The Weitzman overlapping coefficient ∆(X,Y) is the most important and widely used overlapping coefficient, which represents the intersection area between two probability distributions. This research proposes a new general technique to estimate ∆(X,Y) assuming the existence of two independent random samples following normal distributions. In contrast to some studies in this scope that place some restrictions on the parameters of the two populations such as the equality of their means or the equality of their variances, this study did not assume any restrictions on the parameters of normal distributions. Two new estimators for ∆(X,Y) were derived based on the proposed new technique, and then the properties of the estimator resulting from taking their arithmetic mean was studied and compared with some corresponding estimators available in the literature based on the simulation method. An extensive simulation study was performed by assuming two normal distributions with different parameter values to cover most possible cases in practice. The parameter values were chosen taking into account the exact value of ∆(X,Y), which taken to be small (close to zero), medium (close to 0.5) and large (close to 1). The simulation results showed the effectiveness of the proposed technique in estimating ∆(X,Y). By comparing the proposed estimator of ∆(X,Y) with some existing corresponding estimators, its performance was better than the performances of the other estimators in almost all considered cases.
Published in | International Journal of Theoretical and Applied Mathematics (Volume 10, Issue 2) |
DOI | 10.11648/j.ijtam.20241002.11 |
Page(s) | 14-22 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Overlapping Weitzman Coefficient, Maximum Likelihood Method, Parametric Method, Normal Distribution, Expected Value, Relative Bias, Relative Mean Square Error
Normal distributions | ${\mathit{f}}_{\mathit{1}}\left(\mathit{x}\right)$ | ${\mathit{f}}_{\mathit{2}}\left(\mathit{y}\right)$ | $\mathit{\Delta}\left(\mathit{X}\mathit{,}\mathit{Y}\right)$ | |
---|---|---|---|---|
Case 1: Equal variances | A | $N\left(0,1\right)$ | $N\left(-0.5,1\right)$ | 0.8025 |
B | $N\left(0,1\right)$ | $N\left(1,1\right)$ | 0.671 | |
C | $N\left(0,1\right)$ | $N\left(1.5,1\right)$ | 0.4532 | |
D | $N\left(0,1\right)$ | $N\left(3,1\right)$ | 0.1336 | |
Case 2: Equal means | A | $N\left(0,1\right)$ | $N\left(0,1.5\right)$ | 0.8064 |
B | $N\left(0,1\right)$ | $N\left(0,2.5\right)$ | 0.585 | |
C | $N\left(0,1\right)$ | $N\left(0,5\right)$ | 0.3528 | |
D | $N\left(0,1\right)$ | $N\left(0,10\right)$ | 0.2017 | |
Case 3: Different means and different variances | A | $N\left(0,1\right)$ | $N\left(-0.2,1.1\right)$ | 0.9151 |
B | $N\left(0,1\right)$ | $N\left(1,2\right)$ | 0.6099 | |
C | $N\left(0,1\right)$ | $N\left(2.5,4\right)$ | 0.3577 | |
D | $N\left(0,1\right)$ | $N\left(5,2\right)$ | 0.0891 |
