Quantile%20Estimation%20for%20Heavy-Tailed%20Data

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Quantile Estimation for Heavy-Tailed Data

• 23/03/2000
• J. Beirlant
• (jan.beirlant_at_wis.kuleuven.ac.be)
• G. Matthys
• (gunther.matthys_at_ucs.kuleuven.ac.be)

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Introduction Notation

• Contents
• MODEL SPECIFICATION
• EXAMPLE SP500s daily returns
• EXTREME QUANTILE ESTIMATION review of 3 classes
  of methods
• Blocks method
• Peaks-over-treshold (POT) method
• Q(uantile)-based methods
• IMPROVING Q-BASED METHODS ML-estimator
• SIMULATION RESULTS
• CONCLUSIONS

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• Notation
• Sample X1, X2,,Xn from
• Order statistics
• Pareto-type / heavy-tailed model
• Tail decays essentially as a power function
• with l slowly varying, i.e.
• Terminology tail index ?
• extreme value index (EVI)
• Equivalently for some other slowly varying
  function l

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• Examples of heavy-tailed distributions
• Pareto
• Students t with degrees of freedom
• Loggamma
• Fréchet
• Not normal, exponential, lognormal, Weibull !

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• Heavy-tailed distributions in finance
• log-returns and exchange rates mostly do exhibit
  heavy tails
• (often for log-return series)
• often one is interested in a certain high
  level, which will
• be exceeded with only a (very) small
  probability
• search for an extreme quantile of the
  distribution
• Example
• 16/10/1987 Risk manager wants to assess the
  risks his investment is exposed to and
  investigates some worst case scenarios
• estimate fall of the SP500 index that will
  happen only once every 10.000 days (? 40 years)
  on average

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• Formally
• Consider daily falls as random variables X1,
  X2,, Xn, and denote their stationary
  distribution (assuming it exists) with F
• then we look for quantile x of F that satisfies
  
• F(x) 1 - 1/10000.
• Problem of estimating extreme quantiles xp with
  tail probability p ? (0,1), where p will be
  usually very small
• 3 classes of methods
• method of block maxima
• POT-method
• Q(uantile)-based methods

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Peaks Over Treshold (POT) Method

• Limit law for excesses over a high treshold
  (PickandsBalkemade Haan, 1974, 1975)
• If F is a heavy-tailed distribution with EVI
  ? gt0 and excess
• distribution
• then
• where G?,? denotes the Generalised Pareto
  Distribution

(GPD)
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• Thus, for large tresholds u,
• for ? gt0 and some ? gt 0.
• Description of the POT method
• 1 given a sample X1, X2,, Xn, select a high
  treshold u
• let Nu be the number of exceedances
• denote the excesses for j1,,Nu
• 2 fit a GPD G?,? to the excesses Y1,,Yn, e.g.
  with MLE
• parameter estimates

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• 3 as
• estimate F(x) for x gt u with
• where Fn is the empirical distribution
  function Fn(u) 1-Nu/n
• 4 obtain quantile estimates by inverting ()
• Note in general for each different choice of
  treshold other parameter and quantile estimates !

()
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• MLEs for GPD-fit of the excesses above treshold
  u 1.5
• (thus )
• 95CI (4.9, 10)
• compare the fitted excess distribution with the
  empirical one

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Q(uantile)-based Methods

• Hill estimator
• Pickands estimator
• Moment estimator
• MLE for exponential regression model

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• Hill estimator
• Heavy-tailed distribution
• distribution of log(X)
• Quantile function
• Order statistics of X1, X2,,Xn
• is a consistent estimator for
• plot

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• slope estimator
• for k large, denominator 1 Hill estimator
  (1975)
• for each k (number of order statistics)
  different estimator
• k small bias small, variance large
• k large variance small, bias large
• bias-variance trade-off to select
  optimal k

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• estimating a quantile Q(1-p) with Hill
  estimator
• extrapolate along fitted line on Pareto quantile
  plot
• again different estimator for each k
• bias-variance trade-off needed to select
  optimal k

(Weissman, 1978)
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• Pickands estimator
• EVI estimator (1975)
• derived quantile estimator

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• Moment estimator
• EVI estimator (DekkersEinmahlde Haan, 1989)
• where
• derived quantile estimator

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Q-based estimators for EVI of SP500's falls
0.4
0.3
0.2
0.1
0.0
0
100
200
300
400
k
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Q-based estimators for 0.0001-quantile of
SP500's falls
11
10
9
8
7
6
5
0
100
200
300
400
k
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• ML Method improving Q-based methods
• (Beirlant, Dierckx, Goegebeur Matthys,
  Extremes, 1999)
• Hill estimator
• for 0 lt k lt n, consider log-spacings
• Assumption on second-order slow variation
• for x ? ?
• with ? lt 0 and b(x) ? 0 as x ? ?

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with uniform (0,1) order statistics
exponential regression model for log-spacings
with f1,k, f2,k,,fk,k i.i.d.
exponential (1)
with f1, f2,,fk i.i.d. exponential (1)
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Burr(1,1,2), k 200, n 500
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Q-based estimators for EVI of SP500's falls
0.4
0.3
0.2
0.1
0.0
0
100
200
300
400
k
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• quantile estimation try to preserve stability
• first idea, by analogy to Weissman estimator
• improvement, including estimated information on
  l

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Q-based estimators for 0.0001-quantile of
SP500's falls
11
10
9
8
7
6
5
0
100
200
300
400
k
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Loggamma(1,2), p 0.0002
100000
50000
10000
5000
0
50
100
150
k
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Loggamma(2,2), p 0.0002
500
400
300
200
0
50
100
150
k
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Student's t2, p 0.0002
100
50
0
50
100
150
k
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Student's t4, p 0.0002
100
50
10
5
0
50
100
150
k
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Burr(1,1,1), p 0.0002
10000
5000
1000
500
0
50
100
150
k
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Burr(1,0.5,2), p 0.0002
50000
5000
1000
500
0
50
100
150
k
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Arch(1), p 0.001, gamma 0.466
40
30
20
9
8
7
6
0
50
100
150
k
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Arch(1), p 0.001, gamma 0.094
0.07
0.05
0.04
0.03
0.02
0
50
100
150
k
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Garch(1,1), p 0.001, gamma 0.419
40
30
20
9
8
7
6
0
50
100
150
k
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Garch(1,1), p 0.001, gamma 0.096
0.07
0.05
0.04
0.03
0.02
0
50
100
150
k
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Related Problems and Questions

• How to detect heavy-tailedness
• Extension of the Q-based ML method to all classes
  of distributions (not only heavy-tailed ones)
• Further investigation of asymptotic properties of
  the Q-based ML estimators construction of CIs
  for (financial) time series (dependent data)