Parameter Estimation and Hypothesis Testing on Bivariate Log-Normal Regression Models
Abstract
This study aims to introduce a bivariate Log-Normal regression model and to develop a technique for parameter estimation and hypothesis testing. We term the model Bivariate Log-Normal Regression (BLNR). The estimation procedure is conducted by the standard Maximum Likelihood Estimation (MLE) employing the Newton-Raphson method. To perform hypothesis testing, we adapt the Maximum Likelihood Ratio Test (MLRT) for simultaneous testing with test statistics which, for large n, follows Chi-Square distribution with degrees of freedom p. In addition, the partial testing is derived from a central limit theorem which results in a Z-test statistic.
Keywords: parameter estimation, hypothesis testing, bivariate log, normal regression
References
[1] El-Kasem B, Salloum N, Brinz T, Heider Y, Markert B. A multivariate regression parametric study on DEM input parameters of free-flowing and cohesive powders with experimental data-based validation. Comput Part Mech. 2021;8(1):87–111.
[2] Abedi V, Olulana O, Avula V, Chaudhary D, Khan A, Shahjouei S, et al. Racial, economic, and health inequality and COVID-19 infection in the United States. J Racial Ethn Health Disparities. 2021 Jun;8(3):732–42.
[3] Sahoo P. Probability and Mathematical Statistics, Department of Mathematics. University of Louisville; 2013.
[4] Gujarati DN, Porter DC. Basic Econometrics. New York: McGrawHill; 2003.
[5] G.H. Wenur, Purhadi, and A. Suharsono, “Three-parameter bivariate gamma regression model for analyzing under-five mortality rate and maternal mortality rate’.,”Journal of Physics: Conference Series. vol. 1538, p.2020.
[6] Bhuyan MJ, Islam MA, Rahman MS. A bivariate Bernoulli model for analyzing malnutrition data. Health Serv Outcomes Res Methodol. 2018;18(2):109–27.
[7] Benz LS, Lopez MJ. Estimating the change in soccer’s home advantage during the Covid-19 pandemic using bivariate Poisson regression. AStA Adv Stat Anal. 2023;107(1-2):205–32.
[8] Mardalena S, Purhadi JT, Prastyo DD. Bivariate poisson inverse gaussian regression model with exposure variable: infant and maternal death case study. J Phys Conf Ser. 2021;1752(1):012016.
[9] Gustavsson S. Evaluation of Regression Methods for Log-Normal Data. Gothenburg: University of Gothenburg; 2015.
[10] Gustavsson SM, Johannesson S, Sallsten G, Andersson EM. Linear maximum likelihood regression analysis for untransformed log-normally distributed data. Open J Stat. 2012;2(4):389–400.
[11] Diantini NL. Purhadi and A, Choiruddin, “ Parameter estimation and hypothesis testing on three parameters log normal regression”AIP Conference Proceedings.Vol. 2554,p. 2023
[12] Yerel S, Konuk A. Bivariate lognormal distribution model of cutoff grade impurities: A case study of magnesite ore deposit. Sci Res Essays. 2019;4:1500–4.
[13] Yue S. “The bivariate lognormal distribution to model a multivariate flood episode.,” Hydrol. Volume 14. Process; 2000. pp. 2575–88.
[14] Dewi DK. Purhadi and Sutikno.” Geographically weighted bivariate gamma regression in the analysis of maternal mortality rate and infant mortality rate in north sumatra province 2017.,”IOP Conference Series: Materials Science and Engineering.vol. 546. no. 5, p. 2019.
[15] Purhadi AR, Wenur GH. Geographically weighted three-parameters bivariate gamma regression and its application. Symmetry (Basel). 2021;13(2):197–213.
[16] Hayati FN. Purhadi, and B.W. Otok, “Parameter estimation and statistical test of mixed geographically Weighted Bivariate Weibull Regression (MGWBWR): On the Cases of Infant Mortality and Maternal Mortality Rate in East Java 2016,”International Symposium on Advanced Intelligent Informatics (SAIN). pp. 78–83, 2018.
[17] Purhadi S, Berliana SM, Setiawan DI. Geographically weighted bivariate generalized Poisson regression: application to infant and maternal mortality data. Lett Spat Resour Sci. 2021;14(1):79–99.
[18] D.N.S. Purhadi, Q. Aini, and Irhamah, “Geographically weighted bivariate zero inflated generalized Poisson regression model and its application..,”Heliyon. vol. 7, p. 2021.
[19] Nur MS., Purhadi and A.” Parameter estimation and hypothesis testing of geographically weighted bivariate zero inflated poisson inverse gaussian regression models” In: Choiruddin,IOP Conference Series: Materials Science and Engineering, vol.1115, p.2021.