Change Point Analysis of Extreme Values

Transcription

1 Change Point Analysis of Extreme Values TIES 2008 p. 1/? Change Point Analysis of Extreme Values Goedele Dierckx Economische Hogeschool Sint Aloysius, Brussels, Belgium Jef L. Teugels Katholieke Universiteit Leuven, Belgium & EURANDOM, Technical University Eindhoven, the Netherlands

2 Change Point Analysis of Extreme Values TIES 2008 p. 2/? Overview 1. Introduction 2. Test statistic (a) Construction (b) Extreme value situation (c) Asymptotics (d) Practical procedure 3. Examples (a) Simulation (b) Malaysian Stock Index. Classical Approach Improved Approach (c) Nile Data (d) Swiss-Re Catastrophic Data 4. Conclusions 5. References

3 Change Point Analysis of Extreme Values TIES 2008 p. 3/? 1. INTRODUCTION We start with an example where a change point has occurred. 987 measurements of the Daily Stock Market Returns of the Malaysian Stock Index. Jan Dec. 1998, covering the Asian financial crisis, July Changes in distribution? /1/95 10/1/96 14/1/97 15/1/98 31/12/98 in parameters of a distribution? central behavior? tail behavior?

4 Change Point Analysis of Extreme Values TIES 2008 p. 4/? 2. TEST STATISTIC 2.a. Construction of Test Statistic Start with a sample X 1,..., X m, X m +1,... X n, from a density function f(x; θ i, η). Csörgő and Horváth (1997) test whether θ i changes at some point m H 0 : θ 1 = θ 2 =... = θ n versus H 1 : θ 1 =... = θ m θ m +1 =... = θ n for some m. using the test statistic Z n = max 1 m<n ( 2log Λ m), where Λ m = sup θ,η n i=1 f(x i; θ, η) sup θ,τ,η m i=1 f(x i; θ, η) n i=m+1 f(x i; τ, η).

5 Change Point Analysis of Extreme Values TIES 2008 p. 5/? 2.TEST STATISTIC Example For the exponential distribution where X i has mean θ i 2log Λ m = 2 m log 1 m m X i (n m)log i=1 1 n m n i=m+1 X i + n log 1 n n i=1 X i For large n, m and n m one can expect normal behaviour expressed in terms of Brownian motions.

6 Change Point Analysis of Extreme Values TIES 2008 p. 6/? 2.TEST STATISTIC 2.b. Extreme Value Situation Assume that X n,n is the maximum in a sample of independent random variables with a common distribution. Maximum domain of attraction condition lim P n ( Xn,n b n a n ) x Under very weak conditions we get the approximation = G γ (x). P (X n,n y) G γ (b n + a n x) where γ is a real-valued extreme value index and G γ (x) = exp {1 + γx} 1/γ + an extremal law. When γ > 0 we end up with heavy right-tailed distributions, the Pareto-Fréchet Case.

7 Change Point Analysis of Extreme Values TIES 2008 p. 7/? 2.TEST STATISTIC We concentrate on changes of parameters that describe the tail of distributions appearing in extreme value analysis. X has a Pareto-type distribution with parameter θ = γ, when the relative excesses of X over a high threshold u, given that X exceeds u satisfy the condition P ( ) X u > x X > u x 1 γ. u, More generally X follows a Generalized Pareto distribution (GPD) with parameter θ = (γ, σ) if the behavior of the absolute excesses over a high threshold u satisfies the condition P (X u > x X > u) ( 1 + γx ) 1 γ, u. σ

8 Change Point Analysis of Extreme Values TIES 2008 p. 8/? 2.TEST STATISTIC For large values, log of Pareto-type with extreme value index γ i is close to be exponential with mean γ i. The most classical approach for the estimation of the extreme value index γ > 0 is to use the Hill estimator: H k,n = 1 k k log X n i+1,n log X n k,n. i=1 Hence, only a segment of the available data is used. The determination of the quantity k is important. Alternatively, we look at extremes above a threshold u = X n k,n. The Hill estimator has small bias but large variance for small k large bias but small variance for large k. As a compromise we select k such that the empirical mean squared error is minimal.

9 Change Point Analysis of Extreme Values TIES 2008 p. 9/? 2.TEST STATISTIC 1. Pareto-type density Suppose X 1,..., X m, X m+1,... X n are independent and Pareto-type distributed. We denote the extreme value index for X i by γ i. In order to determine whether the index γ changes or not, we perform the following test H 0 : γ 1 = γ 2 =... = γ n = γ versus H 1 : γ 1 = γ m γ m +1 = γ n for some m Hence Z n = where in turn max 1 m<n ( 2log Λ m) log Λ m = [ k 1 log H k1,m + (k k 1 ) log H k k1,n m k log H k,n ] [ ] 1 ( ) + k1 H k1,m + (k k 1 )H k k1,n m kh k,n H k,n.

