L-Moments and TL-Moments as an Alternative Tool of Statistical Data Analysis
Author(s)
Diana Bílková1*
Affiliation(s)
1 University of Economics, Faculty of Informatics and Statistics, Department of Statistics and Probability, Prague, Czech Republic. 2 Department of Information Technology and Analytical Methods, University of Business, Prague, Czech Republic.
1 University of Economics, Faculty of Informatics and Statistics, Department of Statistics and Probability, Prague, Czech Republic. 2 Department of Information Technology and Analytical Methods, University of Business, Prague, Czech Republic.
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.
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.
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
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
Cite this paper
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. doi: 10.4236/jamp.2014.210104.
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. doi: 10.4236/jamp.2014.210104.
References
[1] 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. [2] Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York.
http://dx.doi.org/10.1002/9780470316481 [3] Adamowski, K. (2000) Regional Analysis of Annual Maximum and Partial Duration Flood Data by Nonparametric and L-Moment Methods. Journal of Hydrology, 229, 219-231.
http://dx.doi.org/10.1016/S0022-1694(00)00156-6 [4] Ulrych, T.J., Velis, D.R., Woodbury, A.D. and Sacchi, M.D. (2000) L-Moments and C-Moments. Stochastic Environmental Research and Risk Assessment, 14, 50-68. http://dx.doi.org/10.1007/s004770050004 [5] Kysely, J. and Picek, J. (2007) Regional Growth Curves and Improved Design Value Estimates of Extreme Precipitation Events in the Czech Republic. Climate Research, 33, 243-255.
http://dx.doi.org/10.3354/cr033243 [6] Elamir, E.A.H. and Seheult, A.H. (2003) Trimmed L-Moments. Computational Statistics & Data Analysis, 43, 299-314. http://dx.doi.org/10.1016/S0167-9473(02)00250-5 [7] Marek, L. (2011) Wage Development in the Czech Republic for the Past 16 Years. Proceedings of AIESA—Budovanie spolecnosti zalozenej na vedomostiach, Bratislava, 10-11 November 2011, 1-7. [8] Marek, L. and Vrabec, M. (2013) Model Wage Distribution—Mixture Density Functions. International Journal of Economics and Statistics, 1, 113-121.
[1] 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. [2] Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. John Wiley & Sons, New York.
http://dx.doi.org/10.1002/9780470316481 [3] Adamowski, K. (2000) Regional Analysis of Annual Maximum and Partial Duration Flood Data by Nonparametric and L-Moment Methods. Journal of Hydrology, 229, 219-231.
http://dx.doi.org/10.1016/S0022-1694(00)00156-6 [4] Ulrych, T.J., Velis, D.R., Woodbury, A.D. and Sacchi, M.D. (2000) L-Moments and C-Moments. Stochastic Environmental Research and Risk Assessment, 14, 50-68. http://dx.doi.org/10.1007/s004770050004 [5] Kysely, J. and Picek, J. (2007) Regional Growth Curves and Improved Design Value Estimates of Extreme Precipitation Events in the Czech Republic. Climate Research, 33, 243-255.
http://dx.doi.org/10.3354/cr033243 [6] Elamir, E.A.H. and Seheult, A.H. (2003) Trimmed L-Moments. Computational Statistics & Data Analysis, 43, 299-314. http://dx.doi.org/10.1016/S0167-9473(02)00250-5 [7] Marek, L. (2011) Wage Development in the Czech Republic for the Past 16 Years. Proceedings of AIESA—Budovanie spolecnosti zalozenej na vedomostiach, Bratislava, 10-11 November 2011, 1-7. [8] Marek, L. and Vrabec, M. (2013) Model Wage Distribution—Mixture Density Functions. International Journal of Economics and Statistics, 1, 113-121.