1. Statistical Moeling of Extreme
Values: Basic Theory and Its
Implementation in Open Source
Programing Environment R
Nader Tajvidi
Department of Mathematical Statistics
Lund Institute of Technology
Box 118
SE-22100 Lund
Sweden
August 6, 2010
Khon Kaen University
2. Outline
• Some examples of application of extreme value
theory
• Univariate extreme value distributions
• Characterisation of multivariate extreme value
distributions
• Bivariate extreme value distributions
• Parametric models for the dependence function
• Parametric and nonparametric estimation of the
dependence function
• Monte Carlo approximations to mean integrated
squared errors of parametric and nonparametric
estimators
• Application to Australian temperature data
Khon Kaen University August 6, 2010
3. Annual maximum sea levels at Port Pirie, South
Australia
4.6
4.4
4.2
4.0
Sea−Level (meters)
3.8
3.6
1930 1940 1950 1960 1970 1980
Year
Khon Kaen University August 6, 2010
4. Breaking strengths of glass fibers
Histogram of breaking strengths of glass fibers
Percent of Total 30
20
10
0
0.5 1.0 1.5 2.0
Breaking Strength
Density plot of breaking strengths of glass fibers
1.5
Density
1.0
0.5
0.0
0.5 1.0 1.5 2.0 2.5
Breaking Strength
Khon Kaen University August 6, 2010
5. Annual maximum sea levels at Fremantle, Western
Australia
1.8
1.6
Sea−Level (meters)
1.4
1.2
1900 1920 1940 1960 1980
Year
Khon Kaen University August 6, 2010
6. Annual maximum sea levels at Fremantle, Western
Australia, versus mean annual value of Southern
Oscillation Index
1.8
1.6
Sea−Level (meters)
1.4
1.2
−1 0 1 2
SOI
Khon Kaen University August 6, 2010
7. Comparing Port Pirie and Fremantle datasets
4.6
4.4
4.2
4.0
3.8
Sea−Level (meters)
3.6
1930 1940 1950 1960 1970 1980
Year
1.8
1.6
1.4
Sea−Level (meters)
1.2
1900 1920 1940 1960 1980
Year
Khon Kaen University August 6, 2010
8. Daily closing prices of the Dow Jones Index
dowjones
11000
9000
Index
7000
5000
Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1
1995 1996 1997 1998 1999 2000 2001
Year
Khon Kaen University August 6, 2010
9. Log-daily returns of the Dow Jones Index
log.daily.return
0.04
0.02
0.00
−0.02
Index
−0.04
−0.06
Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1
1995 1996 1997 1998 1999 2000 2001
Year
Khon Kaen University August 6, 2010
10. Dow Jones Index data
dowjones log.daily.return
0.04
11000
0.02
0.00
9000
−0.02
Index
Index
7000
−0.04
−0.06
5000
Q1 Q1 Q1 Q1 Q1 Q1 Q1 Q1
1995 1999 1995 1999
Year Year
Khon Kaen University August 6, 2010
11. Windstorm loss data
• Windstorm losses of the Swedish insurance group
L¨nsf¨rs¨kringar during the period 1982 to 1993
a o a
• The database contains:
– The individual amounts of all claims
– The place and time of the claims
– The type of the claim
• 46 storm events, with a total claimed amount of
510 million Swedish crowns (MSEK)
• Farm insurance comprising of approximately 65% of
the total amount
• All values were corrected for inflation
• No adjustments for portfolio changes
Khon Kaen University August 6, 2010
12. Windstorm losses 1982-1993
Feb
92
Dec 88
4
n8
Ja
Ja
83
n
93
Jan
Questions:
• How can we predict the size of the next very severe
storm?
• How much reinsurance does a company need to
buy?
Khon Kaen University August 6, 2010
13. Windstorm losses which exceed the level
u = 0.9 MSEK, for 1982 – 1993
Jan 93
120
100
storm loss (in MSEK)
80
60
Jan 83
Jan 84
40
Dec 88
Feb 92
20
0
0 10 20 30 40
storm number
Khon Kaen University August 6, 2010
14. Australian temperature data
• A very large dataset on annual maximum and
minimum average daily temperatures at 224 stations
across Australia
Queensland
New South Wales
Victoria
South Australia
West Australia
Northern Territory
Tasmania
Khon Kaen University August 6, 2010
15. Annual maximum temperatures in
Victoria, Australia
• The maximum value, over all 34 weather stations
that were operating in the state of Victoria from
1910 to 1993, of annual temperatures (in degrees
Celsius) during this period.
