*Estimating the Overlapping Coefficient in the Case of Normal Distributions*

**DOI:**10.54647/wjm5071001 63 Downloads 85069 Views

**Author(s)**

**Abstract**

Given that we have two independent random samples, each of which follows a normal distribution, the main objective of this paper is to estimate the overlapping Weitzman coefficient ∆. This coefficient is widely used and is defined as the area under two probability density functions. The proposed estimation technique is based on the rules of integral numerical approximation such as trapezoidal rules and Simpson's rules. Simulation results showed the effectiveness of the proposed technique over some of the methods found in the literature.

**Keywords**

Normal Distribution; Numerical Integration Methods; Maximum Likelihood Method; Relative Bias; Relative Mean Square Error.

**Cite this paper**

Omar M. Eidous, Abeer J. Al-Shourman,
Estimating the Overlapping Coefficient in the Case of Normal Distributions
, *World Journal of Mathematics *.
Volume 1, Issue 1, January 2023 | PP. 1-13.
10.54647/wjm5071001

**References**

[ 1 ] | Atkinson (1989). An introduction to numerical analysis. Second edition, John Wiley & Sons, Hoboken. |

[ 2 ] | Clemons, T. E. and Bradley, E. L. (2000). A nonparametric measure of the overlapping coefficient. Computational Statistics & Data Analysis, 34 (1), 51-61. |

[ 3 ] | Eidous, O. and Al-Daradkeh, S. (2022). Estimation of Matusita overlapping coefficient ρ for pair normal distribution. Jordan Journal of Mathematics and Statistics (JJMS), 15(4B), 1137 – 1151. |

[ 4 ] | Eidous, O. and Al-Daradkeh, S. (2023). Estimation of Weitzman overlapping coefficient for pair normal distributions. Submitted. |

[ 5 ] | Eidous, O. and Al-Hayja’a, M. (2023a). Estimation of overlapping measures using numerical approximations: Weibull distributions. To be appear in Jordan Journal of Mathematics and Statistics (JJMS). |

[ 6 ] | Eidous, O. and Al-Hayja’a, M. (2023b). Numerical integration approximations to estimate the Weitzman overlapping measure: Weibull distributions. Yugoslav Journal of Operation Research, https://doi.org/10.2298/YJOR221215021E. |

[ 7 ] | Eidous, O and 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. |

[ 8 ] | Eidous, O. and Al-Talafha, S. (2022). Kernel method for overlapping coefficients estimation. Communications in Statistics - Simulation and Computation, 51(9), 5139-5156. |

[ 9 ] | Eidous, O. and Ananbeh, E. (2024). Kernel method for estimating overlapping coefficient using numerical integration methods. Applied Mathematics and Computation, 462, https://doi.org/10.1016/j.amc.2023.128339. |

[ 10 ] | 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. |

[ 11 ] | Gastwirth, J. L. (1975). Statistical measures of earnings differentials. The American Statistician, 29 (1), 32-35. |

[ 12 ] | Inman, H. F. and Bradley, E. L. (1989). The overlapping coefficient as a measure of agreement between probability distribution and point estimation of the overlap of two normal densities. Communications in Statistics-Theory and Methods, 18 (10), 3851-3874. |

[ 13 ] | Madhuri, S. M., Sherry, G. and Subhash, A. (2001). Estimating overlap of two exponential populations. Proceedings of the Annual Meeting of the American Statistical Association, 281, 848–851. |

[ 14 ] | Matusita, K. (1955). Decision rules based on the distance for problem of fir, two samples, and estimation, Ann. Math. Statist.,26,631-640. |

[ 15 ] | Morisita, M. (1959). Measuring interspecific association and similarity between communities. Memoirs of the faculty of Kyushu University, Series E, Biology, 3, 65-80. |

[ 16 ] | Mulekar, M. S., and Mishra, S. N, (1994). Overlap coefficient of two normal densities: equal means case. J. Japan. Soc., 24,169-180. |

[ 17 ] | Pianka, E. R. (1973). The structure of lizard communications. Annual review of ecology and systematics, 4(1), 53-74. |

[ 18 ] | Reiser, B. and Faraggi D. (1999). Confidence intervals for the overlapping coefficient: the normal equal variance case. The statistician, 48(3): 413-418. |

[ 19 ] | Wang, D. and Tian, L. (2017). Parametric methods for confidence interval estimation of overlap coefficients. Computational Statistics & Data Analysis, 106, 12-26. |

[ 20 ] | Weitzman, M.S. (1970). Measure of overlap of income distribution of white and Negro families in the United States (Vol. 22). Us Bureau of the Census. |