autocovariance Conclusion
Outline
Robust Scale Estimator, Pn
Covariance
Autocovariance
Long RangeDependence
Short RangeDependence
Inverse CovarianceMatrix Estimation
Covariance Matrix
![Page 4: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/4.jpg)
Scale Autocovariance Conclusion
Outline
The robust scale estimator Pn
Autocovariance estimation using Pn
Conclusion and key references
![Page 5: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/5.jpg)
Scale Autocovariance Conclusion
History of scale efficiencies at the Gaussian (n = 20)R
elat
ive
Effi
cien
cy
Year
SD
1894
1.00
1931 1974
0.38
1993
0.67
2012
0.85
![Page 6: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/6.jpg)
Scale Autocovariance Conclusion
History of scale efficiencies at the Gaussian (n = 20)R
elat
ive
Effi
cien
cy
Year
SD
1894
1.00
IQR
1931
MAD
1974
0.38
1993
0.67
2012
0.85
![Page 7: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/7.jpg)
Scale Autocovariance Conclusion
History of scale efficiencies at the Gaussian (n = 20)R
elat
ive
Effi
cien
cy
Year
SD
1894
1.00
IQR
1931
MAD
1974
0.38
Qn
1993
0.67
2012
0.85
![Page 8: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/8.jpg)
Scale Autocovariance Conclusion
History of scale efficiencies at the Gaussian (n = 20)R
elat
ive
Effi
cien
cy
Year
SD
1894
1.00
IQR
1931
MAD
1974
0.38
Qn
1993
0.67
Pn
2012
0.85
![Page 9: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/9.jpg)
Scale Autocovariance Conclusion
Pairwise mean scale estimator: Pn
• Consider the U -statistic, based on the pairwise mean kernel,
Un(X) :=
(n
2
)−1∑i<j
Xi +Xj
2.
• Let H(t) = P ((Xi +Xj)/2 ≤ t) be the cdf of the kernelswith corresponding empirical distribution function,
Hn(t) :=
(n
2
)−1∑i<j
I{Xi +Xj
2≤ t}, for t ∈ R.
Definition (Interquartile range of pairwise means)
Pn = c[H−1n (0.75)−H−1
n (0.25)],
where c ≈ 1.048 is a correction factor to ensure Pn is consistentfor the standard deviation when the underlying observations areGaussian.
![Page 10: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/10.jpg)
Scale Autocovariance Conclusion
Influence curve
• Hampel (1974) defines the influence curve for a functional Tat distribution F as
IC(x;T, F ) = limε↓0
T ((1− ε)F + εδx)− T (F )
ε
where δx has all its mass at x.• Serfling (1984) outlines the IC for GL-statistics.
Influence curve for Pn (Tarr, Muller and Weber, 2012)
Assuming that F has derivative f > 0 on [F−1(ε), F−1(1− ε)] forall ε > 0,
IC(x;P , F ) = c
[0.75− F (2H−1
F (0.75)− x)∫f(2H−1
F (0.75)− x)f(x) dx
−0.25− F (2H−1
F (0.25)− x)∫f(2H−1
F (0.25)− x)f(x) dx
].
![Page 11: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/11.jpg)
Scale Autocovariance Conclusion
Influence curve
• Hampel (1974) defines the influence curve for a functional Tat distribution F as
IC(x;T, F ) = limε↓0
T ((1− ε)F + εδx)− T (F )
ε
where δx has all its mass at x.• Serfling (1984) outlines the IC for GL-statistics.
Influence curve for Pn (Tarr, Muller and Weber, 2012)
Assuming that F has derivative f > 0 on [F−1(ε), F−1(1− ε)] forall ε > 0,
IC(x;P , F ) = c
[0.75− F (2H−1
F (0.75)− x)∫f(2H−1
F (0.75)− x)f(x) dx
−0.25− F (2H−1
F (0.25)− x)∫f(2H−1
F (0.25)− x)f(x) dx
].
