Sains Malaysiana 52(1)(2023): 295-304

http://doi.org/10.17576/jsm-2023-5201-24

Approximation of the Sum of Independent Lognormal Variates using Lognormal Distribution by Maximum Likelihood Estimation Approached

(Penghampiran terhadap Jumlah Variat Tak Bersandar menggunakan Taburan Lognormal Berdasarkan Pendekatan Penganggaran Kebolehjadian Maksimum)

ABDUL RAHMAN OTHMAN¹,  LAI CHOO HENG²,  SONIA AÏSSA³ & NORA MUDA^4,*

¹School of Distance Education, Universiti Sains Malaysia, 11800 Pulau Pinang, Malaysia

²Kolej Vokasional Nibong Tebal, Jalan Bukit Panchor, 14300 Nibong Tebal, Pulau Pinang, Malaysia 

³Institut National de la Recherche Scientifique, Énergie Matériaux Télécommunications Research Centre, 800, De La Gauchetière Ouest, Bureau 6900, Montréal, Québec H5A 1K6, Canada

⁴Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

Received: 11 March 2022/Accepted: 10 October 2022

Abstract

Three methods of approximating the sum of lognormal variates to a lognormal distribution were studied. They were the Wilkinson approximation, the Monte Carlo version of the Wilkinson approximation and the approximation using estimated maximum likelihood lognormal parameters. The lognormal variates were generated empirically using Monte Carlo simulation based on several conditions such as number of lognormal variates in the sum, number of sample points in the variates, the variates are independent and identically distributed (IID) and also not identically distributed (NIID) with lognormal parameters. Evaluation of all three lognormal approximation methods was performed using the Anderson Darling test. Results show that the approximation using estimated maximum likelihood lognormal parameters produced Type I errors close to the 0.05 target and is considered the best approximation.

Keywords: Anderson-Darling test;  lognormal approximation; maximum likelihood; sum of lognormal variates; Wilkinson

Abstrak

Tiga kaedah penghampiran bagi jumlah variat lognormal terhadap taburan lognormal telah dikaji. Tiga kaedah penghampiran tersebut adalah kaedah penghampiran Wilkinson, kaedah versi Monte Carlo bagi penghampiran Wilkinson dan kaedah penghampiran dengan penganggaran kebolehjadian maksimum bagi parameter lognormal. Pemboleh ubah lognormal dijana secara empirik melalui simulasi Monte Carlo dengan beberapa keadaan simulasi iaitu bilangan jumlah pemboleh ubah lognormal, bilangan sampel bagi pemboleh ubah lognormal, pemboleh ubah lognormal tak bersandar dan tertabur secara secaman mengikut taburan (IID) dan juga tidak secaman mengikut taburan (NIID) berdasarkan parameter lognormal. Penilaian bagi ketiga-tiga kaedah penghampiran lognormal tersebut dijalankan menggunakan ujian Anderson Darling. Hasil menunjukkan penghampiran menggunakan penganggaran kebolehjadian maksimum terhadap parameter lognormal telah menghasilkan ralat Jenis 1 menghampiri nilai sasaran ralat 0.05 dan dikatakan sebagai penghampiran terbaik.

Kata kunci: Jumlah variat lognormal; kebolehjadian maksimum; penghampiran lognormal; ujian Anderson-Darling; Wilkinson

REFERENCES

Abdul Majid, M.H. & Ibrahim, K. 2021. Composite pareto distributions for modelling household income distribution in Malaysia. Sains Malaysiana 50(7): 2047-2058.

Beaulieu, N.C. & Xie, Q. 2004. An optimal lognormal approximation to lognormal sum distributions. IEEE Transactions on Vehicular Technology53: 479-489.

Becker, D.N. 1991. Statistical tests of the lognormal distribution as a basis for interest rate changes. Transactions of the Society of Actuaries 43: 7-72.

Bradley, J.V. 1978. Robustness? British Journal of Mathematical and Statistical Psychology 31: 144-152.

Bromideh, A.A. 2012. Discriminating between Weibull and Log-Normal distributions based on Kullback-Leibler divergence. Istanbul University Econometrics and Statistics e-Journal 16(1): 45-54.

Cardieri, P. & Rappaport, T.S. 2000. Statistics of the sum of lognormal variables in wireless communications. In Spring 2000 Vehicular. Technology Conference: IEEE 51^st Vehicular Technology Conference Proceedings May 15-18, Tokyo, Japan. pp. 1823-1827.

Cobb, B.R., Rumí, R. & Salmerón, A. 2012. Approximating the distribution of a sum of log-normal random variables. In The Proceedings of the Sixth European Workshop on Probabilistic Graphical Models. pp. 67-74.