Khon Kaen University August 6, 2010
16. 39 33.6
33.5
35
33.4
ˆ
μ
33 33.3
Temperature
31 33.2
1920 1940 1960 1980 0.0 0.2 0.4 0.6 0.8 1.0
Year t
0.70 -0.1
0.65 -0.2
0.60
ˆ
σ
-0.3
ˆ
γ
0.55
-0.4
0.50
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
t t
Khon Kaen University August 6, 2010
17. Average annual maximum temperature
• The average annual maximum is derived by taking
the mean of maximum annual temperature readings
at 224 weather stations across Australia in the
period 1890–1993.
Queensland
New South Wales
Victoria
South Australia
West Australia
Northern Territory
Tasmania
Khon Kaen University August 6, 2010
18. Location and scale estimates with Gaussian fit
29.0
28.5
28.5
28.0 28.0
27.5
ˆ
μ
27.0 27.5
Temperature
26.5
26.0 27.0
1900 1940 1980 0.0 0.2 0.4 0.6 0.8 1.0
Year t
-136.3
0.50 j
0.45
0.40 -136.5
ˆ
σ
0.35
C1
0.30
0.25 -136.7
0.0 0.2 0.4 0.6 0.8 1.0 0.12 0.16 0.20 0.24
t h
Khon Kaen University August 6, 2010
19. Another application to Australian
temperature data
• maximum annual values of average daily
temperature measurements at two meteorological
stations, Leonora (latitude 28.53, longitude 121.19)
and Menzies (latitude 29.42, longitude 121.02), in
Western Australia during the period 1898–1993.
27
Menzies
26
25
24
25 26 27 28
Leonora
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20. Annual Maximum Wind Speeds in 1944-1983
80
70
60
50
Annual Maximum Wind Speed (konts) at Hartford (CT)
40 45 50 55 60 65
Annual Maximum Wind Speed (konts) at Albany (NY)
Khon Kaen University August 6, 2010
21. Concurrent measurements of wave and surge height in
south west England
0.8
0.6
0.4
0.2
Surge (m)
0.0
−0.2
0 2 4 6 8 10
Wave Height (m)
Khon Kaen University August 6, 2010
22. The framework
1. A proper mathematical model has to be chosen in
each case.
• parametric; best if the model is correct
• non parametric; can not be used for extrapolation
outside the observed values
• semi parametric; very flexible (main subject of
this talk)
2. Parameters in each model have to be estimated
based on the historical data. Which method should
be used?
3. These estimates are our “best guesses” of the
process which is being analyzed. How to specify
uncertainty in the estimates?.
4. Goodness of fit. Does the model give a good
representation of the historical data?
5. How can we reduce the uncertainties in our models?
How can extra information be incorporated in the
models?
Khon Kaen University August 6, 2010
23. Univariate Extreme Value Distributions
X1, X2, . . ., Xn, iid X ∼ F (x)
Mn = max(X1, X2, . . ., Xn), n ∈ N
an > 0 and bn ∈ R
Mn − bn
lim P ( ≤ x) = lim F n(anx + bn) = G(x)
n→∞ an n→∞
G(x) non-degenerate
F ∈ D(G)
F (x) belongs to domain of attraction of G(x)
Khon Kaen University August 6, 2010
24. Type I:
0 x<0
Φα(x) =
exp(−x−α) x ≥ 0
Type II:
exp(−(−xα)) x < 0
Ψα(x) =
1 x≥0
Type III:
Λ(x) = exp(−e−x) x∈R
Generalised Extreme Value Distribution
x−μ γ
1
G(x; γ, μ, σ) = exp{−(1 − γ )+ }
σ
Khon Kaen University August 6, 2010
25. Multivariate Extreme Value Distributions
(1) (d)
{Xn, n ≥ 1} = {(Xn , . . . , Xn ), n ≥ 1}
X ∼ F (x) iid
n n
(1) (d) (1) (d)
Mn = (Mn , . . . , Mn ) = ( Xj , . . . , Xj )
j=1 j=1
(i) (i)
σn > 0, un ∈ R
P [(Mn − u(i))/σn ≤ x(i), 1 ≤ i ≤ d] =
(i)
n
(i)
F n(σn x(1) + u(1), . . . , σn x(d) + u(d)) → G(x)
(1)
n
(d)
n
marginal Gi of G non-degenerate
F ∈ D(G)
F (x) belongs to domain of attraction of G(x)
Khon Kaen University August 6, 2010
26. Characterisation of Multivariate Extreme
Value Distributions
P [(Mn − u(i))/σn ≤ x(i), 1 ≤ i ≤ d] =
(i)
n
(i)
F n(σn x(1) + u(1), . . . , σn x(d) + u(d)) → G(x)
(1)
n
(d)
n
Definition. A df G in Rd is called max-stable if for
every t > 0
Gt(x) = G(α(1)(t)x(1)+β (1)(t), . . . , α(d)(t)x(d)+β (d)(t)).