![Page 12: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/12.jpg)
Scale Autocovariance Conclusion
Influence curves when F = Φ
−4 −2 0 2 4
−1
01
2
x
IC(x
;T,Φ
)
Pn
![Page 13: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/13.jpg)
Scale Autocovariance Conclusion
Influence curves when F = Φ
−4 −2 0 2 4
−1
01
2
x
IC(x
;T,Φ
)
PnSD
![Page 14: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/14.jpg)
Scale Autocovariance Conclusion
Influence curves when F = Φ
−4 −2 0 2 4
−1
01
2
x
IC(x
;T,Φ
)
Pn
MAD
SD
![Page 15: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/15.jpg)
Scale Autocovariance Conclusion
Influence curves when F = Φ
−4 −2 0 2 4
−1
01
2
x
IC(x
;T,Φ
)
Pn
MAD
Qn
SD
![Page 16: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/16.jpg)
Scale Autocovariance Conclusion
Asymptotic variance and relative efficiency
• Tarr, Muller and Weber (2012) show that when the underlyingobservations are independent, Pn is asymptotically normalwith variance, V , given by the expected square of theinfluence function.
• When the underlying data are independent Gaussian,
V (Pn,Φ) =
∫IC(x;P ,Φ)2 d Φ(x) = 0.579.
• This equates to an asymptotic efficiency of 0.86 as comparedwith 0.82 for Qn and 0.37 for the MAD.
! But how does it compare at heavier tailed distributions?
![Page 17: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/17.jpg)
Scale Autocovariance Conclusion
Asymptotic variance and relative efficiency
• Tarr, Muller and Weber (2012) show that when the underlyingobservations are independent, Pn is asymptotically normalwith variance, V , given by the expected square of theinfluence function.
• When the underlying data are independent Gaussian,
V (Pn,Φ) =
∫IC(x;P ,Φ)2 d Φ(x) = 0.579.
• This equates to an asymptotic efficiency of 0.86 as comparedwith 0.82 for Qn and 0.37 for the MAD.
! But how does it compare at heavier tailed distributions?
![Page 18: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/18.jpg)
Scale Autocovariance Conclusion
Asymptotic variance and relative efficiency
• Tarr, Muller and Weber (2012) show that when the underlyingobservations are independent, Pn is asymptotically normalwith variance, V , given by the expected square of theinfluence function.
• When the underlying data are independent Gaussian,
V (Pn,Φ) =
∫IC(x;P ,Φ)2 d Φ(x) = 0.579.
• This equates to an asymptotic efficiency of 0.86 as comparedwith 0.82 for Qn and 0.37 for the MAD.
! But how does it compare at heavier tailed distributions?
![Page 19: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/19.jpg)
Scale Autocovariance Conclusion
Asymptotic relative efficiency when f = tν for ν ∈ [1, 10]
1 2 3 54 76 98 10
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Degrees of freedom
Asy
mp
toti
cre
lati
veeffi
cien
cy
Pn
![Page 20: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/20.jpg)
Scale Autocovariance Conclusion
Asymptotic relative efficiency when f = tν for ν ∈ [1, 10]
1 2 3 54 76 98 10
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Degrees of freedom
Asy
mp
toti
cre
lati
veeffi
cien
cy
Pn
MAD
![Page 21: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/21.jpg)
Scale Autocovariance Conclusion
Asymptotic relative efficiency when f = tν for ν ∈ [1, 10]
1 2 3 54 76 98 10
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Degrees of freedom
Asy
mp
toti
cre
lati
veeffi
cien
cy
Pn
MAD
Qn
![Page 22: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/22.jpg)
Scale Autocovariance Conclusion
Asymptotic distribution under short range dependence
Assumption 1: Gaussian SRD
Let {Xi}i≥1 be a stationary mean-zero Gaussian process withautocovariances γ(h) = E(X1Xh+1) satisfying
∑h≥1 |γ(h)| <∞.
Result
Under Assumption 1, it can be shown that,
√n(Pn − σ) =
cΦ√n
n∑i=1
IC(Xi, P ,Φ) + op(1),
hence applying a limit theorem from Arcones (1994) we have√n(Pn − σ)
D−→ N (0, σ2),
where σ =√γ(0) and
σ2 = E(IC2(X1, P ,Φ)
)+ 2
∑k≥1
E (IC(X1, P ,Φ)IC(Xk+1, P ,Φ)) .