10 Change Point Analysis of Extreme Values TIES 2008 p. 10/? 2.TEST STATISTIC 2. GPD. Suppose now that X i is GPD with parameters θ i = (γ i, σ i ).To perform the test H 0 : θ 1 = θ 2 =... = θ n versus H 1 : θ 1 =... = θ m θ m +1 =... = θ n for some m we use as test statistic Z n = max 1 m<n ( 2 log Λ m), where 2 log Λ m = 2 L m (ˆθ m ) = m log ˆσ m [ ] L k1 (ˆθ k1 ) + L + k (ˆθ + 1 k ) L 1 k (ˆθ k ) ( ) 1 m + 1 ˆγ m ( 1 L + m(ˆθ m) + = (n m)log ˆσ m + ˆγ + m i=1 + 1 ( ) x log 1 + ˆγ m ˆσ m ) n ( log 1 + ˆγ m + i=m+1 x ˆσ + m ) and likelihood estimators (ˆγ m, ˆσ m ) resp. (ˆγ + m, ˆσ + m) based on X 1, X 2,..., X m and X m+1,... X n are obtained by numerical procedures.

11 Change Point Analysis of Extreme Values TIES 2008 p. 11/? 2. TEST STATISTIC 2.c. Asymptotics Using the procedure suggested by Csörgő and Horváth we have Theorem Suppose X 1,..., X m, X m+1,... X n are independent and identically distributed. We set the threshold at u = X n k,n. Define Z n = max c n m<n d n ( 2 log Λ m ), with 2log Λ m as before. Let n, k such that k/n 0. Let further c n and d n be intermediate sequences for which c n /n 0 and d n /n 0. Then, under H 0 of our test, Z n d sup 0 t<1 sup 0 t<1 B 2 (t) t(1 t) if Pareto-type, B 2 2 (t) t(1 t) if GPD. B(t) is a Brownian bridge, B 2 (t) is a sum of two independent Brownian bridges.

12 Change Point Analysis of Extreme Values TIES 2008 p. 12/? 2. TEST STATISTIC 2.d. Practical Procedure Consecutive steps 1. Check on Pareto-type behavior of the data by Q Q plots. 2. Select a threshold u or the value of k = k opt,n that minimizes the asymptotic mean square error of the Hill estimator. We choose the optimal threshold u = X n kopt,n. 3. (a) Define c n as the smallest number such that at least k min = (log k opt,n ) 3/2 of the data points X 1,, X cn are larger than u. (b) Define d n as the smallest number such that at least k min of the data points X n dn +1,..., X n are larger than u. 4. Repeat the next step for all m from c n up to n d n. (a) Split the data up in two groups X 1, X 2,..., X m and X m+1,,..., X n. (b) Calculate 2log Λ m. 5. Calculate Z n = values for sample size k. max c n m<n d n ( 2 log Λ m ) and compare Z n with the critical

13 Change Point Analysis of Extreme Values TIES 2008 p. 13/? 3.a. Simulation We simulate 1000 data sets of size n (with n = 100, n = 500) from the Burr distribution Burr(β, τ, λ) with parameters as given by P(X > x) = ( β β + x τ ) λ, an example of a GPD with γ = (λτ) 1. The rejection probabilities are given below. H 0 true H 0 false n m γ = 1 γ 1 = 1 γ 1 = 2 γ 1 = 1 γ 1 =.5 γ 2 = 2 γ 2 = 1 γ 2 =.5 γ 2 =

14 Change Point Analysis of Extreme Values TIES 2008 p. 14/? The corresponding median of ˆm is given in the table below. H 0 false n m γ 1 = 1 γ 1 = 2 γ 1 = 1 γ 1 =.5 γ 2 = 2 γ 2 = 1 γ 2 =.5 γ 2 =

15 Change Point Analysis of Extreme Values TIES 2008 p. 15/? Figure shows Boxplot of ˆm for the Burr cases for n = 500 and m = 100.

16 Change Point Analysis of Extreme Values TIES 2008 p. 16/? 3.b. Malaysian Stock Index: Classical approach Figure below indicates that the data are Pareto-type distributed. If we accept that July 1997 was a change point, then the data before that date give an extreme value index γ 1 between 0.1 and 0.2 while those after that date give γ 2 around 0.5. The mean squared error of the Hill estimator based on the whole data set attains a local minimum for the threshold u given by X ,987 = so that k = k opt =

17 Change Point Analysis of Extreme Values TIES 2008 p. 17/? 1. Pareto-type distribution First 2log Λ m,1 m n 1 is plotted below /1/95 10/1/96 14/1/97 15/1/98 31/12/98 Graph of (m, 2 log Λ m ) with critical value indicated with a horizontal line. We see that Z n = max( 2log Λ m ) = 5.8 falls above the critical value 3.14 and we reject H 0. The maximum is attained at m = 635, which corresponds to 1/08/1997, shortly after the beginning of the Asian crisis.