Definition. A df G in Rd is called max-infinitely
divisible (max-id) if F t(x1, . . . , xd) is a df for every
t > 0.
G(∞, ∞, . . . , xi, . . . , ∞) = Φ1(xi) = exp(−x−1 )
i
G∗(x) is a MEVD with Φ1 marginals
Khon Kaen University August 6, 2010
27. Characterisation of Max-id and
Max-Stable Distributions
F max-id iff for a Radon measure μ on
E := [k, ∞] {k}, k ∈ [−∞, ∞)d
exp{−μ[−∞, y]c} y ≥ k
F (y) =
0 otherwise
The measure μ is called an exponent measure.
G(∞, ∞, . . . , xi, . . . , ∞) = Φ1(xi) = exp(−x−1 )
i
G∗(x) is a MEVD with Φ1 marginals if for a finite
measure S on
ℵ = {y : y = 1}
d
a(i)
G∗(x) = exp − (i)
S(da)
ℵ i=1 x
a(i)S(da) = 1, 1 ≤ i ≤ d
ℵ
Khon Kaen University August 6, 2010
28. Bivariate Extreme Value Distributions
−μ∗ [0,(x,y)]c
G∗(x, y) = e
1 1 x
μ∗[0, (x, y)] = ( + )A(
c
)
x y x+y
1
A(w) = max{q(1 − w), (1 − q)w}S(dq)
0
A(w) is called dependence function.
1 1
qS(dq) = (1 − q)S(dq) = 1
0 0
• A(0) = A(1) = 1
• max{w, 1 − w} ≤ A(w) ≤ 1
• A(w) is convex for w ∈ [0, 1]
Khon Kaen University August 6, 2010
29. Some examples of the dependence function
1.0
0.9
0.8
A(w)
0.7
0.6
Mixed
Generalised mixed
0.5 Asym. mixed
0.0 0.2 0.4 0.6 0.8 1.0
w
Khon Kaen University August 6, 2010
30. Parametric Models for the Dependence
Function
1. The mixed model
1 1 θ
μ∗([0, (x, y)] ) = + −
c
, 0≤θ≤1
x y x+y
A(w) = θw2 − θw + 1, 0≤θ≤1
• θ = 0 gives independent case
• Complete dependence is not possible
2. The logistic model
μ∗([0, (x, y)]c) = (x−r + y −r )1/r , r≥1
A(w) = {(1 − w)r + wr }1/r , r≥1
• r = 1 gives independent case
• r = +∞ gives complete dependence
Khon Kaen University August 6, 2010
31. The Generalised Symmetric Mixed Model
1 1
c 1
μ∗([0, (x, y)] ) = + − k( p )1/p, (0 ≤ k ≤ 1, p ≥ 0)
x y x + yp
k
A(w) = 1 − 1
−p p
(1 − w) + w−p
• Independence for k = 0 or p = 0
• Complete dependence can be obtained with k = 1 and p = ∞ (Not
possible in the symmetric or asymmetric mixed model)
Khon Kaen University August 6, 2010
32. The Generalised Symmetric Logistic Model
1 1 k 1
c p,
μ∗([0, (x, y)] ) = ( p + p + p/2
) (0 < k ≤ 2(p − 1), p ≥ 2)
x y (xy)
1
p p
p 2
A(w) = (1 − w) + wp + k ((1 − w) w)
• k = 2 gives the symmetric logistic model
• Independence corresponds to p = 2 and k = 2
• Complete dependence for k = 2 and p = +∞
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33. The Parameter Region for the Generalised Symmetric
Logistic Model
k
logistic model
2
equivalent models
0
2 4 6 p
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34. The Asymmetric Mixed Model
c x3 + 3 x2 y − 2 φ x2 y − θ x2 y + 3 x y 2 − φ x y 2 − θ x y 2 + y 3
μ∗([0, (x, y)] ) =
x y (x + y)2
A(w) = φw3 + θw2 − (θ + φ)w + 1, (θ ≥ 0, θ + 2φ ≤ 1, θ + 3φ ≥ 0)
• Symmetric mixed model for φ = 0
• Independent case for θ = φ = 0 (Complete dependence is not possible)
• The parameter φ stands for non-symmetry in the model
Khon Kaen University August 6, 2010
35. The Asymmetric Logistic Model
φr xr +θ r y r 1 φr xr +θ r y r 1
(1 − φ) x + (1 − θ) y + x ( (x+y)r ) r + y ( (x+y)r ) r
c
μ∗([0, (x, y)] ) =
xy
A(w) = {(θ(1 − w))r + (φw)r }1/r + (θ − φ)w + 1 − θ, (0 ≤ θ, φ ≤ 1, r ≥ 1)
• For θ = φ = 1 this model reduces to the corresponding symmetric logistic
model which gives the diagonal case for r = +∞.