![Page 23: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/23.jpg)
Scale Autocovariance Conclusion
Asymptotic distribution under long range dependence
Assumption 2: Gaussian LRD
Let {Xi}i≥1 be a stationary mean-zero Gaussian process withautocovariance sequence γ(h) = E(X1Xh+1) satisfyingγ(h) = h−DL(h), 0 < D < 1, where L is slowly varying at infinity.
Result
Under Assumption 2,
1. If D > 1/2,√n(Pn − σ)
D−→ N (0, σ2),
σ2 = E IC2(X1, P ,Φ) + 2∑k≥1
E IC(X1, P ,Φ)IC(Xk+1, P ,Φ).
!Note that this is the same as the SRD result, however theproof is somewhat more involved.
![Page 24: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/24.jpg)
Scale Autocovariance Conclusion
Asymptotic distribution under long range dependence
Assumption 2: Gaussian LRD
Let {Xi}i≥1 be a stationary mean-zero Gaussian process withautocovariance sequence γ(h) = E(X1Xh+1) satisfyingγ(h) = h−DL(h), 0 < D < 1, where L is slowly varying at infinity.
Conjecture
Under Assumption 2,
2. If D < 1/2,
k(D)nDL(n)−1(Pn − σ)D−→ σ
2
(Z2,D(1)− Z2
1,D(1)),
where k(D) = B((1−D)/2, D) and B denotes the betafunction Z1,D is the standard fBm process and Z2,D is theRosenblatt process.
!This is consistent with results for other scale estimators,e.g. SD and Qn (Levy-Leduc et. al., 2011) and the IQR...
![Page 25: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/25.jpg)
Scale Autocovariance Conclusion
Interquartile range result
Consider equivalent result for the interquartile range,
Tn(x) = cΦ
(F−1n (3/4)− F−1
n (1/4)).
Result
Under Assumption 2 with D < 1/2, Tn satisfies the following limittheorem as n→∞:
k(D)nDL(n)−1(Tn − σ)D−→ σ
2
(Z2,D(1)− Z2
1,D(1)).
• In establishing this result we used Wu’s (2005) Bahadurrepresentation for sample quantiles under LRD:
F−1n (p)− r =
p− Fn(r)
φ(r)+X2n
2
φ′(r)
φ(r)+O(nh(D)L1(n)).
!It appears to be non-trivial to extend this result to the pairwisemean empirical distribution function.
![Page 26: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/26.jpg)
Scale Autocovariance Conclusion
Interquartile range result
Consider equivalent result for the interquartile range,
Tn(x) = cΦ
(F−1n (3/4)− F−1
n (1/4)).
Result
Under Assumption 2 with D < 1/2, Tn satisfies the following limittheorem as n→∞:
k(D)nDL(n)−1(Tn − σ)D−→ σ
2
(Z2,D(1)− Z2
1,D(1)).
• In establishing this result we used Wu’s (2005) Bahadurrepresentation for sample quantiles under LRD:
F−1n (p)− r =
p− Fn(r)
φ(r)+X2n
2
φ′(r)
φ(r)+O(nh(D)L1(n)).
!It appears to be non-trivial to extend this result to the pairwisemean empirical distribution function.
![Page 27: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/27.jpg)
Scale Autocovariance Conclusion
Interquartile range result
Consider equivalent result for the interquartile range,
Tn(x) = cΦ
(F−1n (3/4)− F−1
n (1/4)).
Result
Under Assumption 2 with D < 1/2, Tn satisfies the following limittheorem as n→∞:
k(D)nDL(n)−1(Tn − σ)D−→ σ
2
(Z2,D(1)− Z2
1,D(1)).
• In establishing this result we used Wu’s (2005) Bahadurrepresentation for sample quantiles under LRD:
F−1n (p)− r =
p− Fn(r)
φ(r)+X2n
2
φ′(r)
φ(r)+O(nh(D)L1(n)).
