Download Likelihood inference for generalized Pareto distribution

Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Likelihood inference for generalized Pareto
distribution
J. del Castillo1 and I. Serra1
1 Departament de Matemàtiques
Universitat Autònoma de Barcelona
EVT2013
Sep de 2013
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Table of contents
1
Introduction
2
Problem: Calibration of the GPD for likelihood inference
3
Solution: A good algorithm and a new methodology approach
4
Examples
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Table of contents
1
Introduction
2
Problem: Calibration of the GPD for likelihood inference
3
Solution: A good algorithm and a new methodology approach
4
Examples
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
History of EVT: onset of GPD
Fisher-Tippett Theorem (1928)
Let (X1 . . . Xn ) be a sequence of i.i.r.v, letMn = max{X
1 . . . Xn }. If
Mn −bn
(an , bn ) and F exist, such that limn→∞ P
≤ x = F(x), then F
an
belongs to the generalized extreme value distribution.
Pickands−Balkema−de Haan theorem (1974)
Let (X1 , . . . , Xn ) be a sequence of i.i.r.v., and let Fu be their
conditional excess distribution function then
Fu (y) → GPD(k, ψ)(y),
as
u→∞
where GPD is the generalized Pareto distribution.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Widely used
Davis, H. T. and Michael L. F. (1979). The Generalized Pareto Law as a Model for
Progressively Censored Survival Data. Biometrika
Davison, A. C.; Smith, R. L. (1990). Models for exceedances over high thresholds.
With discussion and a reply by the authors. JRSS-B
Embrechts, P. Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events
for Insurance and Finance. Springer-Verlag, Berlin.
McNeil, A. J., Frey, R. and Embrechts P. (2005). Quantitative Risk Management:
Concepts, Techniques and Tools. Princeton Univ. Press.
Coles. S. and Sparks (2006). Extreme value methods for modelling historical series
of large volcanic magnitudes. Statistics in Volcanology,
Rachev, S. T., Racheva-Iotova, B., Stoyanov, S. (2010). Capturing fat tails, in Risk.
Risk Management, Derivatives and Regulation,
Papastathopoulos, I. and Tawn, J.A. (2012). Extended generalised Pareto models for
tail estimation. J. Statist. Plann. Inference
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Generalized Pareto distribution
The GPD is a two-parameter distribution for non-negative variables
Probability density function
f (x; k, ψ) =
1
(1 − kx/ψ)1/k−1 ,
ψ
for ψ > 0, k ∈ R
The range of x is (0, ∞) for k ≤ 0 or (0, ψ/k) for k > 0.
Special cases: GPD contains exponential and uniform distribution
f (x; 0, ψ) =
1
exp(−x/ψ),
ψ
Serra, I.
f (x; 1, ψ) = x/ψ
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Submodels
For k < 0 the Pareto submodel (range (0, ∞))
For k = 0 the exponential distribution
For k ∈ (0, 1) a family of decreasing probability densities
functions with finite support (0, ψ/k)
For k = 1 the uniform distribution
For k > 1 a family of increasing densities with finite support.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Main use
From Balkema-DeHaan Theorem, the tails were classified for k < 0,
k = 0 and k > 0 as heavy tails, exponential tails and light tails,
respectively. Hence, a distribution has a class of tail-distribution in the
GPD family. Evidently, this is well defined since for any threshold
u > 0, the shape parameter k is invariant by the tail of GPD, that is
GPDu (k, ψ) = GPD(k, ψ − ku)
(1)
The natural classification of the tails
Heavy-tail, exponential-tail and light-tail
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Table of contents
1
Introduction
2
Problem: Calibration of the GPD for likelihood inference
3
Solution: A good algorithm and a new methodology approach
4
Examples
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Attemps and partial solutions
Hosking, J.R.M.; Wallis, J.R. (1987). Parameter and quantile estimation for the
generalized Pareto distribution. Technometrics
Castillo, E.; Hadi, A. S. (1997). Fitting the generalized Pareto distribution to data. J.
Amer. Statist. Assoc
Juarez, S. F.; Schucany, W. R. (2004). Robust and efficient estimation for the
generalized pareto distribution. Extremes
Luceño, A. (2006). Fitting the generalized Pareto distribution to data using maximum
goodness-of-fit estimators. Comput. Statist. Data Anal.
Castillo, J. del; Daoudi, J. (2009). Estimation of generalized Pareto distribution.
Statistics and Probability Letters
Zhang, J.; Stephens, M. A. (2009). A new and efficient estimation method for the
generalized Pareto distribution. Technometrics
Song, J; Song, S. (2012). A quantile estimation for massive data with generalized
Pareto distribution. Comput. Statist. Data Anal.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Likelihood inference
Goodness-of-Fit Test for GPD
Choulakian, V.; and Stephens, M. A. (2001) . Goodness-of-Fit for the
Generalized Pareto Distribution. Technometrics
BUT it is available for maximum likelihood estimation of parameters.
