A Closer Look at the Hill Estimator: Edgeworth Expansions and Confidence Intervals

Transcription

1 A Closer Look at the Hill Estimator: Edgeworth Expansions and Confidence Intervals Erich HAEUSLER University of Giessen Johan SEGERS Tilburg University EVA 2005, AUGUST 15 - p. 1/34

2 INTRODUCTION Ordered sample X 1:n X n:n from Pareto-type cdf F HILL (1975) estimator for positive extreme-value index γ Ĥ n (k) = 1 k k log X n k+i:n log X n k:n i=1 Simple and popular Asymptotic properties well known ) (Ĥn (k n ) d kn 1 µ n N(0, 1) γ intermediate sequence: k n, k n = o(n) asymptotic bias µ n = O(1), depends on F and k n EVA 2005, AUGUST 15 - p. 2/34

3 Confidence intervals and tests Confidence intervals and hypothesis tests less studied CI of nominal level 1 α: ( symmetric CI : Ĥ n (k) 1 ± z ) / ( k asymmetric CI : Ĥ n (k) 1 z ) with k Φ(z) =1 α/2 Relevance: Existence of moments CI s/tests for exceedance probabilities, quantiles,... [VANDEWALLE 2004] EVA 2005, AUGUST 15 - p. 3/34

4 Questions Which CI to be preferred? Yet other CI s? Which k to use for which CI? Comparisons between CI s requires Edgeworth expansions [ ) ] kn (Ĥn (k) Pr 1 x =Φ(x)+error term γ EVA 2005, AUGUST 15 - p. 4/34

5 Related literature One-term Edgeworth expansions in CHENG &PAN (1998) and CHENG &PENG (2001) Useful for one-sided CI s [CHENG &PENG 2001] Insufficiently accurate to analyse two-sided CI s Expansions in terms of Gamma distributions [CHENG & DE HAAN 2001; GUILLOU &HALL 2001] Insufficiently accurate for two-sided CI s as well Note: If µ n o(1), then these CI s are inconsistent This is the case for AMSE-minimizing k n Bias-corrected CI s in FERREIRA & DE VRIES (2004) EVA 2005, AUGUST 15 - p. 5/34

6 For proper understanding... Won t talk about: Bias reduction Data-driven methods to choose threshold Comparisons with other estimators Bayesian inference Quantiles, exceedance probabilities Other domains of attraction Temporal dependence, non-stationarity, covariates Will talk about: Iid variables Positive extreme-value index Performance of various Hill-based CI s/tests Understanding of impact of intermediate sequence, nominal level, underlying distribution EVA 2005, AUGUST 15 - p. 6/34

7 Outline Inference in Pareto model CI s and hypothesis tests for extreme-value index Edgeworth expansions for normalized Hill estimator Main result Simulations Conclusion EVA 2005, AUGUST 15 - p. 7/34

8 PARETO MODEL Cdf and pdf of Pareto(1/γ): G γ (x) = 1 x 1/γ, p γ (x) = 1 γ x 1 1/γ for x>1 Inference on γ>0 from iid Y 1,...,Y k p γ? Estimation Testing Confidence intervals EVA 2005, AUGUST 15 - p. 8/34

9 Likelihood computations Log-likelihood of γ given Y 1,...,Y k k (Ĥk l k (γ) = log p γ (Y i )= k γ i=1 +log(γ) ) + constant Ĥ k = 1 k k log(y i ) i=1 Score ) (Ĥk l k (γ) = k γ γ 1 Fisher information ( ) I(γ) =Var γ γ log p γ(y ) = 1 γ 2 EVA 2005, AUGUST 15 - p. 9/34

10 MLE and deviance statistic Ĥ k is sufficient statistic and MLE for γ k(ĥk γ) d N(0,γ 2 ), Deviance statistic (likelihood ratio) at γ: ( ) D k (γ) = 2 l k (Ĥ k ) l k (γ) = 2k (Ĥk γ k Ĥk 1 log γ ) d χ 2 1, k EVA 2005, AUGUST 15 - p. 10/34

11 Hypothesis tests (1) Test for H 0 : γ 0 = γ versus H 1 : γ 0 γ at nominal level 1 α z = z 1 α/2 standard-normal quantile Φ(z) =1 α/2 Reject H 0 : γ 0 = γ if T k (γ) >z 2 where Test Test statistic T k (γ) ) 2 (Ĥk γ Wald k Ĥ k ) 2 (Ĥk γ Score k γ ) (Ĥk Ĥk Likelihood ratio D k (γ) =2k 1 log γ γ / ( Bartlett-corrected LR D k (γ) 1+ 1 ) 6k EVA 2005, AUGUST 15 - p. 11/34

12 Hypothesis tests (2) Wald and score tests also Bartlett correctable One-sided tests: similarly Corresponding confidence intervals at nominal level 1 α: {All γ>0for which H 0 : γ 0 = γ is not rejected at level 1 α} EVA 2005, AUGUST 15 - p. 12/34

