Likelihood inference for generalized Pareto distributionevt2013.weebly.com › uploads › 1 › 2...

IntroductionProblem: Calibration of the GPD for likelihood inference

Solution: A good algorithm and a new methodology approachExamples

Likelihood inference for generalized Paretodistribution

J. del Castillo1 and I. Serra1

1Departament de MatematiquesUniversitat Autonoma de Barcelona

EVT2013Sep de 2013

Serra, I. Likelihood inference for generalized Pareto distribution



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




History of EVT: onset of GPD

Fisher-Tippett Theorem (1928)

Let (X1 . . .Xn) be a sequence of i.i.r.v, let Mn = max{X1 . . .Xn}. If(an, bn) and F exist, such that limn→∞ P

(Mn−bn

an≤ x)

= F(x), then Fbelongs 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 theirconditional excess distribution function then

Fu(y)→ GPD(k, ψ)(y), as u→∞

where GPD is the generalized Pareto distribution.




Widely usedDavis, H. T. and Michael L. F. (1979). The Generalized Pareto Law as a Model forProgressively 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. Kluppelberg, C. and Mikosch, T. (1997). Modelling Extremal Eventsfor 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 seriesof 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 fortail estimation. J. Statist. Plann. Inference




Generalized Pareto distribution

The GPD is a two-parameter distribution for non-negative variables

Probability density function

f (x; k, ψ) =1ψ

(1− kx/ψ)1/k−1, 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) =1ψ

exp(−x/ψ), f (x; 1, ψ) = x/ψ




Submodels

For k < 0 the Pareto submodel (range (0,∞))For k = 0 the exponential distributionFor k ∈ (0, 1) a family of decreasing probability densitiesfunctions with finite support (0, ψ/k)

For k = 1 the uniform distributionFor k > 1 a family of increasing densities with finite support.




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 theGPD family. Evidently, this is well defined since for any thresholdu > 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 tailsHeavy-tail, exponential-tail and light-tail




Table of contents

1 Introduction



4 Examples




Attemps and partial solutionsHosking, J.R.M.; Wallis, J.R. (1987). Parameter and quantile estimation for thegeneralized 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 thegeneralized pareto distribution. Extremes

Luceno, A. (2006). Fitting the generalized Pareto distribution to data using maximumgoodness-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 thegeneralized Pareto distribution. Technometrics

Song, J; Song, S. (2012). A quantile estimation for massive data with generalizedPareto distribution. Comput. Statist. Data Anal.




Likelihood inference

Goodness-of-Fit Test for GPDChoulakian, V.; and Stephens, M. A. (2001) . Goodness-of-Fit for theGeneralized Pareto Distribution. Technometrics

BUT it is available for maximum likelihood estimation of parameters.

Zhang, J.; Stephens, M. A. (2009) propose a new estimation methodBUT goodness-of-fit test is not modified

Model selectionThus several models to the same data can be compared throughAkaike and Bayesian information criteria.

BUT the underlying theory uses maximum likelihood estimation.




Computing maximum likelihood estimator

The log-likelihood function is given by

l(k, ψ) = n

(− log(ψ) + (1/k − 1)

1n

n∑i=1

log(1− kxi/ψ)

)where ψ > 0 for k ≤ 0 and ψ > kx(n) for k > 0.

The MLE existsBUT the likelihood equations not always have a solution. It is anon-regular problem.

In generalThe MLE problem for non-regular cases is intensely worked, the mainresults are in Smith (1985), Cheng and Amin (1983), Cheng and Iles(1987) and Hall and Wang (2005).




Best alternatives to MLE: ZSE and SSE

Currently, the best alternatives to MLE are: ZSE proposed by Zhangand 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



Misspecification problem

Percentages of the classification for each submodel for some samplesize and values of k. The model A corresponds to the submodel ofGPD for k < 0, the model B for k ∈ [0, 1] and the model C for k > 1.

A B Ck -0,1 0,1 0,9 1,1

n A B C A B C A B C A B CZSE

15 65.8 34.1 0.1 44.1 55.5 0.4 0.9 76.5 22.6 0.4 57.7 41.9100 84.2 15.8 0.0 19.9 80.1 0.0 0.0 90.7 9.3 0.0 31.0 69.0SSE

15 85.3 14.5 0.3 73.8 25.6 0.7 20.7 50.0 29.4 14.7 40.7 44.6100 70.1 30.0 0.0 29.1 70.9 0.0 0.0 77.1 22.9 0.0 27.3 72.7

MLE

15 41.4 50.2 8.4 18.8 64.6 16.6 0.1 11,2 88.8 0.0 4.1 95.9100 75.0 25.0 0.0 9.7 90.3 0.0 0.0 49.8 50.2 0.0 4.4 95.6

The big-bold values correspond to well classified categories.




Table of contents

1 Introduction



4 Examples




On the existence of MLE

Since the GPD consists in three models: A, B and C, separated byexponential and uniform distribution, the existence of MLE isanalized for each submodel.

Proposition (motivation)

Let f (x) be a probability density function of a random variable withsupport [0, xF] and 0 < f (xF) <∞. Then its tail distribution is theuniform distribution.

That if a family of right-truncated distributions (in a point where thedensity different to 0) is used to model tails, then this model onlycontains a class of tail: the uniform distribution.

This is a motivation for choosing the model previously and and oncechosen, let to show that the existence of MLE is not a problem.




On the existence of MLE. Submodel A: k ≤ 0

For k ≤ 0From Castillo and Daoudi (2009) the MLE exists for the model GPDfor k ∈ (−∞, 0].

