A Closer Look at the Hill Estimator: Edgeworth Expansions and Confidence Intervals Erich HAEUSLER Johan SEGERS University of Giessen Tilburg University http://www.uni-giessen.de http://www.center.nl F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 1/34
INTRODUCTION � Ordered sample X 1: n ≤ · · · ≤ X n : n from Pareto-type cdf F � INTRODUCTION � PARETO MODEL � H ILL (1975) estimator for positive extreme-value index γ � CI’S AND TESTS � EDGEWORTH EXPANSIONS � MAIN RESULT � SIMULATIONS k H n ( k ) = 1 � CONCLUSION ˆ � log X n − k + i : n − log X n − k : n k i =1 � Simple and popular � Asymptotic properties well known � ˆ � H n ( k n ) d � k n − 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 F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 2/34
Confidence intervals and tests � Con fi dence intervals and hypothesis tests less studied � INTRODUCTION � PARETO MODEL � CI of nominal level 1 − α : � CI’S AND TESTS � EDGEWORTH EXPANSIONS � � 1 ± z � MAIN RESULT ˆ symmetric CI : √ H n ( k ) � SIMULATIONS � CONCLUSION k with � � 1 ∓ z � ˆ asymmetric CI : H n ( k ) √ k Φ( z ) = 1 − α/ 2 � Relevance: � Existence of moments � CI’s/tests for exceedance probabilities, quantiles,. . . [V ANDEWALLE 2004] F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 3/34
Questions � Which CI to be preferred? � INTRODUCTION � PARETO MODEL � Yet other CI’s? � CI’S AND TESTS � EDGEWORTH EXPANSIONS � Which k to use for which CI? � MAIN RESULT � SIMULATIONS � CONCLUSION � Comparisons between CI’s requires Edgeworth expansions � ˆ �� � � H n ( k ) = Φ( x ) + error term Pr k n − 1 ≤ x γ F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 4/34
Related literature � One-term Edgeworth expansions in C HENG & P AN (1998) and � INTRODUCTION � PARETO MODEL C HENG & P ENG (2001) � CI’S AND TESTS � EDGEWORTH EXPANSIONS � Useful for one-sided CI’s [C HENG & P ENG 2001] � MAIN RESULT � Insuf fi ciently accurate to analyse two-sided CI’s � SIMULATIONS � CONCLUSION � Expansions in terms of Gamma distributions [C HENG & DE H AAN 2001; G UILLOU & H ALL 2001] � Insuf fi ciently 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 F ERREIRA & DE V RIES (2004) F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 5/34
For proper understanding. . . � Won’t talk about: � INTRODUCTION � PARETO MODEL � Bias reduction � CI’S AND TESTS � EDGEWORTH EXPANSIONS � Data-driven methods to choose threshold � MAIN RESULT � Comparisons with other estimators � SIMULATIONS � CONCLUSION � 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 F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 6/34
Outline � Inference in Pareto model � INTRODUCTION � PARETO MODEL � CI’s and hypothesis tests for extreme-value index � CI’S AND TESTS � EDGEWORTH EXPANSIONS � Edgeworth expansions for normalized Hill estimator � MAIN RESULT � SIMULATIONS � CONCLUSION � Main result � Simulations � Conclusion F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 7/34
PARETO MODEL � Cdf and pdf of Pareto( 1 /γ ): � INTRODUCTION � PARETO MODEL � CI’S AND TESTS 1 − x − 1 /γ , � EDGEWORTH EXPANSIONS G γ ( x ) = � MAIN RESULT � SIMULATIONS 1 � CONCLUSION γ x − 1 − 1 /γ p γ ( x ) = for x > 1 � Inference on γ > 0 from iid Y 1 , . . . , Y k ∼ p γ ? � Estimation � Testing � Con fi dence intervals F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 8/34
Likelihood computations � Log-likelihood of γ given Y 1 , . . . , Y k � INTRODUCTION � PARETO MODEL � ˆ � CI’S AND TESTS k � � EDGEWORTH EXPANSIONS H k � MAIN RESULT � + constant ℓ k ( γ ) = log p γ ( Y i ) = − k γ + log( γ ) � SIMULATIONS � CONCLUSION i =1 k 1 ˆ � H k = log( Y i ) k i =1 � Score � ˆ � ℓ k ( γ ) = k H k ˙ γ − 1 γ � Fisher information � ∂ � = 1 I ( γ ) = Var γ ∂γ log p γ ( Y ) γ 2 F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 9/34
