Wind and Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Plotting positions and approximating first two moments of order statistics for Gumbel distribution: estimating quantiles of wind speed

  • Hong, H.P. (Department of Civil and Environmental Engineering, University of Western Ontario) ;
  • Li, S.H. (Department of Civil and Environmental Engineering, University of Western Ontario)
  • Received : 2013.10.22
  • Accepted : 2014.07.14
  • Published : 2014.10.25
⟨ Previous Next ⟩

Abstract

Probability plotting positions are popular and used as the basis for distribution fitting and for inspecting the quality of the fit because of its simplicity. The plotting positions that lead to excellent approximation to the mean of the order statistics should be used if the objective of the fitting is to estimate quantiles. Since the mean depends on the sample size and is not amenable for simple to use closed form solution, many plotting positions have been presented in the literature, including a new plotting position that is derived based on the weighted least-squares method. In this study, the accuracy of using the new plotting position to fit the Gumbel distribution for estimating quantiles is assessed. Also, plotting positions derived by fitting the mean of the order statistics for all ranks is proposed, and an approximation to the covariance of the order statistics for the Gumbel (and Weibull) variate is given. Relative bias and root-mean-square-error of the estimated quantiles by using the proposed plotting position are shown. The use of the proposed plotting position to estimate the quantiles of annual maximum wind speed is illustrated.

Keywords

References

  1. Balakrishnan, N. and Chan, P.S. (1992), "Order statistics from extreme value distribution, I tables of means, variances and covariances", Commun. Stat.-Simul., 21(4), 1199-1217. https://doi.org/10.1080/03610919208813073
  2. Benard, A. and Bos-Levenbach, E.C. (1953), Het uitzetten van waarnemingen op waarschijnlijkdeids-papier, (The Plotting of Observations on Probability Paper), Statististica Neerlandica, 7, 163-173. https://doi.org/10.1111/j.1467-9574.1953.tb00821.x
  3. Blom, G. (1958), Statistical Estimates and Transformed Beta-Variables, New York, John Wiley & Sons.
  4. Castillo, E. (1988), Extreme Value Theory in Engineering, Academic Press, New York.
  5. Cook, N.J. (2012), "Rebuttal of "Problems in the extreme value analysis", Struct. Saf., 34(1), 418-423. https://doi.org/10.1016/j.strusafe.2011.08.002
  6. Cook, N.J. and Harris, R.I. (2013), "The Gringorten estimator revisited", Wind Struct., 16(4), 355-372. https://doi.org/10.12989/was.2013.16.4.355
  7. Cunnane, C. (1978), "Unbiased plotting-positions-a review", J. Hydrology, 37, 205-222. https://doi.org/10.1016/0022-1694(78)90017-3
  8. David, H.A. and Nagaraja, H.N. (2003), Order Statistics, 3rd Ed., John Wiley & Sons.
  9. Filliben, J.J. (1975), "The probability plot correlation test for normality", Technometrics, 17(1), 111-117. https://doi.org/10.1080/00401706.1975.10489279
  10. Fuglem, M., Parr, G. and Jordaan, I.J. (2013), "Plotting positions for fitting distributions and extreme value analysis", Can. J. Civil. Eng., 40(2), 130-139. https://doi.org/10.1139/cjce-2012-0427
  11. Genschel, U. and Meeker, W.Q. (2010), "A comparison of maximum likelihood and median-rank regression for Weibull estimation", Quality Eng., 22(4), 236-255. https://doi.org/10.1080/08982112.2010.503447
  12. Goda, Y. (2011), "Plotting-position estimator for the L-moment method and quantile confidence interval for the GEV, GPA and Weibull distribution applied for extreme wave analysis", Coast. Eng. J., 53(2) 111-149. https://doi.org/10.1142/S057856341100229X
  13. Gringorten, I.I. (1963), "A plotting rule for extreme probability paper", J. Geophys. Res., 68, 813-814. https://doi.org/10.1029/JZ068i003p00813
  14. Harris, R.I. (1996), "Gumbel re-visited-a new look at extreme value statistics applied to wind speeds", J. Wind Eng. Ind. Aerod., 59, 1-22. https://doi.org/10.1016/0167-6105(95)00029-1
  15. Harter, H.L. (1984), "Another look at plotting positions", Commun. Stat.-Theor. M., 13(13), 1613-1633. https://doi.org/10.1080/03610928408828781
  16. Holmes, J.D. (2001), Wind loading of structures, Spon Press, New York.
  17. Hong, H.P. (2013), "Selection of regressand for fitting the extreme value distributions using the ordinary, weighted and generalized least-squares methods", Reliab. Eng. Syst. Safe., 118, 71-80. https://doi.org/10.1016/j.ress.2013.04.003
  18. Hong, H.P. (1994), "A note on extremal analysis", Struct. Safe., 13(4), 227-233. https://doi.org/10.1016/0167-4730(94)90030-2
  19. Hong, H.P., Li, S.H. and Mara, T. (2013), "Performance of the generalized least-squares method for the extreme value distribution in estimating quantiles of wind speeds", J. Wind Eng. Ind. Aerod., 119, 121-132. https://doi.org/10.1016/j.jweia.2013.05.012
  20. Jordaan, I.J. (2005), Decisions under uncertainty: probabilistic analysis for engineering decisions, Cambridge University Press, New York.
  21. Lieblein, J. (1953), "On the exact evaluation of the variances and covariances of order statistics in samples from the extreme-value distribution", Annal. Math. Stat., 24(2), 282-287. https://doi.org/10.1214/aoms/1177729034
  22. Lieblein, J. (1974), Efficient methods of extreme-value methodology, report NBSIR 74-602, National Bureau of Standards, Washington.
  23. Lieblein, J. and Zelen, M. (1956), "Statistical investigation of the fatigue life of deep-groove ball bearings", J. Res. National Bureau Standards, 57(5), 273-316. https://doi.org/10.6028/jres.057.033
  24. Lloyd, E.H. (1952), "Least-squares estimation of location and scale parameters using order statistics", Biometrika, 39(1-2), 88-95. https://doi.org/10.1093/biomet/39.1-2.88
  25. Pirouzi Fard, M.N. and Holmquist, B. (2008), "Approximations of variances and covariances for order statistics from the standard extreme value distribution", Commun. Stat.-Simul. C., 37(8), 1500-1506. https://doi.org/10.1080/03610910802244059
  26. Pirouzi Fard, M.N. (2010), "Probability plots and order statistics of the standard extreme value distribution", Comput Stat., 25(2), 257-267. https://doi.org/10.1007/s00180-009-0174-8

Cited by

  1. A Mathematical model to estimate the wind power using three parameter Weibull distribution vol.22, pp.4, 2016, https://doi.org/10.12989/was.2016.22.4.393
  2. Mathematical modeling of wind power estimation using multiple parameter Weibull distribution vol.23, pp.4, 2016, https://doi.org/10.12989/was.2016.23.4.351
  3. A probability-based analysis of wind speed distribution and related structural response in southeast China pp.1744-8980, 2018, https://doi.org/10.1080/15732479.2018.1485710