!It appears to be non-trivial to extend this result to the pairwisemean empirical distribution function.
![Page 28: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/28.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
30
40
0.95 1.00 1.05
Scale estimate
Den
sity
Pn
87%SD Relative Efficiency
Qn 83%IQR 39%MAD 39%
![Page 29: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/29.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
30
40
0.95 1.00 1.05
Scale estimate
Den
sity
Pn 87%SD Relative Efficiency
Qn 83%IQR 39%MAD 39%
![Page 30: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/30.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
30
40
0.95 1.00 1.05
Scale estimate
Den
sity
Pn 87%SD Relative Efficiency
Qn 83%
IQR 39%MAD 39%
![Page 31: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/31.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
30
40
0.95 1.00 1.05
Scale estimate
Den
sity
Pn 87%SD Relative Efficiency
Qn 83%IQR 39%
MAD 39%
![Page 32: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/32.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
30
40
0.95 1.00 1.05
Scale estimate
Den
sity
Pn 87%SD Relative Efficiency
Qn 83%IQR 39%MAD 39%
![Page 33: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/33.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0
2
4
1.2 1.4 1.6 1.8 2.0
Scale estimate
Den
sity
Pn
94%SD Relative Efficiency
Qn 94%IQR 82%MAD 83%
![Page 34: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/34.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0
2
4
1.2 1.4 1.6 1.8 2.0
Scale estimate
Den
sity
Pn 94%SD Relative Efficiency
Qn 94%IQR 82%MAD 83%
![Page 35: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/35.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0
2
4
1.2 1.4 1.6 1.8 2.0
Scale estimate
Den
sity
Pn 94%SD Relative Efficiency
Qn 94%
IQR 82%MAD 83%
![Page 36: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/36.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0
2
4
1.2 1.4 1.6 1.8 2.0
Scale estimate
Den
sity
Pn 94%SD Relative Efficiency
Qn 94%IQR 82%
MAD 83%
![Page 37: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/37.jpg)
Scale Autocovariance Conclusion
Empirical densities ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0
2
4
1.2 1.4 1.6 1.8 2.0
Scale estimate
Den
sity
Pn 94%SD Relative Efficiency
Qn 94%IQR 82%MAD 83%
![Page 38: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/38.jpg)
Scale Autocovariance Conclusion
Empirical densities with 2% contamination at 5
0
2
4
1.2 1.4 1.6 1.8 2.0
Scale estimate
Den
sity
Pn
SD
QnIQRMAD
![Page 39: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/39.jpg)
Scale Autocovariance Conclusion
Empirical densities with 2% contamination at 6
0
2
4
1.2 1.4 1.6 1.8 2.0
Scale estimate
Den
sity
Pn
SD
QnIQRMAD
![Page 40: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/40.jpg)
Scale Autocovariance Conclusion
Empirical densities with 2% contamination at 7
0
2
4
1.2 1.4 1.6 1.8 2.0
Scale estimate
Den
sity
Pn
SD
QnIQRMAD
![Page 41: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/41.jpg)
Scale Autocovariance Conclusion
Empirical densities with 2% contamination at 8
0
2
4
1.2 1.4 1.6 1.8 2.0
Scale estimate
Den
sity
Pn
SD
QnIQRMAD
![Page 42: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/42.jpg)
Scale Autocovariance Conclusion
Outline
The robust scale estimator Pn
Autocovariance estimation using Pn
Conclusion and key references
![Page 43: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/43.jpg)
Scale Autocovariance Conclusion
From scale to autocovariance
• Gnanadesikan and Kettenring (1972) highlighted an identitythat relates scale and covariance.
• In the autocovariance setting,
γ(h) = cov(X1, Xh+1) =1
4[var(X1+Xh+1)− var(X1−Xh+1)] .
• For a series of n observations, Xn = {Xt}1≤t≤n,
γP (h) =1
4
[P 2n−h(X1:n−h+Xh+1:n)− P 2
n−h(X1:n−h−Xh+1:n)].
where X1:n−h are the first n− h observations in Xn andXh+1:n are the last n− h observations.