Zhang, J.; Stephens, M. A. (2009) propose a new estimation method
BUT goodness-of-fit test is not modified
Model selection
Thus several models to the same data can be compared through
Akaike and Bayesian information criteria.
BUT the underlying theory uses maximum likelihood estimation.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Computing maximum likelihood estimator
The log-likelihood function is given by
n
1X
l(k, ψ) = n − log(ψ) + (1/k − 1)
log(1 − kxi /ψ)
n
!
i=1
where ψ > 0 for k ≤ 0 and ψ > kx(n) for k > 0.
The MLE exists
BUT the likelihood equations not always have a solution. It is a
non-regular problem.
In general
The MLE problem for non-regular cases is intensely worked, the main
results are in Smith (1985), Cheng and Amin (1983), Cheng and Iles
(1987) and Hall and Wang (2005).
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Best alternatives to MLE: ZSE and SSE
Currently, the best alternatives to MLE are: ZSE proposed by Zhang
and Stephens (2009-2010) and SSE by Song and Song (2012).
The horizontal is the parameter k with which we make the simulation of samples and
the vertical axis is the parameter k estimated for each of the estimation methods: ML,
SS i ZS. The grey intensity corresponds to the observed frequency in 1000 trials of
sample size 50 in each range of k with: width of 0.1 and uniformly distributed.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Misspecification problem
Percentages of the classification for each submodel for some sample
size and values of k. The model A corresponds to the submodel of
GPD for k < 0, the model B for k ∈ [0, 1] and the model C for k > 1.
A
n
B
-0,1
k
C
0,1
0,9
1,1
A
B
C
A
B
C
A
B
C
A
B
C
65.8
84.2
34.1
0.1
44.1
0.9
0.4
57.7
19.9
0.0
0.0
76.5
90.7
22.6
0.0
55.5
80.1
0.4
15.8
9.3
0.0
31.0
41.9
69.0
85.3
70.1
14.5
0.3
73.8
20.7
14.7
40.7
29.1
0.0
0.0
50.0
77.1
29.4
0.0
25.6
70.9
0.7
30.0
22.9
0.0
27.3
41.4
75.0
50.2
8.4
18.8
0.1
0.0
4.1
9.7
0.0
0.0
11,2
49.8
88.8
0.0
64.6
90.3
16.6
25.0
50.2
0.0
4.4
ZSE
15
100
SSE
15
100
44.6
72.7
MLE
15
100
95.9
95.6
The big-bold values correspond to well classified categories.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Table of contents
1
Introduction
2
Problem: Calibration of the GPD for likelihood inference
3
Solution: A good algorithm and a new methodology approach
4
Examples
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On the existence of MLE
Since the GPD consists in three models: A, B and C, separated by
exponential and uniform distribution, the existence of MLE is
analized for each submodel.
Proposition (motivation)
Let f (x) be a probability density function of a random variable with
support [0, xF ] and 0 < f (xF ) < ∞. Then its tail distribution is the
uniform distribution.
That if a family of right-truncated distributions (in a point where the
density different to 0) is used to model tails, then this model only
contains a class of tail: the uniform distribution.
This is a motivation for choosing the model previously and and once
chosen, let to show that the existence of MLE is not a problem.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On the existence of MLE. Submodel A: k ≤ 0
For k ≤ 0
From Castillo and Daoudi (2009) the MLE exists for the model GPD
for k ∈ (−∞, 0].
Remark that the global maximum holds in the interior of the
domain of k if the empirical coefficient of variation is greater
than 1.
If the empirical coefficient of variation is less than 1, then in
k = 0 has a local maximum and the authors remarks that from a
empirical point of view, it is global.
See also Kozubowski et al. (2009).
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On the existence of MLE. Submodel B: 0 ≤ k ≤ 1
For 0 ≤ k ≤ 1
Choulakian and Stephens (2001) shows that given k < 1 fixed a single
solution exists for ∂l/∂ψ = 0 and it’s a maximum denoted by ψ̂(k).
The set (k, ψ̂(k)) for k ∈ (0, 1) is called Choulakian-Stephens curve.
Theorem
Consider the model GPD for 0 ≤ k ≤ 1, then the global MLE exists.
Moreover, x̄ ≤ ψ̂(k) ≤ x(n) ,
lim ψ̂(k) = x̄
k→0
and
lim ψ̂(k) = x(n)
k→1
(Idea:) ψ̂(k), for k ∈ (0, 1) is cont., monotonous incr. and diff.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On the existence of MLE. Submodel C: k ≥ 1
Proposition
Any solution of the likelihood equations for GPD always satisfies
k < 1.
The GPD model for k ≥ 1 has not MLE as solution of log-likelihood
equations. However, this does not contradict the existence of the MLE
for the model GPD for k ≥ 1.
Proposition
The MLE of the model GPD for k ∈ [1, ∞) with k fixed is ψ̂ = kx(n) .
