Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions

Diana Bílková

doi:doi:10.11648/j.pamj.20140302.11

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Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions

Diana Bílková

Published in Pure and Applied Mathematics Journal (Volume 3, Issue 2)

Received: 22 March 2014 Accepted: 10 April 2014 Published: 20 April 2014

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Abstract

Moments and cumulants are commonly used to characterize the probability distribution or observed data set. The use of the moment method of parameter estimation is also common in the construction of an appropriate parametric distribution for a certain data set. The moment method does not always produce satisfactory results. It is difficult to determine exactly what information concerning the shape of the distribution is expressed by its moments of the third and higher order. In the case of small samples in particular, numerical values of sample moments can be very different from the corresponding values of theoretical moments of the relevant probability distribution from which the random sample comes. Parameter estimations of the probability distribution made by the moment method are often considerably less accurate than those obtained using other methods, particularly in the case of small samples. The present paper deals with an alternative approach to the construction of an appropriate parametric distribution for the considered data set using order statistics

Published in	Pure and Applied Mathematics Journal (Volume 3, Issue 2)
DOI	10.11648/j.pamj.20140302.11
Page(s)	14-25
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

L-Moments and Tl-Moments of Probability Distribution, Sample L-Moments and Tl-Moments, Probability Density Function, Distribution Function, Quantile Function, Order Statistics, Income Distribution

References

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[3]	J. R. M. Hosking, “L-moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics,” Journal of the Royal Statistical Society (Series B), 1990, Vol. 52, No. 1, pp. 105–124. ISSN 1467-9868.
[4]	D. Bílková, “Modelling of Wage Distributions Using the Method of L-Moments,” Paper presented at AMSE – Applications of Mathematics and Statistics in Economy held on 25–28 August 2010, Demänovská Dolina, pp. 16–30. ISBN 978-80-89438-02-0.
[5]	D. Bílková, “Use of the L-Moments Method in Modeling the Wage Distribution,” Paper presented at Aplimat held on 01–04 February 2011, Bratislava, pp. 1471–1481. ISBN 978-80-89313-52-5.
[6]	D. Bílková, “L-Moments and Their Use in Modeling the Distribution of Income and Wage,” Paper presented at ISI held on 21–26 August 2011, Dublin, flashdisk, pp. 1–6.
[7]	D. Bílková, “Modeling of Income and Wage Distribution Using the Method of L-Moments of Parameter Estimation,” Paper presented at International Days of Statistics and Economics at VŠE held on 22–23 September 2011, Prague, pp. 1–10. ISBN 978-80-86175-72-0.
[8]	D. Bílková, “Three-Parametric Lognormal Distribution and Estimating Its parameters Using the Method of L-Moments,” Paper presented at RELIK – Reprodukce lidského kapitálu held on 05–06 December 2011, Prague, CD. ISBN 978-80-86175-75-1.
[9]	D. Bílková, “Estimating Parameters of Lognormal Distribution Using the Method of L-Moments,” Research Journal of Economics, Business and ICT, 2011, Vol. 4, No. 1, pp. 4–9. ISSN 2045-3345.
[10]	D. Bílková, “Modelling of Wage and Income Distributions Using the Method of L-Moments,” Journal of Mathematics and System Science, 2012, Vol. 2, No. 1, pp. 13–19. ISSN 2159-5291.
[11]	D. Bílková, “Lognormal Distribution and Using L-Moment Method for Estimating Its Parameters,” International Journal of Mathematical Models and Methods in Applied Sciences [online], 2012, Vol. 6, No. 1, pp. 30–44. ISSN 1998-0140. URL: http://www.naun.org/journals/m3as/17-079.pdf.
[12]	D. Bílková, “Lognormal Distribution Parameter Estimating Using L-Moments,” Journal of Mathematics and Technology, 2012, Vol. 3, No. 1, pp. 33–51. ISSN: 2078-0257.
[13]	D. Bílková and I. Malá, I. “Application of the L-Moment Method when Modelling the Income Distribution in the Czech Republic,” Austrian Journal of Statistics, 2012, Vol. 41, No. 2, pp. 125–132. ISSN: 1026-597X.
[14]	J. Kyselý and J. Picek, “Regional Growth Curves and Improved Design value Estimates of Extreme Precipitation Events in the Czech Republic,” Climate research, 2007, Vol. 33, pp. 243−255. ISSN: 0936-577X.
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Diana Bílková. (2014). Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions. Pure and Applied Mathematics Journal, 3(2), 14-25. https://doi.org/10.11648/j.pamj.20140302.11

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ACS Style

Diana Bílková. Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions. Pure Appl. Math. J. 2014, 3(2), 14-25. doi: 10.11648/j.pamj.20140302.11

