Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

A data-adaptive maximum penalized likelihood estimation for the generalized extreme value distribution

  • Lee, Youngsaeng (Department of Statistics, Chonnam National University) ;
  • Shin, Yonggwan (Department of Statistics, Chonnam National University) ;
  • Park, Jeong-Soo (Department of Statistics, Chonnam National University)
  • Received : 2017.06.12
  • Accepted : 2017.08.21
  • Published : 2017.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

Maximum likelihood estimation (MLE) of the generalized extreme value distribution (GEVD) is known to sometimes over-estimate the positive value of the shape parameter for the small sample size. The maximum penalized likelihood estimation (MPLE) with Beta penalty function was proposed by some researchers to overcome this problem. But the determination of the hyperparameters (HP) in Beta penalty function is still an issue. This paper presents some data adaptive methods to select the HP of Beta penalty function in the MPLE framework. The idea is to let the data tell us what HP to use. For given data, the optimal HP is obtained from the minimum distance between the MLE and MPLE. A bootstrap-based method is also proposed. These methods are compared with existing approaches. The performance evaluation experiments for GEVD by Monte Carlo simulation show that the proposed methods work well for bias and mean squared error. The methods are applied to Blackstone river data and Korean heavy rainfall data to show better performance over MLE, the method of L-moments estimator, and existing MPLEs.

Keywords

References

  1. Busababodhin P, Seo YA, Park JS, and Kumphon B (2016). LH-moment estimation ofWakeby distribution with hydrological applications. Stochastic Environmental Research and Risk Assessment, 30, 1757-1767. https://doi.org/10.1007/s00477-015-1168-4
  2. Castillo E, Hadi AS, Balakrishnan N, and Sarabia JM (2005). Extreme Value and Related Models with Applications in Engineering and Science, Wiley-Interscience, New Jersey.
  3. Choi H (2015). A note on the dependence conditions for stationary normal sequences. Communications for Statistical Applications and Methods, 22, 647-653. https://doi.org/10.5351/CSAM.2015.22.6.647
  4. Coles S (2001). An Introduction to Statistical Modeling of Extreme Values, Springer, New York.
  5. Coles SG and Dixon MJ (1999). Likelihood-based inference for extreme value models. Extremes, 2, 5-23.
  6. Efron B and Tibshirani RJ (1993). An Introduction to the Bootstrap, Chapman & Hall/CRC, Boca Raton.
  7. Hosking JRM (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society Series B (Methodological), 52, 105-124.
  8. Hosking JRM (2015). Package 'lmom', version 2.5, Retrieved June 9, 2016, from: https://cran.rproject.org/web/packages/lmom/lmom.pdf
  9. Hosking JRM andWallis JR (1997). Regional Frequency Analysis: An Approach based on L-Moments, Cambridge University Press, Cambridge.
  10. Hosking JRM, Wallis JR, and Wood EF (1985). Estimation of the generalized extreme-value distribution by the method of probability weighted moments. Technometrics, 27, 251-261. https://doi.org/10.1080/00401706.1985.10488049
  11. Huard D, Mailhot A, and Duchesne S (2010). Bayesian estimation of intensity-duration-frequency curves and of the return period associated to a given rainfall event. Stochastic Environmental Research and Risk Assessment, 24, 337-347. https://doi.org/10.1007/s00477-009-0323-1
  12. Katz RW, Parlange MB, and Naveau P (2002). Statistics of extremes in hydrology. Advances in Water Resources, 25, 1287-1304. https://doi.org/10.1016/S0309-1708(02)00056-8
  13. Korea Meteorological Administration (2016). Annual daily maximum precipitation record for 75 weather stations, Retrieved May 1, 2017, from: http://www.kma.go.kr
  14. Martins ES and Stedinger JR (2000). Generalized maximum-likelihood generalized extreme-value quantile estimators for hydrologic data. Water Resources Research, 36, 737-744. https://doi.org/10.1029/1999WR900330
  15. Meshgi A and Khalili D (2009). Comprehensive evaluation of regional flood frequency analysis by L- and LH-moments. I. A re-visit to regional homogeneity. Stochastic Environmental Research and Risk Assessment, 23, 119-135. https://doi.org/10.1007/s00477-007-0201-7
  16. Mudholkar GS and Hutson AD (1998). LQ-moments: analogs of L-moments. Journal of Statistical Planning and Inference, 71, 191-208. https://doi.org/10.1016/S0378-3758(98)00088-3
  17. Murshed MS, Seo YA, and Park JS (2014). LH-moment estimation of a four parameter kappa distribution with hydrologic applications. Stochastic Environmental Research and Risk Assessment, 28, 253-262. https://doi.org/10.1007/s00477-013-0746-6
  18. Park JS (2005). A simulation-based hyperparameter selection for quantile estimation of the generalized extreme value distribution. Mathematics and Computers in Simulation, 70, 227-234. https://doi.org/10.1016/j.matcom.2005.09.003
  19. Pericchi LR and Rodriguez-Iturbe I (1985). On the statistical analysis of floods. In Atkinson AC, Fienberg SE (Eds), A Celebration of Statistics (pp. 511-541), Springer, New York.
  20. Yoon S, ChoW, Heo JH, and Kim CE (2010). A full Bayesian approach to generalized maximum like-lihood estimation of generalized extreme value distribution. Stochastic Environmental Research and Risk Assessment, 24, 761-770. https://doi.org/10.1007/s00477-009-0362-7
  21. Zhu J, Forsee W, Schumer R, and Gautam M (2013). Future projections and uncertainty assessment of extreme rainfall intensity in the United States from an ensemble of climate models. Climatic Change, 118, 469-485. https://doi.org/10.1007/s10584-012-0639-6