The Bivariate Length Biased - Power Garima Distribution Derived from Copula: Properties and Application to Environmental Data
Keywords:
Bivariate distribution, FGM copula, Dependent variables, Numerical simulationAbstract
Bivariate distributions calculate the probabilities for simultaneous outcomes of two random variables. They are essential for understanding the relationship between two variables. This study proposed a new bivariate distribution called the bivariate length biased power Garima (BLBPG) distribution, created by combining the Farlie-Gumbel-Morgenstern (FGM) copula with a length-biased power Garima distribution. The BLBPG distribution describes lifetime bivariate data with a weak correlation between variables as a flexible alternative to bivariate lifetime distributions for modeling real-valued data in applications. The proposed distribution yielded various properties including joint conditional probability functions, reliability (survival) and hazard functions, and generating a random variable. The maximum likelihood estimation was presented to estimate the parameters of the proposed distribution and a Monte Carlo simulation study was conducted to evaluate the performance of the estimators. A practical application of the proposed bivariate distribution to analyze environmental data was also demonstrated.
References
Gupta RC, Akman HO. On the reliability studies of a weighted inverse Gaussian model. Journal of Statistical Planning and Inference 1995; 48(1): 69-83.
Klinjan K, Aryuyuen S. The lengthbiased power Garima distribution and its application to model lifetime data. Songklanakarin Journal of Science & Technology, 2021; 43(3): 667-76.
Lai CD. Constructions of continuous bivariate distributions. Journal of the Indian Society for Probability and Statistics 2004; 8: 21-43.
Nelsen RB. An introduction to copulas, 2nd ed., Springer, New York, NY.; 2006.
Sklar M. Fonctions de répartition à n dimensions et leurs marges. In Annales de l’ISUP 1959; 8: 229-231.
Sklar A. Random variables, joint distribution functions, and copulas. Kybernetika 1973; 9(6): 449-60.
Morgenstern D. Einfache Beispiele zweidimensionaler Verteilungen. Mitteilingsblatt für Mathematische Statistik, 1956; 8: 234-5.
Kotz S, Balakrishnan N, Johnson NL. Continuous multivariate distributions, Volume 1: Models and applications, volume 1, John Wiley & Sons; 2004.
Nadarajah S. A bivariate Pareto model for drought. Stochastic Environmental Research and Risk Assessment 2009; 23: 811-22.
Abd Elaal MK, Jarwan RS. Inference of bivariate generalized exponential distribution based on copula functions. Applied Mathematical Sciences 2017; 11(24): 1155-86.
Peres MVDO, Achcar JA, Martinez EZ. Bivariate modified Weibull distribution derived from Farlie-Gumbel-Morgenstern copula: a simulation study. Electronic Journal of Applied Statistical Analysis 2018; 11(2): 463-88.
Almetwally EM, Muhammed HZ, ElSherpieny ESA. Bivariate Weibull distribution: properties and different methods of estimation. Annals of Data Science 2020; 7: 163-93.
Zhao J, Faqiri H, Ahmad Z, Emam W, Yusuf M, Sharawy A. The Lomax-claim model: bivariate extension and applications to financial data. Complexity 2021; 2021: 1-17.
Hamed MS, Cordeiro GM, Yousof HM. A new compound Lomax model: Properties, copulas, modeling and risk analysis utilizing the negatively skewed insurance claims data. Pakistan Journal of Statistics and Operation Research 2022; 18(3): 601-31.
Abulebda M, Pandey A, Tyagi S. On bivariate inverse Lindley distribution derived from copula. Thailand Statistician 2023; 21(2): 291 304.
Saminger-Platz S, Kolesárová A, Šeliga A, Mesiar R, Klement EP. The impact on the properties of the EFGM copulas when extending this family. Fuzzy Sets and Systems 2021; 415: 1-26.
Bekrizadeh H. Generalized family of copulas: definition and properties. Thailand Statistician 2021; 19(1): 162-77.
Aryuyuen, S, Panphut W, Pudprommarat, C. Bivariate Sushila distribution based on copulas: properties, simulations, and applications. Lobachevskii Journal of Mathematics 2024; 44(11): 4592-609.
Shenbagaraja R, Rather AA, Subramanian C. On some aspects of length biased technique with real life data. Science, technology &
Development 2019; 8(9): 326-35.
Basu AP. Bivariate failure rate. Journal of the American Statistical Association1971; 66(333): 103-4.
Klinjan K, Aryuyuen S. R package for the length-biased power Garima distribution to analyze lifetime data. International Journal of Mathematics and Computer Science 2022; 17(3): 1351-62.
R Core Team et al. R: A language and environment for statistical computing. [cited 2023 November 1]. Available from: https://www.R-project.org/
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Science & Technology Asia
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