18 Change Point Analysis of Extreme Values TIES 2008 p. 18/? 2. GPD Now 2log Λ m, 1 m n 1 is plotted below /1/95 10/1/96 14/1/97 15/1/98 31/12/98 Since Z n = max( 2 log Λ m ) = 5.93 is above the critical value 3.18 we again reject H 0. Also the instant of change ˆm = 636 is again very close to the value before.

19 Change Point Analysis of Extreme Values TIES 2008 p. 19/? 3.b. Malaysian Stock Index: Improved approach In the above analysis, we assumed that the data were independent. But market data are hardly ever independent. However, it is known that the Hill estimator withstands many forms of dependence. Alternatively, one can proceed as follows. The time series and an estimate of the extremal index are given below /1/95 10/1/96 14/1/97 15/1/98 31/12/ A declusturing scheme cuts the data into clusters that can safely be taken as independent. Apply the previous procedure to the 76 cluster maxima.

20 Change Point Analysis of Extreme Values TIES 2008 p. 20/? 1. Pareto-type distribution /1/95 14/1/97 15/1/98 31/12/98 There is a local maximum for cluster maximum 48 which corresponds to m = 631 on (28/07/1997). However this local maximum is not larger than the critical. The actual maximum Z n is attained for cluster maximum 66 which corresponds to m = 854 (22/6/98). We cannot reject the hypothesis.

21 Change Point Analysis of Extreme Values TIES 2008 p. 21/? 2. GPD /1/95 14/1/97 15/1/98 31/12/98 Now 2log Λ m, 1 m n 1 is plotted in the figure. The maximum Z n is attained for cluster maximum 48 which corresponds to m = 631(28/07/1997). The critical value 2.95 for the test is indicated with a horizontal line. On the basis of this test, we reject the hypothesis of no change.

22 Change Point Analysis of Extreme Values TIES 2008 p. 22/? 3.c. Nile Data Annual flow volume of the Nile River at Aswan from 1871 to

23 Change Point Analysis of Extreme Values TIES 2008 p. 23/? Prior studies indicate 1877 (measurement 813) candidate for additive outlier, 1913 (measurements 456) candidate for additive outlier, 1899 (measurement 774) indicates start of construction of Aswan dam. nile Time

24 Change Point Analysis of Extreme Values TIES 2008 p. 24/? Group 1: first 28 points Group 2: remaining 71 points with Pareto QQ plots for both groups. Optimal values k = 17, resp. k = 13 lead to the estimators 0.07 and Pareto quantile plot Pareto quantile plot log(x) log(x) Quantiles of Standard Exponential Quantiles of Standard Exponential

25 Change Point Analysis of Extreme Values TIES 2008 p. 25/? The change point detection based on the Pareto and the GPD model are given in the figure, both leading to a significant change point at ˆm = 28 at the beginning of construction of the Aswan dam. sqrt(hh$ll) sqrt(hh$ll) Time Time

26 Change Point Analysis of Extreme Values TIES 2008 p. 26/? 3.d. Swiss-Re Catastrophic Data PLACE DATE VICTIMS PLACE DATE VICTIMS Bangladesh 0, Indonesia 34, China 6, Bangladesh 21, Peru 0, Gilan (Iran) 20, Bam (Iran) 33, Tabas (Iran) 8, Armenia 18, Colombia 15, Guatemala 6, Izmit (Turkey) 29, Gujarat (India) 31, India 8, India 29, India 9, India 1, Venezuela 29, Bangladesh 7, Mexico 15, India 23, Honduras 28, Kobe (Japan) 25, Philippines 21, Pakistan 4, Brazil 31,

27 Change Point Analysis of Extreme Values TIES 2008 p. 27/? The Pareto QQ plots with the corresponding Hill estimators. The mean squared error is minimal at k = 39 for Pareto and k = 22 for GPD, both leading to ˆγ = 1.3. Pareto quantile plot Estimates of extreme value index log(x) gamma Quantiles of Standard Exponential k

28 Change Point Analysis of Extreme Values TIES 2008 p. 28/? The likelihood expression 2 log Λ m based on the Pareto model and the GPD model as a function of m where m is indicating where the group is split up in two. Pictures for Pareto model with critical value 1.6 and GPD model with critical value 1.4.

29 Change Point Analysis of Extreme Values TIES 2008 p. 29/? 4. CONCLUSIONS What has been shown are just first attempts Assumption on positive γ Rounded figures make accurate conclusions harder There is a need for sufficiently large data sets Need for studies under specific dependence structures Multivariate extensions should be possible 5. REFERENCES Beirlant, J., Goegebeur Y., Segers, J. and Teugels, J.L. (2004). Statistics of Extremes, Theory and Applications, Wiley, Chichester. Csörgő, M., Horváth, L. (1997). Limit Theorems in Change Point Analysis. Wiley, Chichester.