- Volume 13 Issue 1
DOI QR Code
Derivation of Probability Plot Correlation Coefficient Test Statistics and Regression Equation for the GEV Model based on L-moments
L-모멘트 법 기반의 GEV 모형을 위한 확률도시 상관계수 검정 통계량 유도 및 회귀식 산정
- Ahn, Hyunjun (Civil and Environmental Engineering, Yonsei University) ;
- Jeong, Changsam (Civil and Environmental Engineering, Induk University) ;
- Heo, Jun-Haeng (Civil and Environmental Engineering, Yonsei University)
- Received : 2020.02.07
- Accepted : 2020.02.28
- Published : 2020.03.31
One of the important problem in statistical hydrology is to estimate the appropriated probability distribution for a given sample data. For the problem, a goodness-of-fit test is conducted based on the similarity between estimated probability distribution and assumed theoretical probability distribution. Probability plot correlation coefficient test (PPCC) is one of the goodness-of-fit test method. PPCC has high rejection power and its application is simple. In this study, test statistics of PPCC were derived for generalized extreme value distribution (GEV) models based on L-moments and these statistics were suggested by the multiple and nonlinear regression equations for its usability. To review the rejection power of the newly proposed method in this study, Monte Carlo simulation was performed with other goodness-of-fit tests including the existing PPCC test. The results showed that PPCC-A test which is proposed in this study demonstrated better rejection power than other methods, including the existing PPCC test. It is expected that the new method will be helpful to estimate the appropriate probability distribution model.
- Ahn, H., Shin, H., Kim, S., and Heo, J. -H. (2014). Comparison on Probability Plot Correlation Coefficient Test Considering Skewness of Sample for the GEV Distribution. Journal of Korea Water Resources Association. 47(2): 161-170. https://doi.org/10.3741/JKWRA.2014.47.2.161
- Arnell, N. W., Beran, M., and Hosking, J. R. M. (1986). Unbiased Plotting Positions for the General Extreme Value Distribution. Journal of Hydrology. 86: 59-69. https://doi.org/10.1016/0022-1694(86)90006-5
- Blom, G. (1958). Statistical Estimates and Transformed Beta Variables. John Wiley and Sons. New York.
- Chowdhury, J. D., Stedinger, J. R., and Lu, L. H. (1991). Goodness-of-fit Tests for Regional Generalized ExTreme Value Flood Distributions. Water Resources Research. 27(7): 1765-1776. https://doi.org/10.1029/91WR00077
- Cunnane, C. (1978). Unbiased Plotting Positions-A Review. Journal of Hydrology. 37(3/4): 205-222. https://doi.org/10.1016/0022-1694(78)90017-3
- Filliben, J. J. (1969). Simple and Robust Linear Estimation of the Location Parameter of a Symmetric Distribution. Unpublished Ph.D. Dissertation, Princeton University. Princeton. New Jersey.
- Filliben, J. J. (1975). The Probability Plot Correlation Coefficient Test for Normality. Technometrics. 17(1): 111-117. https://doi.org/10.1080/00401706.1975.10489279
- Goel, N. K. and De, M. (1993). Development of Unbiased Plotting Position Formula for General Extreme Value Distribution. Stochastic Environmental Research and Risk Assessment. 7: 1-13.
- Gringorten, I. I. (1963). A Plotting Rule for Extreme Probability Paper. Journal of Geophysical Research. 68(3): 813-814. https://doi.org/10.1029/JZ068i003p00813
- Gumbel, E. J. (1958). Statistics of Extremes. Columbia University Press. New York, NY.
- Hazen, A. (1914). Storage to Be Provided in Impounding Reservoirs for Municipal Water Supply. Transactions American society of Civil Engineers. 1308(77): 1547-1550.
- Heo, J. -H., Kim, K. -D., and Han, J. -H. (1999). Derivation of Rainfall Intensity-duration-frequency Equation Based on the Approproate Probability Distribution. Journal of Korea Water Resources Association. 32: 247-254.
- Heo, J. -H., Kho, Y., Shin, H., Kim, S., and Kim, T. (2008). Regression Equations of Probability Plot Correlation Coefficient Test Statistics from Several Probability Distributions. Journal of Hydrology. 355: 1-15. https://doi.org/10.1016/j.jhydrol.2008.01.027
- Heo, J. -H., Kho, Y. -W., and Kim, K. -D. (2001). Test Statistics Derivation and Power Test for Probability Plot Correlation Coefficient Goodness of Fit Test. Journal of Korean Society of Civil Engineering. 21(2-b): 85-92.
- In-na, N., and Nguyen, V-T-V. (1989). An Unbiased Plotting Position Formula for the Generalized Extreme Value Distribution. Journal of Hydrology. 106: 193-209. https://doi.org/10.1016/0022-1694(89)90072-3
- Jenkinson, A. F. (1955). The Frequency Distribution of the Annual Maximum (or Minimum) Values of Meteorological Elements. Quarterly Journal of the Royal Meteorological Society. 87: 158-171.
- KICT (2000). Research Report on the Development of Water Management Techniques, Vol. I. Preparation of Korea Probability Rainfall Diagram. Gwacheon: KICT.
- Kim, S., Shin, H., Joo, K., and Heo, J.-H. (2012). Development of Plotting Position for the General Extreme Value Distribution. Journal of Hydrology. 475: 259-269. https://doi.org/10.1016/j.jhydrol.2012.09.055
- Kim, S., Shin, H., Ahn. H., and Heo, J.-H. (2015). Development of an Unbiased Plotting Position Formula Considering the Coefficient of Skewness for the Generalized Logistic Distribution. Journal of Hydrology. 527: 471-481. https://doi.org/10.1016/j.jhydrol.2015.05.002
- Kimball, B. F. (1946). Assignment of Frequencies to a Completely Ordered Set of Sample Data. Transaction on the American Geophysical Union. 27: 843- 846. https://doi.org/10.1029/TR027i006p00843
- Looney, S. W. and Gulledge, T. R. (1985). Use the CorRelation Coefficient with Normal Probability Plots. The American Statistician. 39(1): 75-79. https://doi.org/10.2307/2683917
- Stedinger, J. R., Vogel, R. M., and Foufoula-Georgiou, E. (1993). Frequency Analysis of Extreme Events. Handbook of Hydrology. D. R. Maidment, de., McGraw-Hill. New York, N.Y. pp. 18.24-18.26
- Vogel, R. M. (1986). The Probability Plot Correlation Coefficient Test for the Normal, Lognormal, and Gumbel Distributional Hypotheses. Water Resources Research. 22(4): 587-590. https://doi.org/10.1029/WR022i004p00587
- Vogel, R. M. and McMartin, D. E. (1991). Probability Plot Goodness-of-fit and Skewness Estimation Procedures for the Pearson Type III Distribution. Water Resources Research. 27(12): 3149-3158. https://doi.org/10.1029/91WR02116