Cohen, A.C. 1951. Estimating parameters of logarithmic-normal distributions by maximum likelihood. Journal of the American Statistical Association 46: 206-212.

Di Renzo, M., Imbriglio, L., Graziosi, F. & Santucci, F. 2009. Distributed data fusion over correlated log-normal sensing and reporting channels: Application to cognitive radio networks. IEEE Transactions on Wireless Communications 8: 5813-5821.

Havemann, F., Heinz, M. & Kretschme, H. 2006. Collaboration and distances between German immunological institutes - A trend analysis. Journal of Biomedical Discovery and Collaboration 1: 6.

Keselman, H.J., Othman, A.R. & Wilcox, R. 2014. Preliminary testing for normality in the multi-group problem: Is this a good practice? Clinics in Dermatology 2: 29-43.

Keselman, H.J., Othman, A.R. & Wilcox, R. 2013. Preliminary testing for normality: Is this good practice? Journal of Modern Applied Statistical Methods 2: 2-19.

Limpert, E., Stahel, W.A. & Abbt, M. 2001. Log-normal distribution across the sciences: Keys and clue. Bioscience 51(5): 341-352.

Loewenstein, Y., Kuras, A. & Rumpel, S. 2011. Multiplicative dynamics underlie the emergence of the log-normal distribution of spine sizes in the neocortex. Journal of Neuroscience31: 9481-9488.

Muhammad Farouk, Nazrina Aziz & Zakiyah Zain. 2020. The application of lognormal distribution on the new two-sided group chain sampling plan. Sains Malaysiana 49(5): 1145-1152.

Osborn, J.F., Cattaruzza, M.S., Ferri, A.M., De Angelis, F., Renzi, D., Marani, A. & Vaira, D. 2013. How long it will take to reduce gastric cancer incidence by eradicating Heliobacter pylori infection? Cancer Prevention Research 6: 695-700.

Othman, A.R., Keselman, H.J. & Wilcox, R. 2015. Assessing normality: Applications in multi-group designs. Malaysian Journal of Mathematical Sciences 9: 53-65.

Saleem, M., Sieskul, B.T. & Kaiser, T. 2006. Channel capacity assessments in UWB communication system over lognormal fading. The Institution of Engineering and Technology Seminar on Ultra Wideband Systems, Technologies and Applications. pp. 155-159.

Santos Filho, J.C.S., Yacoub, M.D. & Cardieri, P. 2006. Highly accurate range-Adaptive lognormal approximation to lognormal sum distributions. Electronics Letters 42: 361-363.

Santos Filho, J.C.S., Cardieri, P. & Yacoub, M.D. 2005. Simple accurate lognormal approximation to lognormal sums. Electronics Letters 41: 1016-1017.

SAS Institute Inc. SAS OnlineDoc 9.4. 2015; Cary, NC.

Schwartz, S.C. & Yeh, Y.S. 1982. On the distribution function and moments of power sums with lognormal components. Bell Labs Technical Journal 61: 1441-1462.

Selim, B., Alhussein, O., Muhaidat, S., Karagiannidis, G.K. & Liang, J. 2016. Modeling and analysis of wireless channels via the mixture of Gaussian distribution. IEEE Transactions on Vehicular Technology 65: 8309-8321.

Shafiq, M., Alamgir & Atif, M. 2016. On the estimation of three parameters lognormal distribution based on fuzzy life time data. Sains Malaysiana 45(11): 1773-1777.

Stephens, M.A. 1979. Tests of fit for the logistic distribution based on the empirical distribution function. Biometrika 66: 591-595.

Stephens, M.A. 1977. Goodness of fit for the extreme value distribution. Biometrika 64: 583-588.

Stephens, M.A. 1977a. Goodness of Fit with Special Reference to Tests for Exponentiality. Technical Report No. 262, Department of Statistics, Stanford University, Stanford, CA.

Stephens, M.A. 1976. Asymptotic results for goodness-of-fit statistics with unknown parameters. Annals of Statistics 4: 357-369.

Stephens, M.A. 1974. EDF statistics for goodness of fit and some comparisons. Journal of the American Statistical Association 69: 730-737.

Wagner, P.J. 2011. Modelling rate distributions using character compatibility: Implications for morphological evolution among fossil invertebrates. Biology Letters 8: 143-146.

Wawrik, B., Kutliev, D., Abdivasievna, U.A., Kukor, J.J., Zylstra, G.J. & Kerkhof, L. 2007. Biogeography of actinomycete communities and Type II polyketide synthase genes in soils collected in New Jersey and Central Asia. Applied and Environmental Microbiology 73: 2982-2989.

^*Corresponding author; email: noramuda@ukm.edu.my