• The same technique can be used to turn other scaleestimators into covariance and correlation estimators, e.g. Maand Genton (2000) study γQ based on Qn.
![Page 44: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/44.jpg)
Scale Autocovariance Conclusion
Robustness properties
• Following Genton and Ma (1999), we have shown thatinfluence curve, IC((x, y), γP ,Φ), and therefore the grosserror sensitivity of γP can be derived from the IC of Pn.
x−4 −2 0 2 4
y−4
−20
24
IC(x,y)
−3−2−10123
IC((x,y),Pn,Φ)
! IC is bounded and hence the gross error sensitivity is finite.
![Page 45: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/45.jpg)
Scale Autocovariance Conclusion
Robustness properties
• Following Genton and Ma (1999), we have shown thatinfluence curve, IC((x, y), γP ,Φ), and therefore the grosserror sensitivity of γP can be derived from the IC of Pn.
x−4 −2 0 2 4
y−4
−20
24
IC(x,y)
−3−2−10123
IC((x,y),Pn,Φ)
! IC is bounded and hence the gross error sensitivity is finite.
![Page 46: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/46.jpg)
Scale Autocovariance Conclusion
Robustness properties
• Following Genton and Ma (1999), we have shown thatinfluence curve, IC((x, y), γP ,Φ), and therefore the grosserror sensitivity of γP can be derived from the IC of Pn.
x−4 −2 0 2 4
y−4
−20
24
IC(x,y)
−3−2−10123
IC((x,y),Pn,Φ)
! IC is bounded and hence the gross error sensitivity is finite.
![Page 47: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/47.jpg)
Scale Autocovariance Conclusion
Breakdown value
• Ma and Genton (2000) show that the breakdown value forautocovariance estimators is (roughly) half that of thecorresponding covariance estimator.
• Consider n = 13
, autocovariance at lag h = 2
• Working with X1:11 ±X3:13
• 4 contaminated observations, denoted by
• Leaves only 3 uncontaminated pairs, denoted by
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11X9 X10 X11 X12 X13
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13X1 X2
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13X11 X12 X13
! γP (h) has roughly half the 13.4% breakdown value of Pn.
![Page 48: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/48.jpg)
Scale Autocovariance Conclusion
Breakdown value
• Ma and Genton (2000) show that the breakdown value forautocovariance estimators is (roughly) half that of thecorresponding covariance estimator.
• Consider n = 13, autocovariance at lag h = 2
• Working with X1:11 ±X3:13
• 4 contaminated observations, denoted by
• Leaves only 3 uncontaminated pairs, denoted by
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11X9 X10 X11
X12 X13
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13X1 X2
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13X11 X12 X13
! γP (h) has roughly half the 13.4% breakdown value of Pn.
![Page 49: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/49.jpg)
Scale Autocovariance Conclusion
Breakdown value
• Ma and Genton (2000) show that the breakdown value forautocovariance estimators is (roughly) half that of thecorresponding covariance estimator.
• Consider n = 13, autocovariance at lag h = 2
• Working with X1:11 ±X3:13
• 4 contaminated observations, denoted by
• Leaves only 3 uncontaminated pairs, denoted by
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
X9 X10 X11
X12 X13
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13
X1 X2
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13
X11 X12 X13
! γP (h) has roughly half the 13.4% breakdown value of Pn.
![Page 50: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/50.jpg)
Scale Autocovariance Conclusion
Breakdown value
• Ma and Genton (2000) show that the breakdown value forautocovariance estimators is (roughly) half that of thecorresponding covariance estimator.
• Consider n = 13, autocovariance at lag h = 2
• Working with X1:11 ±X3:13
• 4 contaminated observations, denoted by
• Leaves only 3 uncontaminated pairs, denoted by
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11X9 X10 X11 X12 X13
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13
X1 X2
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13X11 X12 X13
! γP (h) has roughly half the 13.4% breakdown value of Pn.
![Page 51: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/51.jpg)
Scale Autocovariance Conclusion
Breakdown value
• Ma and Genton (2000) show that the breakdown value forautocovariance estimators is (roughly) half that of thecorresponding covariance estimator.
