Published

2015-07-01

Parameter Estimation of Power Function Distribution with TL-moments

Estimación de parámetros de distribuciones de funciones de potencia con momentos TL

DOI:

https://doi.org/10.15446/rce.v38n2.51663

Keywords:

Moments, Monte Carlo Simulation, Order Statistics, Parameter estimation, Power function distribution (en)
Distribución de función de potencias, Estadísticas de orden, Estimación de parámetros, Momentos, Simulación de Monte Carlo. (es)

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Authors

  • Mirza Naveed-Shahzad University of Gujrat, Gujrat, Pakistan
  • Zahid Asghar Quaid-i-Azam University, Islamabad, Pakistan
  • Farrukh Shehzad COMSATS Institute of Information Technology, Lahore, Pakistan
  • Mubeen Shahzadi University of Gujrat, Gujrat, Pakistan
Accurate estimation of parameters of a probability distribution is of immense importance in statistics. Biased and imprecise estimation of parameters can lead to erroneous results. Our focus is to estimate the parameter of Power function distribution accurately because this density is now widely used for modelling various types of data.  In this study, L-moments, TL-moments, LL-moments and LH-moments of Power function distribution are derived. In addition, the coefficient of variation, skewness and kurtosis are obtained by method of moments, L-moments and TL-moments. Parameters of the density are estimated using linear moments and compared with method of moments and MLE on the basis of bias, root mean square error and coefficients through simulation study. L-moments proved to be superior for the parameter estimation and this conclusion is equally true for different parametric values and sample size.

La distribución de función de potencias es ampliamente usada. Dada su importancia, es necesario estimar sus parámetros de manera precisa. En este artículo, los momentos TL de la distribución de función de potencias son derivados así como sus casos especiales tales como los momentos L, LL y LH. Los coeficientes de variación, sesgo y curtosis son obtenidos a partir de los momentos L y TL. Los parámetros desconocidos son estimados y los momentos lineales son comparados con el método de momentos y estimadores máximo verosímiles en la base del sesgo, raíz del error cuadrático medio a través de un estudio de simulación. Los momentos L permiten obtener estimaciones más precisas y esta conclusión es verdad para diferentes valores paramétricos y tamaño de muestra.

https://doi.org/10.15446/rce.v38n2.51663

Parameter Estimation of Power Function Distribution with TL-moments

Estimación de parámetros de distribuciones de funciones de potencia con momentos TL

MIRZA NAVEED-SHAHZAD1, ZAHID ASGHAR2, FARRUKH SHEHZAD3, MUBEEN SHAHZADI4

1University of Gujrat, Department of Statistics, Gujrat, Pakistan. Lecturer. Email: nvd.shzd@uog.edu.com
2Quaid-i-Azam University, Department of Statistics, Islamabad, Pakistan. Assistant Professor. Email: zahid.g@gmail.com
3COMSATS Institute of Information Technology, Department of Statistics, Lahore, Pakistan. Assistant Professor. Email: fshehzad.stat@gmail.com
4University of Gujrat, Department of Statistics, Gujrat, Pakistan. School Teacher. Email: mbn.shzdi@gmail.com

Abstract

Accurate estimation of parameters of a probability distribution is of immense importance in statistics. Biased and imprecise estimation of parameters can lead to erroneous results. Our focus is to estimate the parameter of Power function distribution accurately because this density is now widely used for modelling various types of data. In this study, L-moments, TL-moments, LL-moments and LH-moments of power function distribution are derived. In addition, the coefficient of variation, skewness and kurtosis are obtained by method of moments, L-moments and TL-moments. Parameters of the density are estimated using linear moments and compared with method of moments and MLE on the basis of bias, root mean square error and coefficients through simulation study. L-moments proved to be superior for the parameter estimation and this conclusion is equally true for different parametric values and sample size.

Key words: Moments, Monte Carlo Simulation, Order Statistics, Parameter estimation, Power function distribution.

