THREE-PARAMETER LOGNORMAL DISTRIBUTION: PARAMETRIC ESTIMATION USING L-MOMENT AND TL-MOMENT APPROACH

Authors

  • Nur Amalina Mat Jan Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • Ani Shabri Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • Shuhaida Ismail Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • Basri Badyalina Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • Siti Sarah Abadan Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • Norhafizah Yusof Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia

DOI:

https://doi.org/10.11113/jt.v78.9202

Keywords:

L-moments, TL-moments, three-parameter lognormal distribution

Abstract

The three-parameter lognormal (LN3) distribution has been applied to the frequency analysis of flood events. L-moment and TL-moment methods are applied in estimating parameters of the LN3 distribution which are L-moment, η = 0 and TL-moment, η = 1, 2, 3, and 4 to the LN3 distribution. A simulation study is conducted in this paper by fitting this distribution to generate LN3 and non LN3 samples. Relative Root Mean Square Error (RRMSE) and relative bias are evaluated to illustrate the performance of this distribution. The performance of TL-moments approach was compared with L-moments based on the streamflow data from Sg. Trolak and Sg. Slim which are located in Perak, Malaysia. The results showed that TL-moments approach produced a better result at high quantile estimation compared to L-moments. 

References

Levy, H. and Y. Kroll. 1976. Stochastic Dominance With Riskless Assets. Journal of Financial and Quantitative Analysis. 11: 743-777.

Limpert, E., W. A. Stahel, and M. Abbt. 2001. Log-Normal Distributions Across The Sciences: Keys And Clues. Bioscience. 51(5): 341-352.

Takara, K. and T. Takasao. 1990. Comparison for Parameter Estimation Methods for Hydrologic Frequency Analysis Models. Proceedings of Hydraulic Engineering, JSCE. 34: 7-12.

Yamaguchi, M. 1996. Intercomparison of Parameter Estimation Methods in Extreme Wave Analysis. 25th International Conference on Coastal Engineering. Florida, USA. 2-6 September 1996. 900-913.

Takeuchi, K. and K. Tsuchiya. 1988. On Relative Accuracy of PWM Estimates of Normal and 3-parameter Log-normal Distribution. Proceeding of Japan Society of Civil Engineers. 393(11-9): 103-112.

Sankarasubramanian, A. and K. Srinivasan. 1999. Investigation and Comparison of Sampling Properties of L-moment and Conventional Moments. Journal of Hydrology. 218: 13-34.

Chen, Y. F., S. Xu, Z. Sha, P. V. Gelder, and S. H. Gu. 2004. Study on L-moment Estimations for Log-normal Distibution with Historical Flood Data. GIS and Remote Sensing in Hydrology, Water Resources and Environment (Proceeding of ICGRHWE. Three Gorges Dam, China. September 2003. 289.

Bílková, D. 2011. Estimating Parameters of Lognormal Distribution Using the Method of L-Moments. Research Journal of Economics, Business and ICT. 4(1): 4-9.

Bílková, D. 2012. Modelling of Wage and Income Distributions Using the Method of L-Moments. Journal of Mathematics and System Science. 2(1): 13-19.

Nedvìd, J. and I. Malá. 2011. Analysis of Incomes in the Czech Republic Using the Lognormal Distribution. Research Journals of Economics, Business and ICT. 4: 58-64.

Hosking, J. R. M. 1990. L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics. Journal of the Royal Statistical Society: Series B. 52: 105-124.

Elamir, E. A. H. and A. H. Seheult. 2003. Trimmed L-Moments. Computational Statistics & Data Analysis. 43: 299-314.

Asquith, W. H. 2007. L-moments and TL-Moments of The Generalized Lambda Distribution. Computational Statistics & Data Analysis. 51: 4484-4496.

Hosking, J. R. M. 2007. Some Theory and Practical Uses of Trimmed L-Moments. Journal of Statistical Planning and Inference.137: 3024-3039.

Abdul Moniem, I. B. and Y. M. Selim. 2009. TL-Moments and L-Moments estimation for the Generalized Pareto Distribution. Applied Mathematical Sciences. 3(1): 43–52.

Abdul Moniem, I. B. 2010. TL-moments and L-moments Estimation for the Weibull distribution. Advances and Applications in Statistics. 15(1): 83-99.

Ahmad, U. N., A. Shabri, and Z. A. Zakaria. 2011. Trimmed L-Moments (1,0) For The Generalized Pareto Distribution. Hydrological Sciences Journal. 56(6): 1053-1060.

Ahmad, U. N., A. Shabri, and Z. A. Zakaria. 2011. Flood Frequency Analysis of Annual Maximum Stream Flows Using L-Moments and TL-Moments Approach. Applied Mathematical Sciences. 5(5): 243-253.

Shabri, A., Z. M. Daud, and N. M. Ariff. 2011. Regional Analysis of Annual Maximum Rainfall Using TL-Moments Method. Theoretical and Applied Climatology. 104(3): 561-570.

Noura, A. T. and Abu El-Magd. 2010. TL-Moments of The Exponentiated Generalized Extreme Value Distribution. Journal of Advanced Research (JARE). 1: 351-359.

Shahzad, M. N. and Z. Asghar. 2013. Comparing TL-Moments, L-Moments and Conventional Moments of Dagum Distribution by Simulated Data. Revista Colombiana de Estadística. 36(1): 79-93.

Bílková, D. 2014. Robust Parameter Estimations Using L-Moments, TL-Moments and the Order Statistics. American Journal of Applied Mathematics. 2(2): 36-53.

Bílková, D. 2014. L-Moments and TL-Moments as an Alternative Tool of Statistical Data Analysis. Journal of Applied Mathematics and Physics. 2: 919-929.

Bílková, D. 2014. Trimmed L-Moments: Analogy of Classical L-Moments. American Journal of Mathematics and Statistics. 4(2): 80-106.

Bílková, D. 2014. Alternative Means of Statistical Data Analysis: L-Moments and TL-Moments of Probability Distributions. Statistika. 94(2): 77-94.

Hosking, J. R. M. 1986. The Theory of Probability Weighted Moments. New York: IBM Thomas J. Watson Research Center.

Munro, A. H. and R. A. J. Wixley. 1970. Estimators Based on Order Statistics of Small Samples from a Three-Parameter Lognormal Distribution. Journal of the American Statistical Association. 65: 212-225.

Tolikas, K. and S. Heravi. 2008. The Anderson-Darling Goodness-Of-Fit Test Statistic for the Three-Parameter Lognormal Distribution. Communications in Statistics-Theory and Methods. 37(19): 3135-3143.

Song, D. and Y. Hou. 1988. A New Method for Estimating Parameters of Log-normal Distribution. Nanjing Hydraulic Research Institute Hydrology and Water Resources.

Tukey, J. W. 1977. Expolratory Data Analysis. Massachusetts: Addison-Wesley.

Downloads

Published

2016-06-23

Issue

Vol. 78 No. 6-11: Innovations in Science and Technology I

Section

Science and Engineering

License

Copyright of articles that appear in Jurnal Teknologi belongs exclusively to Penerbit Universiti Teknologi Malaysia (Penerbit UTM Press). This copyright covers the rights to reproduce the article, including reprints, electronic reproductions, or any other reproductions of similar nature.

How to Cite

THREE-PARAMETER LOGNORMAL DISTRIBUTION: PARAMETRIC ESTIMATION USING L-MOMENT AND TL-MOMENT APPROACH. (2016). Jurnal Teknologi, 78(6-11). https://doi.org/10.11113/jt.v78.9202