MLE and deviance statistic � ˆ H k is suf fi cient statistic and MLE for γ � INTRODUCTION � PARETO MODEL � CI’S AND TESTS √ � EDGEWORTH EXPANSIONS d k ( ˆ → N (0 , γ 2 ) , H k − γ ) k → ∞ � MAIN RESULT � SIMULATIONS � CONCLUSION � Deviance statistic (likelihood ratio) at γ : � � ℓ k ( ˆ D k ( γ ) = 2 H k ) − ℓ k ( γ ) � ˆ � ˆ H k H k = 2 k γ − 1 − log γ d χ 2 → 1 , k → ∞ F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 10/34
Hypothesis tests (1) � Test for H 0 : γ 0 = γ versus H 1 : γ 0 � = γ at nominal level 1 − α � INTRODUCTION � PARETO MODEL � z = z 1 − α/ 2 standard-normal quantile Φ( z ) = 1 − α/ 2 � CI’S AND TESTS � EDGEWORTH EXPANSIONS � Reject H 0 : γ 0 = γ if T k ( γ ) > z 2 where � MAIN RESULT � SIMULATIONS � CONCLUSION Test Test statistic T k ( γ ) � ˆ � 2 H k − γ Wald k ˆ H k � ˆ � 2 H k − γ Score k γ � ˆ � ˆ H k H k Likelihood ratio D k ( γ ) = 2 k γ − 1 − log γ � � 1 + 1 � Bartlett-corrected LR D k ( γ ) 6 k F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 11/34
Hypothesis tests (2) � Wald and score tests also Bartlett correctable � INTRODUCTION � PARETO MODEL � One-sided tests: similarly � CI’S AND TESTS � EDGEWORTH EXPANSIONS � Corresponding confidence intervals at nominal level 1 − α : � MAIN RESULT � SIMULATIONS � CONCLUSION { All γ > 0 for which H 0 : γ 0 = γ is not rejected at level 1 − α } F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 12/34
CI’S AND TESTS FOR EVI � Pareto domain of attraction � INTRODUCTION � PARETO MODEL � cdf F has extreme-value index γ > 0 iff � CI’S AND TESTS � EDGEWORTH EXPANSIONS � MAIN RESULT 1 − F ( ux ) � SIMULATIONS � CONCLUSION Pr[ X/u > x | X > u ] = 1 − F ( u ) x − 1 /γ , → u → ∞ � Relative excesses over high thresholds are asymptotically Pareto( 1 /γ ) distributed F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 13/34
Hill estimator � Heuristic: � INTRODUCTION � PARETO MODEL 1. Take large threshold u = X n − k : n � CI’S AND TESTS � EDGEWORTH EXPANSIONS 2. Relative excesses Y i : k = X n − k + i : n /X n − k : n for i = 1 , . . . , k � MAIN RESULT � SIMULATIONS 3. Pretend Y 1: k , . . . , Y k : k are order statistics from iid � CONCLUSION Pareto( 1 /γ ) sample � Pseudo-likelihood inference: H ILL (1975) k H n ( k ) = 1 log X n − k + i : n ˆ � k X n − k : n i =1 � Other interpretations [E MBRECHTS ET AL . 1997; B EIRLANT ET AL . 2004] F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 14/34
Hypothesis tests and CI’s (1) � Fix k � INTRODUCTION � PARETO MODEL � Reject H 0 : γ 0 = γ at nominal level 1 − α if T n,k ( γ ) > z 2 � CI’S AND TESTS 1 − α/ 2 � EDGEWORTH EXPANSIONS � MAIN RESULT � SIMULATIONS Test statistic T n,k ( γ ) Test � CONCLUSION � ˆ � 2 H n ( k ) − γ Wald k ˆ H n ( k ) � ˆ � 2 H n ( k ) − γ Score k γ � ˆ � ˆ H n ( k ) H n ( k ) Likelihood ratio D n,k ( γ ) = 2 k − 1 − log γ γ � � 1 + 1 � Bartlett-corrected LR D n,k ( γ ) 6 k F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 15/34
Hypothesis tests and CI’s (2) � Con fi dence intervals � INTRODUCTION � PARETO MODEL � CI’S AND TESTS { All γ > 0 for which H 0 : γ 0 = γ is not rejected } � EDGEWORTH EXPANSIONS � MAIN RESULT � SIMULATIONS � False rejection of H 0 : γ 0 = γ (type I error) � CONCLUSION 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 F A C U L T Y O F E C O N O M I C S A N D B U S I N E S S A D M I N I S T R A T I O N EVA 2005, AUGUST 15 - p. 16/34
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