Modeling of Annual  Maximum Temperature Using Weibull Distribution With Application:  A Case Study in Sebha City

H. A.  ALASWED; Ibrahim Suliman Hanaish

المؤلفون

H. A. ALASWED Assistant Professor at Statistics Department, Faculty of Science, Sebha University-Libya haf.alaswed.sebhau.edu.ly haf.alaswed@sebhau.edu.ly
Ibrahim Suliman Hanaish , Assistant Professor at Statistics Department, Libyan Academy- Misurata,

الكلمات المفتاحية:

Domain of Attraction، Max-Weibull، Constant، Maximum

الملخص

Weibull distribution appears very frequently in practical when we observed data that represent extreme values. In this paper, the first objective is to identify the domain of attraction of a given distribution to belong in the domain attraction of maximum Weibull distribution by using Castillo and de Haan, necessary and sufficient condition. The second objective is used diagnostics plot by using three different plots, probability paper (P-P) plot, Quantile Quantile (Q-Q) plot and Return level (R-L) plot to detect an appropriate distribution of extreme. The third objective of the present study is to model the behavior of annual maximum temperature data by using maximum Weibull distribution. Maximum Likelihood Estimation (MLE) is used to estimate the parameters of distribution. The results show that the maximum temperature is significant to be fitted by max-Weibull model and it is the better choice basis on LR-test and graphical diagnostics. For the parameter estimation, LR-test and diagnostic plots calculations, we use the R programming language with packages of fExtreme and ismev to perform these calculations.

المراجع

Fisher, Ronald Aylmer, and Leonard Henry Caleb Tippett. "Limiting forms of the frequency distribution of the largest or smallest member of a sample." In Mathematical Proceedings of the Cambridge Philosophical Society Cambridge University Press, vol. 24, no. 2, (1928) : PP180-190.

R. Von Mises, “Über allgemeine Quadraturformeln.,” J. für die reine und Angew. Math., vol. 174, pp. 56–67, 1936.

B. Gnedenko, “Sur la distribution limite du terme maximum d’une serie aleatoire,” Ann. Math., pp. 423–453, 1943.

De Haan, Laurens. "Weak limits of sample range." Journal of Applied Probability 11.4 (1974): 836-841.‏

Reiss, Rolf-Dieter, Michael Thomas, and R. D. Reiss. Statistical analysis of extreme values. Vol. 2. Basel: Birkhäuser, 1997.‏

Reiss, Thomas. "Drug discovery of the future: the implications of the human genome project." Trends in biotechnology 19.12 (2001): 496-499.‏

Kotz, Samuel, and Saralees Nadarajah. Extreme value distributions: theory and applications. World Scientific, ISBN 1860942245, (2000):P1.

Coles, Stuart, Joanna Bawa, Lesley Trenner, and Pat Dorazio. An introduction to statistical modeling of extreme values. Vol. 208. London: Springer, ISBN 1852334592, (2001):PP4-12,59-64.

Beirlant, Jan, Yuri Goegebeur, Johan Segers, and Jozef Teugels. Statistics of extremes: theory and applications. John Wiley & Sons, ISBN 0-471-97647-4, (2004):PP19-42.

E. Castillo, A. S. Hadi, N. Balakrishnan, and J.-M. Sarabia, Extreme value and related models with applications in engineering and science. Wiley Hoboken, NJ, 2005:P108,P148

Reiss, Henning, and Ingrid Kröncke. "Seasonal variability of benthic indices: an approach to test the applicability of different indices for ecosystem quality assessment." Marine Pollution Bulletin 50.12 (2005): 1490-1499.‏

J. Beirlant, Y. Goegebeur, J. Segers, and J. L. Teugels, Statistics of extremes: theory and applications. John Wiley & Sons, 2006:PP56-59.

Matas, Jiri, Charles Galambos, and Josef Kittler. "Robust detection of lines using the progressive probabilistic hough transform." Computer vision and image understanding 78.1 (2000): 119-137.

Galambos, Janos, and Nicholas Macri. "The life length of humans does not have a limit." Journal of Applied Statistical Science 9.4 (2000): 253-264.‏

Nadarajah, Saralees. "The exponentiated Gumbel distribution with climate application." Environmetrics: The official journal of the International Environmetrics Society 17.1 (2006): 13-23.

K. Abbas and T. Yincai, “Comparison of Estimation Methods for Frechet Distribution with Known Shape.,” Casp. J. Appl. Sci. Res., vol. 1, no. 10, 2012.

E. Omey, F. Mallor, and E. Nualart, “An introduction to statistical modelling of extreme values. Application to calculate extreme wind speeds,” 2009.

Bardsley, Earl. "The Weibull distribution as an extreme value model for transformed annual maxima." Journal of Hydrology (New Zealand) 58.2 (2019): 123-131.‏

Wuertz, Diethelm. "fextremes: Rmetrics-extreme financial market data." R package version 2100 (2013).‏

Gilleland, Eric, Mathieu Ribatet, and Alec G. Stephenson. "A software review for extreme value analysis." Extremes 16