Point Estimation of Pianka and Kullback-Leibler Overlapping Coefficients: Two Normal Distributions
Department of Statistics, Yarmouk University, Irbid 21163, Jordan
| Abstract: | The well-known two overlapping coefficients; Pianka and Kullback-Leibler are considered to be important measures that focus on measuring the closeness or similarity between two continuous or discrete probability distributions. A general expression for each of them has been developed for some statistical distributions, such as exponential distribution and Weibull distribution. In this research, we focused on the most famous statistical distribution, which is the normal probability distribution.Our main contribution in this paper is to derivation and estimation the two overlapping coefficients Pianka (PI) and Kullback-Leibler (KL) under the assumption of existing a pair of normal distributions. The mathematical formulas (parameters) for each of the PI and KL coefficients were obtained without imposing any restrictions on the parameters of the pair normal distributions. To complete the estimation process of each resulting parameter, an estimator was proposed for each of them by deriving the maximum likelihood estimator based on the assumption of existing two independent random samples, each following the statistical normal distribution. |
| Keywords: | Overlapping Coefficient; Pianka Coefficient; Kullback-Leibler Coefficient; Maximum Likelihood Estimation Method; Normal Distribution |
| DOI: | 10.57237/j.wjms.2025.01.001 |
View:
Downloads:
收稿日期: 2024-10-05
接受日期: 2024-11-19
在线出版日期: 2025-01-02
参考文献
| [1] | Pianka, E. R. (1973). The structure of lizard communications. Annual review of ecology and systematics, 4(1), 53-74. |
| [2] | Federer, W. T., Powers, L. & Payne, M. G. (1963). Studies on statistical procedures applied to chemical genetic data from sugar beets. Technical Bulletin, Agricultural Experimentation Station, Colorado State University 77. |
| [3] | Dhaker, H., Deme, E. and El-Adlouni, S. (2021). On inference of overlapping coefficients in two inverse Lomax populations. Statistical Theory and Applications, 20 (1), 61-75. https://doi.org/10.2991/jsta.d.210107.002 |
| [4] | Alodat, M. T., Al Fayez, M. and Eidous, O. (2021). On the asymptotic distribution of Matusita's overlapping measure. Communications in Statistics - Theory and Methods. 51(20), 6963-6977. https://doi.org/10.1080/03610926.2020.186926 |
| [5] | Samawi H. M., Helu A. and Vogel R. (2011). A Nonparametric Test of Symmetry Based on the Overlapping Coefficient. Journal of Applied Statistics, 38(5), 885-898. |
| [6] | Eidous, O. M., and AL-Talafha, S. A. (2022). Kernel method for overlapping coefficients estimation. Communications in Statistics: Simulation and Computation, 51(9), 5139–5156. https://doi.org/10.1080/03610918.2020.1757711 |
| [7] | Eidous, O., and Abu Al-Hayja’s, M. (2023). Numerical integration approximations to estimate the Weitzman overlapping measure: Weibull distributions. Yugoslav Journal of Operations Research, 33 (4), 699-712. http://dx.doi.org/10.2298/YJOR221215021E |
| [8] | Eidous, O. and Maqableh, H. (2023). Estimation of Matusita Overlapping Coefficient ρ for Two Weibull Distributions. Journal of Statistics and Computer Science (JSCS), 2 (1), 1-16. |
| [9] | Eidous, O. and Abu Al-Hayja’a, M. (2023). Weibull distributions for estimation of Pianka and Kullback-Leibler overlapping measures. Journal of Mathematics and Statistics Research, 5 (1), 165. https://www.researchgate.net/publication/376681761 |
| [10] | Eidous, O. and Magableh, H. (2024). Estimation of Morisita overlapping measure λ for two Weibull distributions. World Journal of Mathematics and Statistics, 3(1), 13-21. |
| [11] | Maqableh, H. and Eidous, O. (2024). On inference of Weitzman overlapping coefficient in two Weibull distributions. Journal of Health Statistics Reports, 3 (3), 1-6. |
| [12] | Eidous, O. and Daradkeh, S. (2024). On inference of Weitzman overlapping coefficient Δ(X,Y) in the case for two normal distributions. International Journal of Theoretical and Applied Mathematics, 10 (2), 14-22. |
| [13] | Chaubey, Y. P., Sen, D. and Mishra, S. N. (2008). Inference on overlap for two inverse Gaussian populations: Equal means case. Communications in Statistics—Theory and Methods, 37 (12), 1880-1894. |
| [14] | Helu, A. and Samawi, H. (2011). On inference of overlapping coefficients in two Lomax populations using different sampling methods. Statistical Theory and Practice, 5(4), 683–696. |
| [15] | Mulekar, M. S., and Mishra, S. N. (1994). Overlap coefficients of two normal densities: equal means case. Journal of the Japan Statistical Society, Japanese Issue, 24, 169-180. |
| [16] | Eidous, O. M. and Abu Al-Hayja, a, M. (2023). Estimation of overlapping measures using numerical approximations: Weibull distributions. Jordan Journal of Mathematics and Statistics (JJMS), 16(4), 741–761. https://doi.org/10.47013/16.4.8 |
| [17] | Eidous, O., and Al-Daradkeh, S. (2022). Estimation of Matusita overlapping coefficient for pair normal distributions. Jordan Journal of Mathematics and Statistics (JJMS), 15(4B), 1137-1151. https://doi.org/10.47013/15.4.23 |
| [18] | Eidous, O. M., Al-Shourman, A. (2022). Numerical integral approximation to estimate Matusita overlapping coefficient for normal distributions. Journal of Mathematical Techniques and Computational Mathematics, 1(3), 264-270. https://api.semanticscholar.org/CorpusID:255699518 |
| [19] | Eidous, O. M. and Al-Shourman, A. (2023). Estimating the overlapping coefficient in the case of normal distributions. World Journal of Mathematics. 1(1), 1–13. https://doi.org/10.54647/wjm5071001 |
| [20] | Reiser, B. and Faraggi D. (1999). Confidence intervals for the overlapping coefficient: the normal equal variance case. The statistician, 48(3): 413-418. |
| [21] | Inácio V. and Garrido Guillén J. E. (2022). Bayesian nonparametric inference for the overlap coefficient: With an application to disease diagnosis. Statistics in Medicine, 41(20), 3879-3898. https://doi.org/10.1002/sim.9480 |
| [22] | Clemons, T. E. and Bradley, E. L. (2000). A nonparametric measure of the overlapping coefficient. Computational Statistics & Data Analysis, 34, 51-61. |
| [23] | Clemons, T. E. (2001). Erratum to a nonparametric measure of the overlapping coefficient. Computational Statistics & Data Analysis, 36, 243. |
| [24] | Eidous, O. M., and Ananbeh, E. A. (2024). Kernel method for estimating overlapping coefficient using numerical integration methods. Applied Mathematics and Computation, 462, 128339. https://doi.org/10.1016/j.amc.2023.128339 |
| [25] | Eidous, O. M., and Ananbeh, E. A. (2024). Kernel method for estimating Matusita overlapping coefficient using numerical approximations. Annals of Data Science. https://doi.org/10.1007/s40745-024-00563-y |
| [26] | Dhaker, H., Ngom, P., & Mbodj, M. (2017). Overlap coefficients based on Kullback-Leibler divergence: exponential populations case. International Journal of Applied Mathmatical Research, 6(4), 135-140. |