13 CI S AND TESTS FOR EVI Pareto domain of attraction cdf F has extreme-value index γ>0 iff Pr[X/u > x X>u] = 1 F (ux) 1 F (u) x 1/γ, u Relative excesses over high thresholds are asymptotically Pareto(1/γ) distributed EVA 2005, AUGUST 15 - p. 13/34

14 Hill estimator Heuristic: 1. Take large threshold u = X n k:n 2. Relative excesses Y i:k = X n k+i:n /X n k:n for i =1,...,k 3. Pretend Y 1:k,...,Y k:k are order statistics from iid Pareto(1/γ) sample Pseudo-likelihood inference: HILL (1975) Ĥ n (k) = 1 k k i=1 log X n k+i:n X n k:n Other interpretations [EMBRECHTS ET AL. 1997; BEIRLANT ET AL. 2004] EVA 2005, AUGUST 15 - p. 14/34

15 Hypothesis tests and CI s (1) Fix k Reject H 0 : γ 0 = γ at nominal level 1 α if T n,k (γ) >z 2 1 α/2 Test Test statistic T n,k (γ) ) 2 (Ĥn (k) γ Wald k Ĥ n (k) ) 2 (Ĥn (k) γ Score k γ (Ĥn (k) Likelihood ratio D n,k (γ) =2k γ / ( Bartlett-corrected LR D n,k (γ) 1+ 1 ) 6k 1 log Ĥn(k) γ ) EVA 2005, AUGUST 15 - p. 15/34

16 Hypothesis tests and CI s (2) Confidence intervals {All γ>0 for which H 0 : γ 0 = γ is not rejected} False rejection of H 0 : γ 0 = γ (type I error) Pr[False rejection] = α + error term? Not considered here but similar: false acceptance of wrong value (type II error) Will depend on: type of interval intermediate sequence k = k n nominal level underlying distribution EVA 2005, AUGUST 15 - p. 16/34

17 EDGEWORTH EXPANSIONS We work under H 0 : γ 0 = γ for fixed γ>0 Intermediate sequence k n All test statistics can be expressed in terms of H n = ) (Ĥn (k n ) k n 1 γ We ll need expansions of the form Pr[H n x] =Φ(x)+error term Edgeworth expansions for the Hill estimator EVA 2005, AUGUST 15 - p. 17/34

18 Two sources of error Two reasons why Pr[H n x] Φ(x) 1. Relative excesses only asymptotically Pareto(1/γ) 2. Even for Pareto(1/γ), H n is standardized Gamma These sources of error may work in equal or in opposite directions To quantify first effect: higher-order regular variation EVA 2005, AUGUST 15 - p. 18/34

19 Tail quantile function Tail quantile function V (y) =inf{x R : F (x) 1 1/y}, y > 1 Domain-of-attraction condition equivalent to log V (ty) log V (t) =γ log y + o(1), t for y>0 Quantify o(1) to capture deviations from Pareto(1/γ) model EVA 2005, AUGUST 15 - p. 19/34

20 Higher-order regular variation Refine domain-of-attraction condition Second-order regular variation: ast, log V (ty) log V (t) = γ log y + a(t)ch ρ (y) + o(1), with ρ 0, a RV ρ with a( ) =0, c 0, h ρ (y) = y 1 uρ 1 du [BINGHAM ET AL. 1987; GELUK & DE HAAN 1987] Third-order regular variation: ast, log V (ty) log V (t) = γ log y + a(t)ch ρ (y) + a(t)b(t){b(y)+o(1)} with b RV τ for some τ 0 with b( ) =0and some specified form for B(y) [DE HAAN &STADTMÜLLER 1996] EVA 2005, AUGUST 15 - p. 20/34

21 One-term Edgeworth expansion (1) Assume Second-order regular variation k n a(n/k n )=o(1) Expansion of cdf of H n = k n {Ĥn(k n ) γ}/γ: ( ) 1 Pr[H n x] = Φ(x) ϕ(x) 3 (1 x 2 )+µ n k n ( ) 1 + o + o(µ n ) kn where µ n = c kn a(n/k n ) E [H n ] γ(1 ρ) EVA 2005, AUGUST 15 - p. 21/34

22 One-term Edgeworth expansion (2) Slight generalization of CHENG &PAN (1998) and CHENG &PENG (2001) Useful to analyze one-sided tests [CHENG &PENG 2001] Insufficient to compare two-sided tests: it only gives Pr[False rejection] = α + o ( 1 kn ) + o(µ n ) Need for higher-order expansions of Pr[H n x] EVA 2005, AUGUST 15 - p. 22/34