• Independence is obtained for θ = 0 and for φ = 0 or r = 1.
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36. Estimation of the dependence function
• Nonparametric methods
1. Pickands estimator (1981)
2. Cap´ra`, Foug`res and Genest’s estimator (1997)
e a e
• Maximum likelihood based on parametirc models
New Nonparametric Methods:
1. Convex hull of modified Pickands estimator
2. Constrained smoothing splines
Khon Kaen University August 6, 2010
37. Pickands estimator
• Suppose (X, Y ) has a bivariate extreme value
distribution with exponential margins.
• min{X/(1 − w), Y /w} has an exponential
distribution with mean 1/A(w).
• the maximum likelihood estimator of A(w) is
n −1
An(w) = n min {Xi/(1 − w), Yi/w}
i=1
• For each 0 ≤ w ≤ 1, 1/An(w) is an unbiased and
strongly consistent estimator of 1/A(w).
• δn(w) = n1/2 1/An(w) − 1/A(w) satisfies the
central limit theorem in C(0, 1), B ; see Deheuvels,
P. (1991).
Khon Kaen University August 6, 2010
38. Pickands estimates for 100 simulated data from Logistic
model with r = 1.1 and r = 1.3
1.2
1.0
1.0
A(w)
A(w)
0.8
0.8
0.6 0.6
Pickands Pickands
Logistic, r = 1.1 Logistic, r = 1.3
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
w w
Khon Kaen University August 6, 2010
39. Pickands estimates for 100 simulated data from Logistic
model with r = 1.6 and r = 2
1.0
1.0
0.9
0.9
0.8 0.8
A(w)
A(w)
0.7 0.7
0.6 0.6
Pickands Pickands
0.5 Logistic, r = 1.6 0.5 Logistic, r = 2
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
w w
Khon Kaen University August 6, 2010
40. Cap´ra`, Foug`res and Genest’s
e a e
estimator
• Copula for a bivariate extreme value distribution
with marginals F (x) and G(y)
C(u, v) = P {F (x) ≤ u, G(y) ≤ v}
log(u)
= exp log(uv)A
log(uv)
• Ui, Vi ≡ {F (Xi), G(Yi)}(1 ≤ i ≤ n)
log(Ui)
• Pseudo-observations Zi = log(UiVi ) (1 ≤ i ≤ n)
• H(z) = P (Zi ≤ z) = z + z(1 − z)D(z) where
D(z) = A (z)/A(z) for all 0 ≤ z ≤ 1
t H(z)−z 1 H(z)−z
• A(t) = exp 0 z(1−z)
dz = exp − t z(1−z)
dz
t Hn (z)−z
1. A0 (t) = exp
n 0 z(1−z)
dz
Khon Kaen University August 6, 2010
41. 1 Hn(z)−z
2. A1 (t) = exp −
n t z(1−z)
dz
• log An(t) = p(t) log A0 (t) + {1 − p(t)} log A1 (t)
n n
definition of the estimator:
Denote the ordered values of Zi by Z(1), . . . , Z(n) and
define
i 1/n
Qi = Z(k)/(1 − Z(k)) (1 ≤ i ≤ n).
k=1
Then An can be written as
⎧
⎪ (1 − t)Q1−p(t)
⎨ n 0 ≤ t ≤ Z(1)
1−p(t) −1
An(t) = ti/n(1 − t)1−i/nQn Qi Z(i) ≤ t ≤ Z(i+1)
⎪
⎩ −p(t)
tQn Z(n) ≤ t ≤ 1
• An(0) = An(1) = 1 if p(0) = 1 − p(1) = 1.
Khon Kaen University August 6, 2010
42. Cap´ra`’s estimates for 100 simulated data from Logistic
e a
model with r = 1.1 and r = 1.3
1.0
1.0
0.9
0.9
0.8 0.8
A(w)
A(w)
0.7 0.7
0.6 0.6
Caperaa Caperaa
0.5 Logistic, r = 1.1 0.5 Logistic, r = 1.3
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
w w
Khon Kaen University August 6, 2010
43. Cap´ra`’s estimates for 100 simulated data from Logistic
e a
model with r = 1.6 and r = 2
1.0 1.0
0.9 0.9
0.8 0.8
A(w)
A(w)
0.7 0.7
0.6 0.6
Caperaa Caperaa
0.5 Logistic, r = 1.6 0.5 Logistic, r = 2
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
w w
Khon Kaen University August 6, 2010
44. Modified Pickands estimator
(1) (2)
• Let Yi = (Yi , Yi ) for 1 ≤ i ≤ n be independent
and identically extreme value distributed random
variables with exponential margins.
