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Parameter Estimation of the Inverted Kumaraswamy Distribution by Using L-Moments Method: An Application on Precipitation Data

Year 2024, Volume: 45 Issue: 3, 629 - 635, 30.09.2024

Kübra Bağcı

https://doi.org/10.17776/csj.1294596

Abstract

Modeling precipitation data plays a critical role in water resource and flood management. Statistical distributions are frequently used in describing hydrological variables. Different distributions and estimation methods have been presented in previous studies for modeling precipitation data. In this study, the inverted Kumaraswamy distribution is considered for its advantageous properties, and the L-moments and maximum likelihood methods are employed in estimating the parameters of the inverted Kumaraswamy distribution. In the application part, the annual maximum monthly precipitations recorded in the Rize, Türkiye are modeled with the inverted Kumaraswamy distribution. To the best of the author’s knowledge, the L-moment method is considered for the first time to estimate the parameters of the inverted Kumaraswamy distribution. In addition, the efficiencies of the estimation methods are compared with a Monte-Carlo simulation study. For evaluating the performances of the estimation methods, the goodness of fit criteria including root mean square error, Kolmogorov Smirnov test, and coefficient of determination (R^2) are used in the application part of the study. The results show that for the data considered, the L-moments method yields more accurate results than the maximum likelihood method in estimating the parameters when the sample size is small. Accordingly, the corresponding distribution with L-moments estimations provides a better fit to precipitation data obtained from the Rize station.

Keywords

L-moments Parameter estimation Inverted kumaraswamy Precipitation

References

  • [1] European Environmental Agency, EEA, (2023) Available at: https://www.eea.europa.eu/tr.
  • [2] Hosking J.R.M., L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics, Journal of the Royal Statistical Society: Series B (Methodological), 52(1) (1990) 105–124.
  • [3] Hosking J. R. M., Wallis J.R., Regional Frequency Analysis, (1997).
  • [4] Amin M.T., Rizwan M., Alazba A.A., A best-fit probability distribution for the estimation of rainfall in northern regions of Pakistan, Open Life Sciences, 11(1) (2016) 432–440.
  • [5] Abd AL-Fattah A.M., El-Helbawy A.A., Al-Dayian G.R., Inverted Kumaraswamy Distribution: Properties and Estimation, Pakistan Journal of Statistics, 33 (1) (2017).
  • [6] Usman R.M., Ahsan ul Haq M., The Marshall-Olkin extended inverted Kumaraswamy distribution: Theory and applications, J. King Saud Univ. Sci., 32(1) (2020) 356–365.
  • [7] Bagci K., Arslan T., Celik H.E., Inverted Kumarswamy distribution for modeling the wind speed data: Lake Van, Turkey, Renewable and Sustainable Energy Reviews, 135 (2021) 110110.
  • [8] Şen O., Kahya E., Impacts of climate change on intensity–duration–frequency curves in the rainiest city (Rize) of Turkey, Theor Appl Climatol, 144(3) (2021) 1017–1030.
  • [9] Yozgatlıgil C., Türkeş M., Extreme value analysis and forecasting of maximum precipitation amounts in the western Black Sea subregion of Turkey, International Journal of Climatology, 38(15) (2018) 5447–5458.