• Consider n = 13, autocovariance at lag h = 2
• Working with X1:11 ±X3:13
• 4 contaminated observations, denoted by
• Leaves only 3 uncontaminated pairs, denoted by
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11X9 X10 X11 X12 X13
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13
X1 X2
h = 2
X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13X11 X12 X13
! γP (h) has roughly half the 13.4% breakdown value of Pn.
![Page 52: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/52.jpg)
Scale Autocovariance Conclusion
Asymptotic distribution under short range dependence
Result
If {Xi}i≥1 satisfy Assumption 1 (SRD) then the sequences{Xi +Xi+h}i≥1 and {Xi −Xi+h}i≥1 also satisfy Assumption 1,hence we can use the previous results and apply the functionaldelta method to show,
√n(γP (h)− γ(h))
D−→ N (0, σ2(h)),
where
σ2(h) = E[IC2((X1, X1+h), γP ,Φ)
]+ 2
∑k≥1
E [IC((X1, X1+h), γP ,Φ) IC((Xk+1, Xk+1+h), γP ,Φ)] .
![Page 53: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/53.jpg)
Scale Autocovariance Conclusion
Asymptotic distribution under long range dependence
Result
Under Assumption 2 (LRD),
1. D > 1/2 (same limiting distribution as SRD):
√n(γP (h)− γ(h))
D−→ N (0, σ2(h)).
2. D < 1/2 (based on conjectured result):
k(D)nD
L(n)(γP (h)−γ(h))
D−→ γ(0) + γ(h)
2(Z2,D(1)− Z2
1,D(1)),
where k(D) = Beta((1−D)/2, D), Z1,D is the standard fBmprocess, Z2,D is the Rosenblatt process and
L(n) = 2L(n)+L(n+h)(1+h/n)−D+L(n−h)(1−h/n)−D.
![Page 54: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/54.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
0.95 1.00 1.05 1.10First order autocovariance estimate
Den
sity
Pn
87%SD Relative Eff.
Qn 83%IQR 39%MAD 39%
![Page 55: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/55.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
0.95 1.00 1.05 1.10First order autocovariance estimate
Den
sity
Pn 87%SD Relative Eff.
Qn 83%IQR 39%MAD 39%
![Page 56: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/56.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
0.95 1.00 1.05 1.10First order autocovariance estimate
Den
sity
Pn 87%SD Relative Eff.
Qn 83%
IQR 39%MAD 39%
![Page 57: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/57.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
0.95 1.00 1.05 1.10First order autocovariance estimate
Den
sity
Pn 87%SD Relative Eff.
Qn 83%IQR 39%
MAD 39%
![Page 58: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/58.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.1, 0); D = 0.8; n = 5000
0
10
20
0.95 1.00 1.05 1.10First order autocovariance estimate
Den
sity
Pn 87%SD Relative Eff.