¯
• Put Y ( ) = n−1
( )
Yi and Yi
( ) ( )¯
= Yi /Y ( )
for
i
= 1, 2.
(1) (2)
• B(u) ≡ n−1 i=1 min Yi /(1 − u), Yi /u is
n
uniformly root-n consistent for B(u) ≡ A(u)−1.
1. The estimator of the dependence function passes
through the points (0, 1) and (1, 1), and has
gradients −1 and 1 at these respective points.
ˆ
2. B(u) ≤ min{1/(1−u), 1/u} so A ≡ B −1 lies above
the lower boundary of the trianglur area.
˜ ˆ
3. The greatest convex minorant, A, of A satisfies all
necessary conditions for a dependence function.
Khon Kaen University August 6, 2010
45. Modified Pickands estimates for 100 simulated data
from Logistic model with r = 1.1 and r = 1.3
1.0 1.0
0.9 0.9
0.8 0.8
A(w)
A(w)
0.7 0.7
0.6 0.6
chull of modified Pickand chull of modified Pickand
Modified Pickands Modified Pickands
0.5 Logistic, r = 1.1 0.5 Logistic, r = 1.3
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
w w
Khon Kaen University August 6, 2010
46. Modified Pickands estimates for 100 simulated data
from Logistic model with r = 1.6 and r = 2
1.0 1.0
0.9 0.9
0.8 0.8
A(w)
A(w)
0.7 0.7
0.6 0.6
chull of modified Pickand chull of modified Pickand
Modified Pickands Modified Pickands
0.5 Logistic, r = 1.6 0.5 Logistic, r = 2
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
w w
Khon Kaen University August 6, 2010
47. Constrained smoothing splines
ˆ
• A may be approximated by a spline that is
constrained to satisfy all the necessary conditions
on the dependence function.
• Choose regularly spaced points 0 = t0 < . . . <
tm = 1 in the interval [0, 1].
˜
• Given a smoothing parameter s > 0, take As to be
a polynomial smoothing spline of degree 3 or more
which minimises
m 1
ˆ ˜
{A(tj ) − As(tj )}2 + s ˜
As (t)2 dt ,
j=1 0
subject to
1. ˜ ˜
As(0) = As(1) = 1
2. ˜ ˜
As(0) ≥ −1 and As(1) ≤ 1
3. ˜
As ≥ 0 on [0, 1].
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48. Smoothed spline of modified Pickands estimates for 100
simulated data from Logistic model with r = 1.1 and
r = 1.3
1.0 1.0
0.9 0.9
0.8 0.8
A(w)
0.7 A(w) 0.7
0.6 0.6
Smoothed spline Smoothed spline
Modified Pickands Modified Pickands
0.5 Logistic, r = 1.1 0.5 Logistic, r = 1.3
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
w w
Khon Kaen University August 6, 2010
49. Smoothed spline of modified Pickands estimates for 100
simulated data from Logistic model with r = 1.6 and
r=2
1.0 1.0
0.9 0.9
0.8 0.8
A(w)
0.7 A(w) 0.7
0.6 0.6
Smoothed spline Smoothed spline
Modified Pickands Modified Pickands
0.5 Logistic, r = 1.6 0.5 Logistic, r = 2
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
w w
Khon Kaen University August 6, 2010
50. Which model to use in practice?
• maximum likelihood of parametric models, e.g.