Resumen

La distribución de función de potencias es ampliamente usada. Dada su importancia, es necesario estimar sus parámetros de manera precisa. En este artículo, los momentos TL de la distribución de función de potencias son derivados así como sus casos especiales tales como los momentos L, LL y LH. Los coeficientes de variación, sesgo y curtosis son obtenidos a partir de los momentos L y TL. Los parámetros desconocidos son estimados y los momentos lineales son comparados con el método de momentos y estimadores máximo verosímiles en la base del sesgo, raíz del error cuadrático medio a través de un estudio de simulación. Los momentos L permiten obtener estimaciones más precisas y esta conclusión es verdad para diferentes valores paramétricos y tamaño de muestra.

Palabras clave: distribución de función de potencias, estadísticas de orden, estimación de parámetros, momentos, simulación de Monte Carlo.

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References

1. Abdul-Moniem, I. B. (2007), 'L-moments and TL-moments estimation for the exponential distribution', Far East Journal of Theoretical Statistics 23(1), 51-61.

2. Abdul-Moniem, I. B. (2009), 'TL-moments and L-moments estimation for the Weibull distribution', Advances and Application in Statistics 15(1), 83-99.

3. Abdul-Moniem, I. B. (2012), 'L-moments coefficients of variations', International Journal of Statistics and Systems 7(2), 97-100.

4. Abdul-Moniem, I. & Selim, Y. M. (2009), 'TL-moments and L-moments estimation for the generalized Pareto distribution', Applied Mathematical Sciences 3(1), 43-52.

5. Ahsanullah, M. & Kabir, A. L. (1974), 'A characterization of the power function distribution', The Canadian Journal of Statistics 2(1), 95-98.

6. Ali, M. M., Woo, J. & Nadarajah, S. (2005), 'On the Ratic X/(X+ Y) for the Power Function Distribution', Pakistan Journal of Statistics 21(2), 131-138.

7. Asquith, W. H. (2007), 'L-moments and TL-moments of the generalized lambda distribution', Computational Statistics and Data Analysis 51(9), 4484-4496.

8. Bayazit, M. & Onoz, B. (2002), 'LL-moments for estimating low flow quantiles', Hydrological sciences journal 47(5), 707-720.

9. Chang, S. (2007), 'Characterization of the power function by the independence of record values', Journal of the Chungcheong Mathematical Society 20(2), 140-145.

10. El-Magd, A. & Noura, A. T. (2010), 'TL-moments of the exponentiated generalized extreme value distribution', Journal of Advanced Research 1(4), 351-359.

11. Elamir, E. A. & Seheult, A. H. (2003), 'Trimmed L-moments', Computational Statistics and Data Analysis 43(3), 299-314.

12. Hirano, K. & Porter, J. R. (2003), 'Asymptotic Efficiency in Parametric Structural Models with Parameter-Dependent Support', Econometrica 71(5), 1307-1338.

13. 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.

14. Hosking, J. R. M. & Wallis, J. R. (1995), 'A comparison of unbiased and plotting-position estimators of L moments', Water Resources Research 31(8), 2019-2025.

15. Karvanen, J. (2006), 'Estimation of quantile mixtures via L-moments and trimmed L-moments', Computational Statistics and Data Analysis 51(2), 947-959.

16. Meniconi, M. & Barry, D. M. (1996), 'The power function distribution: A useful and simple distribution to assess electrical component reliability', Microelectronics Reliability 36(9), 1207-1212.

17. Rahman, H., Roy, M. K. & Baizid, A. R. (2012), 'Bayes estimation under conjugate prior for the case of power function distribution', American Journal of Mathematics and Statistics 2(3), 44-48.

18. Saran, J. & Pandey, A. (2004), 'Estimation of parameters of a power function distribution and its characterization by kth record values', Statistica 64(3), 523-536.

19. Shabri, A., Ahmad, U. N. & Zakaria, Z. A. (2009), 'TL-moments and L-moments estimation for the generalized Pareto distribution', Journal of Mathematics Research 10(10), 97-106.

20. Shahzad, M. & Asghar, Z. (2013), 'Comparing TL-Moments, L-Moments and conventional moments of dagum distribution by simulated data', Revista Colombiana de Estadistica 36(1), 79-93.

21. Sinha, S., Singh, P., Singh, D. & Singh, R. (2008), 'Power function distribution characterized by dual generalized order statistics', Journal of Reliability and Statistics Studies 1(1), 18-27.