  • [10] Cengiz T.M. et al., Combined Use of Graphical and Statistical Approaches for Analyzing Historical Precipitation Changes in the Black Sea Region of Turkey, Water (Basel), 12(3) (2020).
  • [11] Aksu H. et al., Spatial and temporal characterization of standard duration-maximum precipitation over Black Sea Region in Turkey, Natural Hazards, 111(3) (2022) 2379–2405.
  • [12] Lee S.H. and Maeng S.J., Estimation of drought rainfall using L-moments, Irrigation and Drainage, 54(3) (2005) 279–294.
  • [13] Shabri A. and Jemain A.A., LQ-moments: Parameter estimation for kappa distribution, Sains Malays, 39(5) (2010) 845–850.
  • [14] Wan Zin W.Z., Jemain A.A., and Ibrahim K., The best fitting distribution of annual maximum rainfall in Peninsular Malaysia based on methods of L-moment and LQ-moment, Theor Appl Climatol, 96(3) (2009) 337–344.
  • [15] Ngongondo C.S. et al., Regional frequency analysis of rainfall extremes in Southern Malawi using the index rainfall and L-moments approaches, Stochastic Environmental Research and Risk Assessment, 25(7) (2011) 939–955.
  • [16] Galoie M., Zenz G., Eslamian S., Application of L-moments for IDF determination in an Austrian basin, International Journal of Hydrology Science and Technology, 3(1) (2013) 30–48.
  • [17] Rahman A.S. et al., A study on selection of probability distributions for at-site flood frequency analysis in Australia, Natural Hazards, 69(3) (2013) 1803–1813.
  • [18] Malekinezhad H., Zare-Garizi A., Regional frequency analysis of daily rainfall extremes using L-moments approach, Atmósfera, 27(4) (2014) 411–427.
  • [19] Li Z. et al., Frequency analysis of precipitation extremes in Heihe River basin based on generalized Pareto distribution, Stochastic Environmental Research and Risk Assessment, 28(7) (2014) 1709–1721.
  • [20] Zhou Z. et al., Frequency Analysis for Predicting Extreme Precipitation in Changxing Station of Taihu Basin, China, J Coast Res,(68(10068)) (2014) 144–151.
  • [21] ul R. Khan M.S., Hussain Z., and Ahmad I., Effects of L-Moments, Maximum Likelihood and Maximum Product of Spacing Estimation Methods in Using Pearson Type-3 Distribution for Modeling Extreme Values, Water Resources Management, 35(5) (2021) 1415–1431.
  • [22] Anli A.S., Apaydin H., and Öztürk F., Regional frequency analysis of the annual maximum precipitation observed in Trabzon Province, Tarim Bilimleri Dergisi, 15(3) (2009) 240–248.
  • [23] Seckin N., Haktanir T., Yurtal R., Flood frequency analysis of Turkey using L-moments method, Hydrol Process, 25(22) (2011) 3499–3505.
  • [24] Aydoğan D., Kankal M., Önsoy H., Regional flood frequency analysis for Çoruh Basin of Turkey with L-moments approach, J Flood Risk Manag, 9(1) (2016) 69–86.
  • [25] Topcu E., Seckin N., Drought analysis of the Seyhan Basin by using standardized precipitation index SPI and L-moments, J Agric Sci,(Belihuloya) 22(2) (2016) 196–215.
  • [26] Ghiaei F. et al., Regional intensity–duration–frequency analysis in the Eastern Black Sea Basin, Turkey, by using L-moments and regression analysis, Theor Appl Climatol, 131(1) (2018) 245–257.
  • [27] Bagci K. et al., Alpha power inverted Kumaraswamy distribution: Definition, different estimation methods, and application, Pakistan Journal of Statistics and Operation Research,(2022) 13–25.
  • [28] Greenwood J.A. et al., Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form, Water Resour Res, 15(5) (1979) 1049–1054.