1. symmetric mixed model
2. symmetric logistic model
3. asymmetric mixed model
4. asymmetric logistic model
5. generalised symmetric logistic model
6. generalised asymmetric mixed model
• Nonparametirc methods including
1. the Pickands (1981, 1989) estimator
2. the convex hull of Pickands’ estimator
3. the estimator proposed by Cap´ra`, Foug`res and
e a e
Genest (1997)
4. the convex hull of the latter
5. our modification of Pickands’ estimator
6. the convex hull of the latter
• constrained smoothing splines fitted to any of these
nonparametric estimators
Khon Kaen University August 6, 2010
51. Khon Kaen University
1.0
0.9
0.8
A(w)
Smoothed spline
Logistic
Mixed
0.7 Generalised logistic
Generalised mixed
Asym. logistic
Asym. mixed
0.6 Pickands
chull of modified Pickand
Caperaa
Modified Pickands
0.5 Logistic, r = 1.1
0.0 0.2 0.4 0.6 0.8 1.0
w
August 6, 2010
52. Khon Kaen University
1.0
0.9
0.8
A(w)
Smoothed spline
Logistic
Mixed
0.7 Generalised logistic
Generalised mixed
Asym. logistic
Asym. mixed
0.6 Pickands
chull of modified Pickand
Caperaa
Modified Pickands
0.5 Logistic, r = 1.3
0.0 0.2 0.4 0.6 0.8 1.0
w
August 6, 2010
53. 1.0
Khon Kaen University
0.9
0.8
A(w)
Smoothed spline
Logistic
0.7 Mixed
Generalised logistic
Generalised mixed
Asym. logistic
Asym. mixed
0.6 Pickands
chull of modified Pickand
Caperaa
Modified Pickands
0.5 Logistic, r = 1.6
0.0 0.2 0.4 0.6 0.8 1.0
w
August 6, 2010
54. 1.0
Khon Kaen University
0.9
0.8
A(w)
Smoothed spline
Logistic
0.7 Mixed
Generalised logistic
Generalised mixed
Asym. logistic
Asym. mixed
0.6 Pickands
chull of modified Pickand
Caperaa
Modified Pickands
0.5 Logistic, r = 2
0.0 0.2 0.4 0.6 0.8 1.0
w
August 6, 2010
55. Monte Carlo approximations to mean integrated squared
errors, multiplied by 105
n = 25 n = 50 n = 100
method r=1 r=2 r=3 r=1 r=2 r=3 r=1 r=2 r=3
logistic 197 64 14 110 34 8 42 14 4
Pickands 5614 2829 3331 2034 1547 1261 1172 712 567
convex hull of Pickands 7229 2611 2775 2588 1388 1049 1430 671 477
Cap´ra` et. al.
e a 889 102 35 568 49 20 307 29 10
convex hull of Cap´ra` et. al.
e a 1188 95 41 666 57 25 373 32 12
modified Pickands 1351 138 33 614 77 18 366 37 11
convex hull of modified Pickands 1861 139 46 815 70 24 453 38 15
smoothed spline of
Pickands 784 919 1020 396 728 487 215 334 220
convex hull of Pickands 525 769 1055 282 637 490 135 327 230
Cap´ra` et. al.
e a 303 82 21 177 37 12 97 24 8
convex hull of Cap´ra` et. al.
e a 286 66 22 167 39 14 97 26 11
modified Pickands 447 104 21 240 62 12 130 31 9
convex hull of modified Pickands 401 73 24 232 49 16 107 30 15
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56. Australian temperature data
• A very large dataset on annual maximum and
minimum average daily temperatures at 224 stations
across Australia
Queensland
New South Wales
Victoria
South Australia
West Australia
Northern Territory
Tasmania
Khon Kaen University August 6, 2010
57. Application to Australian temperature
data
• maximum annual values of average daily
temperature measurements at two meteorological
stations, Leonora (latitude 28.53, longitude 121.19)
and Menzies (latitude 29.42, longitude 121.02), in
Western Australia during the period 1898–1993.
27
Menzies
26
25
24
25 26 27 28
Leonora
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58. Logistic models for the dependence function fitted by
maximum likelihood to the temperature data
1.0
0.9
0.8
A(w)
0.7
0.6
Logistic
Generalised logistic
Asym. logistic
0.5 Modified Pickands
0.0 0.2 0.4 0.6 0.8 1.0
w
Khon Kaen University August 6, 2010
59. Mixed models for the dependence function fitted by
maximum likelihood to the temperature data
1.0
0.9
0.8
A(w)
0.7
0.6
Mixed
Generalised mixed
Asym. mixed
0.5 Modified Pickands
0.0 0.2 0.4 0.6 0.8 1.0
w
Khon Kaen University August 6, 2010
60. Estimating a bivariate extreme-value
distribution function
• Let X = (X (1), X (2)) have a bivariate extreme-
value distribution F .
• There exist monotone increasing transformations
Tj = Tj (·|θj ) such that (T1(X (1)), T2(X (2))) has
distribution function G0.
(1) (2)
• Given a sample {Xi = (Xi , Xi ), 1 ≤ i ≤ n},
ˆ
compute a root-n consistent estimator θj of θj from
(j)
the marginal data Xi , 1 ≤ i ≤ n.