23 Two-term Edgeworth expansion Assume Third-order regular variation k n a(n/k n )=o(1) Expansion of cdf of normalized Hill estimator: Pr[H n x] ( ) 1 = Φ(x) ϕ(x) 3 (1 x 2 )+µ n k n { 1 xϕ(x) P 1 (x)+ µ ( n P 2 (x)+ 1 ) k n kn 1 ρ ( ) 1 + o + o(µ 2 n)+o( µ n b(n/k n )) k n µ2 n } for known polynomials P 1 and P 2 Special case of CUNTZ, HAEUSLER &SEGERS (2003) EVA 2005, AUGUST 15 - p. 23/34

24 MAIN RESULT Assume Third-order regular variation k n a(n/k n )=o(1) For the four tests considered earlier: Pr[False rejection] { 1 = α + zϕ(z) Q 1 (z)+ µ ( n Q 2 (z)+ k n kn ( ) 1 + o + o(µ 2 n)+o ( µ n b(n/k n )) k n 2ρ ) 1 ρ + µ 2 n } where Φ(z) =1 α/2 known polynomials Q 1 and Q 2, depending on the test EVA 2005, AUGUST 15 - p. 24/34

25 Comments Error term may disappear for zero, one or two values of k n Finding such k n requires estimation of second-order parameters LR and Bartlett-corrected LR tests very close Small k n : LR tests most accurate Larger k n : Cancellation effect may favor Wald or score tests k n too large: too large bias makes tests inconsistent Bias-corrected intervals: see FERREIRA & DE VRIES (2004) EVA 2005, AUGUST 15 - p. 25/34

26 SIMULATIONS Burr distribution, parameters ρ<0 <γ: F (x) =1 ( x ρ/γ +1) 1/ρ, x > 0 Third-order regularly varying: c = γ, τ = ρ, a(t) =b(t) =t ρ Predicted and simulated type I errors of Wald test Score test Likelihood ratio test Bartlett-corrected LR test Settings: γ =1and ρ { 1, 0.5} nominal type I error α {0.1, 0.05} sample size , 000 samples EVA 2005, AUGUST 15 - p. 26/34

27 Simulations: ρ = 1, α = Burr, (γ,ρ)=(1, 1), α=0.1, n=500, N=10000 Type I error Wald Score LR LR Bartlett k γ =1, ρ = 1, α =0.1 EVA 2005, AUGUST 15 - p. 27/34

28 Simulations: ρ = 1, α = Burr, (γ,ρ)=(1, 1), α=0.05, n=500, N=10000 Type I error Wald Score LR LR Bartlett k γ =1, ρ = 1, α =0.05 EVA 2005, AUGUST 15 - p. 28/34

29 Simulations: ρ = 0.5, α = Burr, (γ,ρ)=(1, 0.5), α=0.1, n=500, N=10000 Type I error Wald Score LR LR Bartlett k γ =1, ρ = 0.5, α =0.1 EVA 2005, AUGUST 15 - p. 29/34

30 Simulations: ρ = 0.5, α = Burr, (γ,ρ)=(1, 0.5), α=0.05, n=500, N=10000 Type I error Wald Score LR LR Bartlett k γ =1, ρ = 0.5, α =0.05 EVA 2005, AUGUST 15 - p. 30/34

31 CONCLUSION Inference on positive extreme-value index Pseudo-likelihood in Pareto model for relative excesses Various CI s/tests: Wald Score Likelihood ratio Bartlett corrections Expansions for type I error of two-sided tests Tool: two-term Edgeworth expansion for Hill estimator Good match between predicted and simulated type I error Performance of CI s and tests depends on nominal level threshold underlying distribution type of CI/test EVA 2005, AUGUST 15 - p. 31/34

32 Thank you! EVA 2005, AUGUST 15 - p. 32/34

33 References (1) Beirlant, J, Goegebeur, Y, Segers, J & Teugels, J (2004) Statistics of Extremes: Theory and Applications. Wiley Bingham, NH, Goldie, CM & Teugels, JL (1987) Regular Variation. Cambridge University Press Cheng, S & Pan, J (1998) Scand. J. Statist. 25, Cheng, S & Peng, L (2001) Bernoulli 7, Cuntz, A, Haeusler, E & Segers, J (2003) CentER discussion paper , center.uvt.nl/pub/dp Embrechts, P, Klüppelberg, C & Mikosch, T (1997) Modelling Extremal Events for Insurance and Finance. Springer-Verlag Ferreira, A & de Vries, CG (2004), Geluk, J&deHaan, L (1987) Regular variation, extensions and Tauberian theorems. CWI Tract 40 EVA 2005, AUGUST 15 - p. 33/34

34 References (2) Guillou, A & Hall, P (2001) J. R. Statist. Soc. B 63, de Haan, L & Stadtmüller, U (1996) J. Austral. Math. Soc., Ser. A 61, Hall, P (1992) The Bootstrap and Edgeworth Expansion. Springer-Verlag Hill, BM (1975) Ann. Statist. 3, Vandewalle, B (2004) PhD thesis, Catholic University Leuven EVA 2005, AUGUST 15 - p. 34/34