22. Tavangar, M. (2011), 'Power function distribution characterized by dual generalized order statistics', Journal of Iranian Statistical Society 10(1), 13-27.

23. Vogel, R. M. & Fennessey, N. M. (1993), 'L-moment diagrams should replace product moment diagrams', Water Resources Research 29(6), 1745-1752.

24. Wang, Q. J. (1997), 'LH moments for statistical analysis of extreme events', Water Resources Research 33(12), 2841-2848.

[Recibido en marzo de 2014. Aceptado en noviembre de 2014]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv38n2a01,
    AUTHOR  = {Naveed-Shahzad, Mirza and Asghar, Zahid and Shehzad, Farrukh and Shahzadi, Mubeen},
    TITLE   = {{Parameter Estimation of Power Function Distribution with TL-moments}},
    JOURNAL = {Revista Colombiana de Estadística},
    YEAR    = {2015},
    volume  = {38},
    number  = {2},
    pages   = {321-334}
}

References

Abdul-Moniem, I. B. (2007), ‘L-moments and TL-moments estimation for the exponential distribution’, Far East Journal of Theoretical Statistics 23(1), 51– 61.

Abdul-Moniem, I. B. (2009), ‘TL-moments and L-moments estimation for the Weibull distribution’, Advances and Application in Statistics 15(1), 83–99.

Abdul-Moniem, I. B. (2012), ‘L-moments coefficients of variations’, International Journal of Statistics and Systems 7(2), 97–100.

Abdul-Moniem, I. & Selim, Y. M. (2009), ‘TL-moments and L-moments estimation for the generalized Pareto distribution’, Applied Mathematical Sciences 3(1), 43–52.

Ahsanullah, M. & Kabir, A. L. (1974), ‘A characterization of the power function distribution’, The Canadian Journal of Statistics 2(1), 95–98.

Ali, M. M., Woo, J. & Nadarajah, S. (2005), ‘On the Ratic X/(X+ Y) for the Power Function Distribution’, Pakistan Journal of Statistics 21(2), 131–138.

Asquith, W. H. (2007), ‘L-moments and TL-moments of the generalized lambda distribution’, Computational Statistics and Data Analysis 51(9), 4484–4496.

Bayazit, M. & Onoz, B. (2002), ‘LL-moments for estimating low flow quantiles’, Hydrological sciences journal 47(5), 707–720.

Chang, S. (2007), ‘Characterization of the power function by the independence of record values’, Journal of the Chungcheong Mathematical Society 20(2), 140– 145.

El-Magd, A. & Noura, A. T. (2010), ‘TL-moments of the exponentiated generalized extreme value distribution’, Journal of Advanced Research 1(4), 351–359.

Elamir, E. A. & Seheult, A. H. (2003), ‘Trimmed L-moments’, Computational Statistics and Data Analysis 43(3), 299–314.

Hirano, K. & Porter, J. R. (2003), ‘Asymptotic Efficiency in Parametric Structural Models with Parameter-Dependent Support’, Econometrica 71(5), 1307– 1338.

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.

Hosking, J. R. M. & Wallis, J. R. (1995), ‘A comparison of unbiased and plotting-position estimators of L moments’, Water Resources Research 31(8), 2019– 2025.

Karvanen, J. (2006), ‘Estimation of quantile mixtures via L-moments and trimmed L-moments’, Computational Statistics and Data Analysis 51(2), 947–959.

Meniconi, M. & Barry, D. M. (1996), ‘The power function distribution: A useful and simple distribution to assess electrical component reliability’, Microelectronics Reliability 36(9), 1207-1212.

Rahman, H., Roy, M. K. & Baizid, A. R. (2012), ‘Bayes estimation under conjugate prior for the case of power function distribution’, American Journal of Mathematics and Statistics 2(3), 44–48.

Saran, J. & Pandey, A. (2004), ‘Estimation of parameters of a power function distribution and its characterization by kth record values’, Statistica 64(3), 523–536.

Shabri, A., Ahmad, U. N. & Zakaria, Z. A. (2009), ‘TL-moments and L-moments estimation for the generalized Pareto distribution’, Journal of Mathematics Research 10(10), 97–106.