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Procedure to calibration
This methodology provides the tools to obtain the more realistic
model for the tail of data.
1st step: To calculate MLE for some thresholds.
2nd(a) step: If the value of k̂ is varying near zero, then consider
the possibility that k = 0. To contrast this hypothesis.
2nd(b) step: If the value of k̂ is varying around k = 1, then
consider this possibility. To contrast this hypothesis.
2nd(c) step: In other case, the submodel is clear and our
algorithm give the MLE.
Remark
To compute the exact confidence interval for k consider the property
that the coefficient of variation of GPD only depends on k.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Table of contents
1
Introduction
2
Problem: Calibration of the GPD for likelihood inference
3
Solution: A good algorithm and a new methodology approach
4
Examples
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
Clarification
To illustrate the advantages of the new methodology for estimation
procedure, we will present two real-world example and controversial
examples. Every one of them is extensively studied in the literature.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On Nidd data
Many authors explore this data from Hosking (1987) and
Davison and Smith (1990) since Papastathopoulos and Tawn
(2012).
These data consists in 154 observations of high river levels of the
River Nidd in Yorkshire above a threshold value of 65.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On Nidd data
The MLE of Nidd data and the exact and confidence intervals for the
parameters: k and ψ.
70
80
90
100
110
120
130
140
MLE
k
ψ
-0.32 21.64
-0.34 25.22
-0.24 33.55
0.00
50.62
0.07
56.38
0.25
71.64
0.14
59.43
0.24
65.58
e.i. for k = 0
n
99%
138
(-0.23 , 0.4)
86
(-0.22 , 0.35)
57
(-0.21 , 0.33)
39
(-0.2 , 0.31)
31
(-0.2 , 0.3)
24
(-0.19 , 0.28)
22
(-0.18 , 0.27)
18
(-0.18 , 0.26)
Serra, I.
c.i. for ψ in exponential cas
99%
(25.43 , 39.48)
(28.43 , 49.66)
(31.48 , 62.53)
(34.78 , 79.9)
(34.55 , 88.01)
(35.47 , 102.97)
(31.7 , 96.63)
(30.66 , 105.54)
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On Nidd data
1.2
1.0
0.8
Coefficient of variation
1.4
For exponential tail tested, there is an other methodology from
Castillo, et al. (2012) and the conclusions are the same.
For data larger than 70, the value of statistic of exponentiality
test is T3 = 16.95 and the exponentiality is just rejected.
For data larger than 90, T2 = 1.42
CV-plotdoes not reject it.
0
20
40
60
80
100
120
Sample
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On Bilbao waves data
This data is originally analyzed in Castillo and Hadi (1997),
which consists of the zero-crossing hourly mean periods (in
seconds) of the sea waves measured in the Bilbao bay, Spain.
Later on, this data set was revisited in Luceño (2006) and in
Zhang and Stephens (2009).
Only the 197 observations with periods above 7 seconds were
taken into consideration.
We model this data by the GPD using thresholds at
t = 7, 7.5, 8, 8.5, 9, 9.5 following the above mentioned authors.
They all agree to say that the MLE not exists for the last three cases.
We note that the MLE exists as we have seen but as solution of a
non-regular problem.
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On Bilbao waves data
The estimation of GPD parameters for Bilbao waves data, and for
each threshold, using different estimators. The maximum, M, of the
distribution is given.
7
7.5
8
8.5
9
9.5
k
0.84
0.56
0.63
0.77
0.80
-0.63
SSE
ψ
2.44
1.60
1.42
1.18
0.81
0.22
M
9.90
10.32
10.25
10.04
10.02
k
0.81
0.71
0.77
0.83
0.88
1.01
Serra, I.
ZSE
ψ
2.38
1.75
1.51
1.21
0.83
0.43
M
9.95
9.99
9.96
9.95
9.94
9.93
k
0.86
0.77
0.86
1.06
1.19
1.53
MLE
ψ
2.50
1.86
1.65
1.49
1.07
0.61
M
9.91
9.92
9.91
9.90
9.90
9.90
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On Bilbao waves data
For each threshold of Bilbao waves data, the table provides the
estimated GPD parameters using our algorithm for the MLE.
7
7.5
8
8.5
9
9.5
n
179
154
106
69
41
17
e.i. for k = 1
95%
99%
0.72 1.36 0.64 1.51
0.70 1.41 0.62 1.55
0.64 1.50 0.55 1.72
0.57 1.65 0.47 1.93
0.47 1.93 0.35 2.37
0.27 2.83 0.13 4.09
Serra, I.
Likelihood inference for generalized Pareto distribution
Introduction
Problem: Calibration of the GPD for likelihood inference
Solution: A good algorithm and a new methodology approach
Examples
On Bilbao waves data
An example of Choulakian Stephens curve for Bilbao waves data with
threshold in 7.5
Serra, I.
Likelihood inference for generalized Pareto distribution