Qn 83%IQR 39%MAD 39%
![Page 59: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/59.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0.0
0.5
1.0
1.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5First order autocovariance estimate
Den
sity
Pn
94%SD Relative Efficiency
Qn 94%IQR 82%MAD 83%
![Page 60: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/60.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0.0
0.5
1.0
1.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5First order autocovariance estimate
Den
sity
Pn 94%SD Relative Efficiency
Qn 94%IQR 82%MAD 83%
![Page 61: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/61.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0.0
0.5
1.0
1.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5First order autocovariance estimate
Den
sity
Pn 94%SD Relative Efficiency
Qn 94%
IQR 82%MAD 83%
![Page 62: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/62.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0.0
0.5
1.0
1.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5First order autocovariance estimate
Den
sity
Pn 94%SD Relative Efficiency
Qn 94%IQR 82%
MAD 83%
![Page 63: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/63.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) ARFIMA(0, 0.45, 0); D = 0.1; n = 5000
0.0
0.5
1.0
1.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5First order autocovariance estimate
Den
sity
Pn 94%SD Relative Efficiency
Qn 94%IQR 82%MAD 83%
![Page 64: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/64.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) with 2% contamination at 5
0.0
0.5
1.0
1.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5First order autocovariance estimate
Den
sity
Pn
SD
QnIQRMAD
![Page 65: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/65.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) with 2% contamination at 6
0.0
0.5
1.0
1.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5First order autocovariance estimate
Den
sity
Pn
SD
QnIQRMAD
![Page 66: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/66.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) with 2% contamination at 7
0.0
0.5
1.0
1.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5First order autocovariance estimate
Den
sity
Pn
SD
QnIQRMAD
![Page 67: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/67.jpg)
Scale Autocovariance Conclusion
Empirical densities γ•(1) with 2% contamination at 8
0.0
0.5
1.0
1.5
1.5 2.0 2.5 3.0 3.5 4.0 4.5First order autocovariance estimate
Den
sity
Pn
SD
QnIQRMAD
![Page 68: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/68.jpg)
Scale Autocovariance Conclusion
Outline
The robust scale estimator Pn
Autocovariance estimation using Pn
Conclusion and key references
![Page 69: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/69.jpg)
Scale Autocovariance Conclusion
Summary
1. Aim
• Efficient and robust scale and autocovariance estimation.
2. Method
• Pn scale estimator transformed using the GK identity.
3. Results
• 86% asymptotic efficiency at the Gaussian and highasymptotic efficiency at heavier tailed distributions.
• Robustness properties transfer from Pn to γP .
• In certain LRD settings robust estimators have very highefficiencies relative to the standard deviation.
![Page 70: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/70.jpg)
Scale Autocovariance Conclusion
Summary
1. Aim
• Efficient and robust scale and autocovariance estimation.
2. Method
• Pn scale estimator transformed using the GK identity.
3. Results
• 86% asymptotic efficiency at the Gaussian and highasymptotic efficiency at heavier tailed distributions.
• Robustness properties transfer from Pn to γP .
• In certain LRD settings robust estimators have very highefficiencies relative to the standard deviation.
![Page 71: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/71.jpg)
Scale Autocovariance Conclusion
Summary
1. Aim
• Efficient and robust scale and autocovariance estimation.
2. Method
• Pn scale estimator transformed using the GK identity.
3. Results
• 86% asymptotic efficiency at the Gaussian and highasymptotic efficiency at heavier tailed distributions.
• Robustness properties transfer from Pn to γP .
• In certain LRD settings robust estimators have very highefficiencies relative to the standard deviation.
![Page 72: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/72.jpg)
Scale Autocovariance Conclusion
References
Genton, M. and Ma, Y. (1999).
Robustness properties of dispersion estimators.
Statistics & Probability Letters, 44(4):343–350.
Gnanadesikan, R. and Kettenring J. R. (1972).
Robust estimates, residuals and outlier detection with multiresponse data
Biometrics, 28(1):81–124.
Hampel, F. (1974).
The influence curve and its role in robust estimation.
Journal of the American Statistical Association, 69(346):383–393.
Levy-Leduc, C., Boistard, H., Moulines, E., Taqqu, M.S. and Reisen, V.A.(2011).
Robust estimation of the scale and of the autocovariance function ofGaussian short- and long-range dependent processes
Journal of Time Series Analysis, 32(2):135–156.
![Page 73: Robust scale and autocovariance estimation](https://reader030.fdocuments.us/reader030/viewer/2022012414/616dfbd404eea8208310b7d8/html5/thumbnails/73.jpg)
Scale Autocovariance Conclusion
References
Ma, Y. and Genton, M. (2000).
Highly robust estimation of the autocovariance function
Journal of Time Series Analysis, 21(6):663–684.
Serfling, R. J. (1984).
Generalized L-, M -, and R-statistics.
The Annals of Statistics, 12(1):76–86.
Tarr, G., Muller, S. and Weber, N.C., (2012).
A robust scale estimator based on pairwise means.
Journal of Nonparametric Statistics, 24(1):187–199.
Wu, W.B., (2005).
On the Bahadur representation of sample quantiles for dependentsequences.
Annals of Statistics, 33(4):1934–1963.