ˆ
• Put Tj = Tj (·|θj ) and
n
( ) ( ) ( )
Yj = T (Xj ) n−1 T (Xi ) .
i=1
ˆ ˆ ˆ
• F x(1), x(2)) = G0 T1 x(1) θ1 , T2 x(2) θ2 is
root-n consistent for F .
Khon Kaen University August 6, 2010
61. Distribution function estimate and semi-infinite
prediction regions corresponding to nominal levels
α = 0.9, 0.95 and 0.99
(a)
28.5
28.0
0.99
fs Menzies 27.5 0.95
0.9
27.0
26.5
Menzies
Leonora
27.0 27.5 28.0 28.5 29.0 29.5 30.0
Leonora
Khon Kaen University August 6, 2010
62. Compact bivariate prediction regions
Construct compact prediction regions by profiling the estimator
˜ ∂2 ˆ ˆ
fs(x) = G1 t(1), t(2)
∂x(1) ∂x(2)
= ˆ ˆ ˆ ˆ
T1 x(1) θ1) T2 x(2) θ2) G1 t(1), t(2)
ˆ
t(2) ˆ
t(1) ˆ
t(2)
× ˜
As (1) + (1) ˜
A
ˆ
t + t(2)ˆ ˆ
t +t ˆ ˆ ˆ
(2) s t(1) + t(2)
ˆ
t(2) ˆ
t(2) ˆ
t(2)
× ˜s
A (1) − (1) ˜s
A (1)
ˆ
t + t(2)ˆ t
ˆ + t(2)ˆ ˆ
t + t(2)ˆ
ˆ ˆ
t(1) t(2) ˆ
t(2)
+ A ˜
ˆ
(t ˆ
(1) + t(2) )3 s ˆ ˆ
t(1) + t(2)
of the density, f , of X.
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63. How to choose s
• CV (s) = ˜
fs(x)2 dx − 2 n−1
n ˜
f−i,s(Xi).
i=1
• CV (s) is an almost-unbiased approximation to
˜2 ˜
E(fs − 2fsf ).
• The value of s that results from minimising CV (s)
˜
will asymptotically minimise E(fs − f )2.
• To construct prediction regions, define
˜
R(u) ≡ x : fs(x) ≥ u , β(u) = ˜
fs(x) dx .
e
R(u)
• Given a prediction level α, let u = uα denote the
˜
solution of β(u) = α. Then, R(˜α) is a nominal
u
α-level prediction region for a future value of X.
Khon Kaen University August 6, 2010
64. Cross-validation criterion CV (s) and spline-smoothed
˜
dependence function estimate As for s = 0.05, with the
unsmoothed, modified Pickands estimate
(a)
-0.228 1.0
-0.230
0.9
-0.232
-0.234 0.8
A(w)
CV
-0.236
0.7
-0.238
-0.240 0.6
Smoothed spline
-0.242 chull of modified Pickand
0.5 Modified Pickands
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.2 0.4 0.6 0.8 1.0
s w
Khon Kaen University August 6, 2010
65. Plot of spline-smoothed density estimate
˜
fs
fs
Menzies
Leonora
Khon Kaen University August 6, 2010
66. Compact bootstrap calibrated prediction regions with
nominal levels α = 0.85 and 0.90
28.0
27.5
27.0
26.5
Menzies
26.0
25.5
0.85
25.0
0.90
26 27 28 29
Leonora
Khon Kaen University August 6, 2010
67. Bootstrap calibration
¯ ˜ ˜
• Take A = A or Aλ in
¯ 1 1 ¯ x(1)
F (x) = exp − (1) + (2) A (1) .
x x x + x(2)
¯
• Compute the chosen region Rα, with nominal
coverage α, from the data X = {X1, . . . , Xn}.
¯
• By resampling from F conditional on X , compute
a new dataset X ∗ = {X1 , . . . , Xn}, and from it
∗ ∗
¯ ¯
calculate the analogue F ∗ of F , and then the
¯α ¯
analogue R∗ of Rα.
• Let γ(α) equal the probability, conditional on the
¯
data X , that a random 2-vector drawn from F lies
¯α
in R∗ .
• Let a = a(α) be the solution of γ(a) = α. Then,
ˆ
¯ˆ ¯
Ra(α) is the bootstrap-calibrated form of Rα.
Khon Kaen University August 6, 2010
68. Theoretical Properties
ˆ ˜
• A and its greatest convex minorant, A, are uniformly root-n consistent
for A:
ˆ ˜
sup |A(u) − A(u)| + sup |A(u) − A(u)| = Op n−1/2 .