$\mathit{\Delta}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | ${\mathit{(}\mathit{n}}_{\mathbf{1}}\mathbf{,}{\mathit{n}}_{\mathbf{2}}\mathbf{)}$ | ${\widehat{\mathbf{\u2206}}}_{\mathit{k}}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | ${\widehat{\mathbf{\u2206}}}_{\mathit{IN}}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | ${\widehat{\mathit{\Delta}}}_{\mathit{Prop}}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | |
---|---|---|---|---|---|
0.8025 | (10,10) | RB | 0.0364 | -0.033 | -0.0995 |
RMSE | 0.246 | 0.1951 | 0.2037 | ||
EFF | 1 | 1.5899 | 1.459 | ||
(50,50) | RB | -0.004 | -0.0038 | -0.0164 | |
RMSE | 0.0914 | 0.0973 | 0.095 | ||
EFF | 1 | 0.8821 | 0.9255 | ||
(100,200) | RB | 0.0018 | 0.002 | -0.0022 | |
RMSE | 0.0615 | 0.0606 | 0.0601 | ||
EFF | 1 | 1.031 | 1.049 | ||
0.617 | (10,10) | RB | -0.0024 | -0.0008 | -0.0484 |
RMSE | 0.2803 | 0.2784 | 0.268 | ||
EFF | 1 | 1.0141 | 1.094 | ||
(50,50) | RB | 0.002 | 0.0029 | -0.0051 | |
RMSE | 0.1342 | 0.1297 | 0.1286 | ||
EFF | 1 | 1.07 | 1.0894 | ||
(100,200) | RB | -0.0012 | -0.0004 | -0.0031 | |
RMSE | 0.0812 | 0.0726 | 0.0726 | ||
EFF | 1 | 1.2507 | 1.2501 | ||
0.4532 | (10,10) | RB | 0.0009 | -0.0036 | -0.041 |
RMSE | 0.3567 | 0.3461 | 0.3368 | ||
EFF | 1 | 1.0619 | 1.1214 | ||
(50,50) | RB | 0.0029 | 0.0011 | -0.0049 | |
RMSE | 0.1698 | 0.1565 | 0.1556 | ||
EFF | 1 | 1.1759 | 1.1899 | ||
(100,200) | RB | -0.0004 | -0.001 | -0.0032 | |
RMSE | 0.1061 | 0.089 | 0.0902 | ||
EFF | 1 | 1.4206 | 1.3833 | ||
0.1336 | (10,10) | RB | 0.0607 | 0.0166 | -0.0229 |
RMSE | 0.7837 | 0.6801 | 0.6807 | ||
EFF | 1 | 1.328 | 1.3257 | ||
(50,50) | RB | 0.0295 | 0.0213 | 0.0158 | |
RMSE | 0.3642 | 0.3073 | 0.3062 | ||
EFF | 1 | 1.4052 | 1.4148 | ||
(100,200) | RB | 0.0009 | -0.0043 | -0.0057 | |
RMSE | 0.2093 | 0.1738 | 0.1768 | ||
EFF | 1 | 1.4494 | 1.4006 |
$\mathit{\Delta}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | ${\mathit{(}\mathit{n}}_{\mathbf{1}}\mathbf{,}{\mathit{n}}_{\mathbf{2}}\mathbf{)}$ | ${\widehat{\mathbf{\u2206}}}_{\mathit{k}}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | ${\widehat{\mathbf{\u2206}}}_{\mathit{MM}}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | ${\widehat{\mathit{\Delta}}}_{\mathit{Prop}}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | |
---|---|---|---|---|---|
$0.8064$ | (10,10) | RB | -0.1686 | 0.0014 | -0.0093 |
RMSE | -0.2447 | 0.1481 | 0.1606 | ||
EFF | 1 | 2.7332 | 2.3214 | ||
(50,50) | RB | -0.042 | 0.0014 | -0.0007 | |
RMSE | 0.0954 | 0.0825 | 0.0844 | ||
EFF | 1 | 1.3367 | 1.2793 | ||
(100,200) | RB | -0.01 | 0.0001 | 0.0008 | |
RMSE | 0.0537 | 0.0463 | 0.047 | ||
EFF | 1 | 1.3505 | 1.3077 | ||
$0.5850$ | (10,10) | RB | -0.1213 | 0.0688 | 0.0152 |
RMSE | 0.297 | 0.2353 | 0.2523 | ||
EFF | 1 | 1.5925 | 1.386 | ||
(50,50) | RB | -0.0274 | 0.0083 | -0.004 | |
RMSE | 0.1266 | 0.0985 | 0.1074 | ||
EFF | 1 | 1.653 | 1.3906 | ||