Remark that the global maximum holds in the interior of thedomain of k if the empirical coefficient of variation is greaterthan 1.

If the empirical coefficient of variation is less than 1, then ink = 0 has a local maximum and the authors remarks that from aempirical point of view, it is global.

See also Kozubowski et al. (2009).




On the existence of MLE. Submodel B: 0 ≤ k ≤ 1

For 0 ≤ k ≤ 1Choulakian and Stephens (2001) shows that given k < 1 fixed a singlesolution exists for ∂l/∂ψ = 0 and it’s a maximum denoted by ψ(k).

The set (k, ψ(k)) for k ∈ (0, 1) is called Choulakian-Stephens curve.

TheoremConsider the model GPD for 0 ≤ k ≤ 1, then the global MLE exists.Moreover, x ≤ ψ(k) ≤ x(n),

limk→0

ψ(k) = x and limk→1

ψ(k) = x(n)

(Idea:) ψ(k), for k ∈ (0, 1) is cont., monotonous incr. and diff.




On the existence of MLE. Submodel C: k ≥ 1

PropositionAny solution of the likelihood equations for GPD always satisfiesk < 1.

The GPD model for k ≥ 1 has not MLE as solution of log-likelihoodequations. However, this does not contradict the existence of the MLEfor the model GPD for k ≥ 1.

Proposition

The MLE of the model GPD for k ∈ [1,∞) with k fixed is ψ = kx(n).




Procedure to calibration

This methodology provides the tools to obtain the more realisticmodel for the tail of data.

1st step: To calculate MLE for some thresholds.

2nd(a) step: If the value of k is varying near zero, then considerthe possibility that k = 0. To contrast this hypothesis.

2nd(b) step: If the value of k is varying around k = 1, thenconsider this possibility. To contrast this hypothesis.

2nd(c) step: In other case, the submodel is clear and ouralgorithm give the MLE.

RemarkTo compute the exact confidence interval for k consider the propertythat the coefficient of variation of GPD only depends on k.




Table of contents

1 Introduction



4 Examples




Clarification

To illustrate the advantages of the new methodology for estimationprocedure, we will present two real-world example and controversialexamples. Every one of them is extensively studied in the literature.




On Nidd data

Many authors explore this data from Hosking (1987) andDavison and Smith (1990) since Papastathopoulos and Tawn(2012).

These data consists in 154 observations of high river levels of theRiver Nidd in Yorkshire above a threshold value of 65.




On Nidd data

The MLE of Nidd data and the exact and confidence intervals for theparameters: k and ψ.

MLE e.i. for k = 0 c.i. for ψ in exponential cask ψ n 99% 99%

70 -0.32 21.64 138 (-0.23 , 0.4) (25.43 , 39.48)80 -0.34 25.22 86 (-0.22 , 0.35) (28.43 , 49.66)90 -0.24 33.55 57 (-0.21 , 0.33) (31.48 , 62.53)100 0.00 50.62 39 (-0.2 , 0.31) (34.78 , 79.9)110 0.07 56.38 31 (-0.2 , 0.3) (34.55 , 88.01)120 0.25 71.64 24 (-0.19 , 0.28) (35.47 , 102.97)130 0.14 59.43 22 (-0.18 , 0.27) (31.7 , 96.63)140 0.24 65.58 18 (-0.18 , 0.26) (30.66 , 105.54)




On Nidd data

For exponential tail tested, there is an other methodology fromCastillo, et al. (2012) and the conclusions are the same.

For data larger than 70, the value of statistic of exponentialitytest is T3 = 16.95 and the exponentiality is just rejected.For data larger than 90, T2 = 1.42 does not reject it.

0 20 40 60 80 100 120

0.8

1.0

1.2

1.4

CV-plot

Sample

Coe

ffici

ent

of v

aria

tion




On Bilbao waves data

This data is originally analyzed in Castillo and Hadi (1997),which consists of the zero-crossing hourly mean periods (inseconds) of the sea waves measured in the Bilbao bay, Spain.

Later on, this data set was revisited in Luceno (2006) and inZhang and Stephens (2009).

Only the 197 observations with periods above 7 seconds weretaken into consideration.

We model this data by the GPD using thresholds att = 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 anon-regular problem.





The estimation of GPD parameters for Bilbao waves data, and foreach threshold, using different estimators. The maximum, M, of thedistribution is given.

SSE ZSE MLEk ψ M k ψ M k ψ M

7 0.84 2.44 9.90 0.81 2.38 9.95 0.86 2.50 9.917.5 0.56 1.60 10.32 0.71 1.75 9.99 0.77 1.86 9.92

8 0.63 1.42 10.25 0.77 1.51 9.96 0.86 1.65 9.918.5 0.77 1.18 10.04 0.83 1.21 9.95 1.06 1.49 9.90

9 0.80 0.81 10.02 0.88 0.83 9.94 1.19 1.07 9.909.5 -0.63 0.22 1.01 0.43 9.93 1.53 0.61 9.90





For each threshold of Bilbao waves data, the table provides theestimated GPD parameters using our algorithm for the MLE.

e.i. for k = 1n 95% 99%

7 179 0.72 1.36 0.64 1.517.5 154 0.70 1.41 0.62 1.55

8 106 0.64 1.50 0.55 1.728.5 69 0.57 1.65 0.47 1.93

9 41 0.47 1.93 0.35 2.379.5 17 0.27 2.83 0.13 4.09





An example of Choulakian Stephens curve for Bilbao waves data withthreshold in 7.5


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