Shahzad, M. & Asghar, Z. (2013), ‘Comparing TL-Moments, L-Moments and conventional moments of dagum distribution by simulated data’, Revista Colombiana de Estadistica 36(1), 79–93.

Sinha, S., Singh, P., Singh, D. & Singh, R. (2008), ‘Power function distribution characterized by dual generalized order statistics’, Journal of Reliability and Statistics Studies 1(1), 18–27.

Tavangar, M. (2011), ‘Power function distribution characterized by dual generalized order statistics’, Journal of Iranian Statistical Society 10(1), 13–27.

Vogel, R. M. & Fennessey, N. M. (1993), ‘L-moment diagrams should replace product moment diagrams’, Water Resources Research 29(6), 1745–1752.

Wang, Q. J. (1997), ‘LH moments for statistical analysis of extreme events’, Water Resources Research 33(12), 2841–2848.

How to Cite

APA

Naveed-Shahzad, M., Asghar, Z., Shehzad, F. and Shahzadi, M. (2015). Parameter Estimation of Power Function Distribution with TL-moments. Revista Colombiana de Estadística, 38(2), 321–334. https://doi.org/10.15446/rce.v38n2.51663

ACM

[1]
Naveed-Shahzad, M., Asghar, Z., Shehzad, F. and Shahzadi, M. 2015. Parameter Estimation of Power Function Distribution with TL-moments. Revista Colombiana de Estadística. 38, 2 (Jul. 2015), 321–334. DOI:https://doi.org/10.15446/rce.v38n2.51663.

ACS

(1)
Naveed-Shahzad, M.; Asghar, Z.; Shehzad, F.; Shahzadi, M. Parameter Estimation of Power Function Distribution with TL-moments. Rev. colomb. estad. 2015, 38, 321-334.

ABNT

NAVEED-SHAHZAD, M.; ASGHAR, Z.; SHEHZAD, F.; SHAHZADI, M. Parameter Estimation of Power Function Distribution with TL-moments. Revista Colombiana de Estadística, [S. l.], v. 38, n. 2, p. 321–334, 2015. DOI: 10.15446/rce.v38n2.51663. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/51663. Acesso em: 22 jan. 2025.

Chicago

Naveed-Shahzad, Mirza, Zahid Asghar, Farrukh Shehzad, and Mubeen Shahzadi. 2015. “Parameter Estimation of Power Function Distribution with TL-moments”. Revista Colombiana De Estadística 38 (2):321-34. https://doi.org/10.15446/rce.v38n2.51663.

Harvard

Naveed-Shahzad, M., Asghar, Z., Shehzad, F. and Shahzadi, M. (2015) “Parameter Estimation of Power Function Distribution with TL-moments”, Revista Colombiana de Estadística, 38(2), pp. 321–334. doi: 10.15446/rce.v38n2.51663.

IEEE

[1]
M. Naveed-Shahzad, Z. Asghar, F. Shehzad, and M. Shahzadi, “Parameter Estimation of Power Function Distribution with TL-moments”, Rev. colomb. estad., vol. 38, no. 2, pp. 321–334, Jul. 2015.

MLA

Naveed-Shahzad, M., Z. Asghar, F. Shehzad, and M. Shahzadi. “Parameter Estimation of Power Function Distribution with TL-moments”. Revista Colombiana de Estadística, vol. 38, no. 2, July 2015, pp. 321-34, doi:10.15446/rce.v38n2.51663.

Turabian

Naveed-Shahzad, Mirza, Zahid Asghar, Farrukh Shehzad, and Mubeen Shahzadi. “Parameter Estimation of Power Function Distribution with TL-moments”. Revista Colombiana de Estadística 38, no. 2 (July 1, 2015): 321–334. Accessed January 22, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/51663.

Vancouver

1.
Naveed-Shahzad M, Asghar Z, Shehzad F, Shahzadi M. Parameter Estimation of Power Function Distribution with TL-moments. Rev. colomb. estad. [Internet]. 2015 Jul. 1 [cited 2025 Jan. 22];38(2):321-34. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/51663

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