0≤u≤1 0≤u≤1
• if the distribution H of Y (1)/(Y (1) + Y (2)) has a bounded density then,
for each ∈ (0, 1 ],
2
sup ˆ
|A (u) − A (u)| + sup ˜
|A (u) − A (u)| = Op n−1/2
≤u≤1− ≤u≤1−
ˆ ˆ ˜ ˜
• if A has three bounded derivatives then the biases of A, A , A, A are
O(n−1).
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69. Shape constrained smoothing using
smoothing splines
Given data {(ti, yi)}, ti ∈ [a, b] for i = 1, . . . , n, what
ˆ
is the behaviour of the solution g of the following
minimisation problem?
n b 2
2 (m)
minimise yi − g(ti ) +λ g (u) du,
i=1 a
(1a)
where g (r)(t) ≥ 0 t ∈ [a, b]. (1b)
References
[1] Mammen, E. and Thomas-Agnan, C. (1999),
Smoothing splines and shape restrictions,
Scandinavian Journal of Statististics, 26, 239–
252.
Khon Kaen University August 6, 2010
70. Proposed Estimator for m = 2 and r ≤ 2
For m = 2, the piecewise polynomial representation of a natural cubic
C 2-spline g is:
n
g(t) = I[ti,ti+1)(t)Si(t), (2a)
i=0
where Si(t) = ai + bi(t − ti) + ci(t − ti)2 + di(t − ti)3, 1 ≤ i ≤ n − 1,
(2b)
S0(t) = a1 + b1(t − t1) and Sn(t) = Sn−1(tn) + Sn−1(tn)(t − tn).
The coefficients in (2b) have to fulfill the following equations for g to be a
Khon Kaen University August 6, 2010
71. natural cubic C 2-spline:
Si−1(ti) = Si(ti) for i = 1, . . . , n
Si−1(ti) = Si(ti) for i = 1, . . . , n (3)
Si−1(ti) = Si (ti) for i = 1, . . . , n
A direct implementation would lead to an unnecessarily large quadratic
programming problem and we propose to use the value-second derivative
representation (see Green and Silverman, 1994, chapter2)for the actual
implementation.
For i = 1, . . . , n, define gi = g(ti) and γi = g (ti). By definition, a natural
cubic C 2-spline has γ1 = γn = 0. Let g denote the vector (g1, . . . , gn)T
and γ = (γ2, . . . , γn−1 )T . Note that for notational simplicity later on the
entries of γ are numbered in a non-standard way, starting at i = 2. The
vectors g and γ specify the natural cubic spline g completely.
Khon Kaen University August 6, 2010
72. However, not all possible vectors g and γ represent natural cubic splines.
To derive sufficient (and necessary) conditions for g and γ to represent a
cubic spline we define the following matrices Q and R. Define
hi = ti+1 − ti for i = 1, . . . , n − 1. Let Q be the n × (n − 2) matrix with
entries qi,j , for i = 1, . . . , n and j = 2, . . . , n − 1, given by
qj−1,j = h−1 ,
j−1 qj,j = −h−1 − h−1 ,
j−1 j and qj,j+1 = h−1 ,
j
for j = 2, . . . , n − 1, and qi,j = 0 for |i − j| ≥ 2. Note, that the columns
of Q are numbered in the same non-standard way as the entries of γ.
The (n − 2) × (n − 2) matrix R is symmetric with elements {ri,j }n−1
i,j=2
given by
ri,i = 1 (hi−1 + hi) for i = 2, · · · , n − 1,
3
ri,i+1 = ri+1,i = 1 hi
6 for i = 2, · · · , n − 2,
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73. and ri,j = 0 for |i − j| ≥ 2. Note, that R is strictly diagonal dominant
and, hence, it follows from standard arguments in numerical linear algebra,
that R is strictly positive-definite.
We are now able to state the following key result.
Proposition. The vectors g and γ specify a natural cubic spline g if and
only if the condition
QT g = Rγ (4)
is satisfied. If (4) is satisfied then we have
b
2
{g (t)} dt = γ T Rγ. (5)
a
For a proof see Green and Silverman (1994, section 2.5).
Khon Kaen University August 6, 2010
74. This result allows us to state problem (1a) as a quadratic programming
problem. Let y denote the (2n − 2)-vector (y1, . . . , yn, 0, . . . , 0)T , g the
T
(2n − 2)-vector g T , γ T , A the (2n − 2) × (n − 2)-matrix Q −RT ,
In the n × n unit matrix and
In 0
D= . (6)
0 λR
Then the solution of (1a) is given by the solution of the following
quadratic program:
minimise − yT g + 1 gT Dg,
2 (7a)
where AT g = 0. (7b)
We propose to use the algorithm of Goldfrab and Idnani (1982, 1983) to
solve (7).
Khon Kaen University August 6, 2010