Year 2024, Volume: 45 Issue: 3, 629 - 635, 30.09.2024

Kübra Bağcı

https://doi.org/10.17776/csj.1294596

Abstract

References

  • [1] European Environmental Agency, EEA, (2023) Available at: https://www.eea.europa.eu/tr.
  • [2] Hosking J.R.M., L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics, Journal of the Royal Statistical Society: Series B (Methodological), 52(1) (1990) 105–124.
  • [3] Hosking J. R. M., Wallis J.R., Regional Frequency Analysis, (1997).
  • [4] Amin M.T., Rizwan M., Alazba A.A., A best-fit probability distribution for the estimation of rainfall in northern regions of Pakistan, Open Life Sciences, 11(1) (2016) 432–440.
  • [5] Abd AL-Fattah A.M., El-Helbawy A.A., Al-Dayian G.R., Inverted Kumaraswamy Distribution: Properties and Estimation, Pakistan Journal of Statistics, 33 (1) (2017).
  • [6] Usman R.M., Ahsan ul Haq M., The Marshall-Olkin extended inverted Kumaraswamy distribution: Theory and applications, J. King Saud Univ. Sci., 32(1) (2020) 356–365.
  • [7] Bagci K., Arslan T., Celik H.E., Inverted Kumarswamy distribution for modeling the wind speed data: Lake Van, Turkey, Renewable and Sustainable Energy Reviews, 135 (2021) 110110.
  • [8] Şen O., Kahya E., Impacts of climate change on intensity–duration–frequency curves in the rainiest city (Rize) of Turkey, Theor Appl Climatol, 144(3) (2021) 1017–1030.
  • [9] Yozgatlıgil C., Türkeş M., Extreme value analysis and forecasting of maximum precipitation amounts in the western Black Sea subregion of Turkey, International Journal of Climatology, 38(15) (2018) 5447–5458.
  • [10] Cengiz T.M. et al., Combined Use of Graphical and Statistical Approaches for Analyzing Historical Precipitation Changes in the Black Sea Region of Turkey, Water (Basel), 12(3) (2020).
  • [11] Aksu H. et al., Spatial and temporal characterization of standard duration-maximum precipitation over Black Sea Region in Turkey, Natural Hazards, 111(3) (2022) 2379–2405.
  • [12] Lee S.H. and Maeng S.J., Estimation of drought rainfall using L-moments, Irrigation and Drainage, 54(3) (2005) 279–294.
  • [13] Shabri A. and Jemain A.A., LQ-moments: Parameter estimation for kappa distribution, Sains Malays, 39(5) (2010) 845–850.
  • [14] Wan Zin W.Z., Jemain A.A., and Ibrahim K., The best fitting distribution of annual maximum rainfall in Peninsular Malaysia based on methods of L-moment and LQ-moment, Theor Appl Climatol, 96(3) (2009) 337–344.
  • [15] Ngongondo C.S. et al., Regional frequency analysis of rainfall extremes in Southern Malawi using the index rainfall and L-moments approaches, Stochastic Environmental Research and Risk Assessment, 25(7) (2011) 939–955.
  • [16] Galoie M., Zenz G., Eslamian S., Application of L-moments for IDF determination in an Austrian basin, International Journal of Hydrology Science and Technology, 3(1) (2013) 30–48.
  • [17] Rahman A.S. et al., A study on selection of probability distributions for at-site flood frequency analysis in Australia, Natural Hazards, 69(3) (2013) 1803–1813.
  • [18] Malekinezhad H., Zare-Garizi A., Regional frequency analysis of daily rainfall extremes using L-moments approach, Atmósfera, 27(4) (2014) 411–427.
  • [19] Li Z. et al., Frequency analysis of precipitation extremes in Heihe River basin based on generalized Pareto distribution, Stochastic Environmental Research and Risk Assessment, 28(7) (2014) 1709–1721.
  • [20] Zhou Z. et al., Frequency Analysis for Predicting Extreme Precipitation in Changxing Station of Taihu Basin, China, J Coast Res,(68(10068)) (2014) 144–151.
  • [21] ul R. Khan M.S., Hussain Z., and Ahmad I., Effects of L-Moments, Maximum Likelihood and Maximum Product of Spacing Estimation Methods in Using Pearson Type-3 Distribution for Modeling Extreme Values, Water Resources Management, 35(5) (2021) 1415–1431.
  • [22] Anli A.S., Apaydin H., and Öztürk F., Regional frequency analysis of the annual maximum precipitation observed in Trabzon Province, Tarim Bilimleri Dergisi, 15(3) (2009) 240–248.
  • [23] Seckin N., Haktanir T., Yurtal R., Flood frequency analysis of Turkey using L-moments method, Hydrol Process, 25(22) (2011) 3499–3505.
  • [24] Aydoğan D., Kankal M., Önsoy H., Regional flood frequency analysis for Çoruh Basin of Turkey with L-moments approach, J Flood Risk Manag, 9(1) (2016) 69–86.
  • [25] Topcu E., Seckin N., Drought analysis of the Seyhan Basin by using standardized precipitation index SPI and L-moments, J Agric Sci,(Belihuloya) 22(2) (2016) 196–215.
  • [26] Ghiaei F. et al., Regional intensity–duration–frequency analysis in the Eastern Black Sea Basin, Turkey, by using L-moments and regression analysis, Theor Appl Climatol, 131(1) (2018) 245–257.
  • [27] Bagci K. et al., Alpha power inverted Kumaraswamy distribution: Definition, different estimation methods, and application, Pakistan Journal of Statistics and Operation Research,(2022) 13–25.
  • [28] Greenwood J.A. et al., Probability weighted moments: Definition and relation to parameters of several distributions expressable in inverse form, Water Resour Res, 15(5) (1979) 1049–1054.
There are 28 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Natural Sciences
Authors

Kübra Bağcı 0000-0002-6679-9738

Publication Date September 30, 2024
Submission Date May 9, 2023
Acceptance Date September 10, 2024
Published in Issue Year 2024Volume: 45 Issue: 3

Cite

APA Bağcı, K. (2024). Parameter Estimation of the Inverted Kumaraswamy Distribution by Using L-Moments Method: An Application on Precipitation Data. Cumhuriyet Science Journal, 45(3), 629-635. https://doi.org/10.17776/csj.1294596
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