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AMA Style

Diana Bílková. Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions. Pure Appl Math J. 2014;3(2):14-25. doi: 10.11648/j.pamj.20140302.11

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@article{10.11648/j.pamj.20140302.11,
  author = {Diana Bílková},
  title = {Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {2},
  pages = {14-25},
  doi = {10.11648/j.pamj.20140302.11},
  url = {https://doi.org/10.11648/j.pamj.20140302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140302.11},
  abstract = {Moments and cumulants are commonly used to characterize the probability distribution or observed data set. The use of the moment method of parameter estimation is also common in the construction of an appropriate parametric distribution for a certain data set. The moment method does not always produce satisfactory results. It is difficult to determine exactly what information concerning the shape of the distribution is expressed by its moments of the third and higher order. In the case of small samples in particular, numerical values of sample moments can be very different from the corresponding values of theoretical moments of the relevant probability distribution from which the random sample comes. Parameter estimations of the probability distribution made by the moment method are often considerably less accurate than those obtained using other methods, particularly in the case of small samples. The present paper deals with an alternative approach to the construction of an appropriate parametric distribution for the considered data set using order statistics},
 year = {2014}
}

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TY  - JOUR
T1  - Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions
AU  - Diana Bílková
Y1  - 2014/04/20
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140302.11
DO  - 10.11648/j.pamj.20140302.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
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PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140302.11
AB  - Moments and cumulants are commonly used to characterize the probability distribution or observed data set. The use of the moment method of parameter estimation is also common in the construction of an appropriate parametric distribution for a certain data set. The moment method does not always produce satisfactory results. It is difficult to determine exactly what information concerning the shape of the distribution is expressed by its moments of the third and higher order. In the case of small samples in particular, numerical values of sample moments can be very different from the corresponding values of theoretical moments of the relevant probability distribution from which the random sample comes. Parameter estimations of the probability distribution made by the moment method are often considerably less accurate than those obtained using other methods, particularly in the case of small samples. The present paper deals with an alternative approach to the construction of an appropriate parametric distribution for the considered data set using order statistics
VL  - 3
IS  - 2
ER  -

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Author Information

Diana Bílková

Department of Statistics and Probability, Faculty of Informatics and Statistics, University of Economics, Prague, Czech Republic

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APA Style

Diana Bílková. (2014). Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions. Pure and Applied Mathematics Journal, 3(2), 14-25. https://doi.org/10.11648/j.pamj.20140302.11

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ACS Style

Diana Bílková. Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions. Pure Appl. Math. J. 2014, 3(2), 14-25. doi: 10.11648/j.pamj.20140302.11

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AMA Style

Diana Bílková. Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions. Pure Appl Math J. 2014;3(2):14-25. doi: 10.11648/j.pamj.20140302.11

Copy | Download

@article{10.11648/j.pamj.20140302.11,
  author = {Diana Bílková},
  title = {Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {2},
  pages = {14-25},
  doi = {10.11648/j.pamj.20140302.11},
  url = {https://doi.org/10.11648/j.pamj.20140302.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140302.11},
  abstract = {Moments and cumulants are commonly used to characterize the probability distribution or observed data set. The use of the moment method of parameter estimation is also common in the construction of an appropriate parametric distribution for a certain data set. The moment method does not always produce satisfactory results. It is difficult to determine exactly what information concerning the shape of the distribution is expressed by its moments of the third and higher order. In the case of small samples in particular, numerical values of sample moments can be very different from the corresponding values of theoretical moments of the relevant probability distribution from which the random sample comes. Parameter estimations of the probability distribution made by the moment method are often considerably less accurate than those obtained using other methods, particularly in the case of small samples. The present paper deals with an alternative approach to the construction of an appropriate parametric distribution for the considered data set using order statistics},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Alternative Tools of Statistical Analysis: L-moments and TL-moments of Probability Distributions
AU  - Diana Bílková
Y1  - 2014/04/20
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140302.11
DO  - 10.11648/j.pamj.20140302.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 14
EP  - 25
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140302.11
AB  - Moments and cumulants are commonly used to characterize the probability distribution or observed data set. The use of the moment method of parameter estimation is also common in the construction of an appropriate parametric distribution for a certain data set. The moment method does not always produce satisfactory results. It is difficult to determine exactly what information concerning the shape of the distribution is expressed by its moments of the third and higher order. In the case of small samples in particular, numerical values of sample moments can be very different from the corresponding values of theoretical moments of the relevant probability distribution from which the random sample comes. Parameter estimations of the probability distribution made by the moment method are often considerably less accurate than those obtained using other methods, particularly in the case of small samples. The present paper deals with an alternative approach to the construction of an appropriate parametric distribution for the considered data set using order statistics
VL  - 3
IS  - 2
ER  -

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