(100,200) | RB | -0.0029 | 0.0053 | 0.0029 | |
RMSE | 0.0723 | 0.0593 | 0.0609 | ||
EFF | 1 | 1.4848 | 1.4057 | ||
$0.3528$ | (10,10) | RB | -0.1349 | 0.2267 | 0.0105 |
RMSE | 0.3798 | 0.4077 | 0.3296 | ||
EFF | 1 | 0.8684 | 1.3284 | ||
(50,50) | RB | -0.03 | 0.0536 | 0.0032 | |
RMSE | 0.1613 | 0.1278 | 0.1323 | ||
EFF | 1 | 1.5931 | 1.4858 | ||
(100,200) | RB | -0.013 | 0.0187 | 0.0014 | |
RMSE | 0.0949 | 0.0754 | 0.0781 | ||
EFF | 1 | 1.5846 | 1.4756 | ||
$0.2017$ | (10,10) | RB | -0.187 | 0.5953 | -0.0112 |
RMSE | 0.4733 | 0.8444 | 0.4167 | ||
EFF | 1 | 0.3141 | 1.2901 | ||
(50,50) | RB | -0.0524 | 0.1694 | 0.0012 | |
RMSE | 0.2116 | 0.278 | 0.1732 | ||
EFF | 1 | 0.5795 | 1.4938 | ||
(100,200) | RB | -0.0325 | 0.0833 | 0.0028 | |
RMSE | 0.1226 | 0.1533 | 0.0983 | ||
EFF | 1 | 0.6402 | 1.5566 |
$\mathit{\Delta}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | (${\mathit{n}}_{\mathbf{1}}\mathbf{,}{\mathit{n}}_{\mathbf{2}}\mathbf{)}$ | ${\widehat{\mathbf{\u2206}}}_{\mathit{k}}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | ${\widehat{\mathit{\Delta}}}_{\mathit{Prop}}\left(\mathit{X}\mathbf{,}\mathbf{}\mathit{Y}\right)$ | |
---|---|---|---|---|
0.9151 | (10,10) | RB | $-0.2384$ | $-0.1537$ |
RMSE | $0.2801$ | $0.2029$ | ||
EFF | 1.0000 | $1.9056$ | ||
(50,50) | RB | $-0.0793$ | $-0.0404$ | |
RMSE | $0.1020$ | $0.0777$ | ||
EFF | 1.0000 | $1.722$ | ||
(100,200) | RB | $-0.0359$ | $-0.0132$ | |
RMSE | $0.0551$ | $0.048$ | ||
EFF | 1.0000 | $1.3183$ | ||
$0.6099$ | (10,10) | RB | $-0.1117$ | $-0.0682$ |
RMSE | 0.2813 | $0.2452$ | ||
EFF | 1.0000 | $1.316$ | ||
(50,50) | RB | $-0.0281$ | $-0.0192$ | |
RMSE | $0.1209$ | $0.1037$ | ||
EFF | 1.0000 | $1.3596$ | ||
(100,200) | RB | $-0.0041$ | $-0.0063$ | |
RMSE | $0.069$ | $0.0598$ | ||
EFF | 1.0000 | $1.3324$ | ||
$0.3577$ | (10,10) | RB | $-0.1141$ | $-0.0594$ |
RMSE | $0.3804$ | $0.3234$ | ||
EFF | 1.0000 | $1.3833$ | ||
(50,50) | RB | $-0.0353$ | $-0.017$0 | |
RMSE | $0.1756$ | $0.1438$ | ||
EFF | 1.0000 | $1.4914$ | ||
(100,200) | RB | $-0.0064$ | $-0.0023$ | |
RMSE | $0.0944$ | $0.0791$ | ||
EFF | 1.0000 | $1.4263$ | ||
$0.0891$ | (10,10) | RB | $-0.0098$ | $-0.0464$ |
RMSE | $0.8987$ | $0.8097$ | ||
EFF | 1.0000 | $1.2317$ | ||
(50,50) | RB | $0.0671$ | $-0.0014$ | |
RMSE | $0.4347$ | $0.366$0 | ||
EFF | 1.0000 | $1.4105$ | ||
(100,200) | RB | $0.0578$ | $0.0118$ | |
RMSE | $0.2547$ | $0.2103$ | ||
EFF | 1.0000 | $1.4662$ |
OVL | Overlapping |
$\mathit{pdf}$ | Probability Density Function |
ML | Maximum Likelihood |
RB | Relative Bias |
MSE | Mean Square Error |
RMSE | Relative Mean Square Error |
EFF | Efficiency |
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APA Style
Eidous, O. M., Daradkeh, S. K. (2024). On Inference of Weitzman Overlapping Coefficient ∆(X,Y) in the Case of Two Normal Distributions. International Journal of Theoretical and Applied Mathematics, 10(2), 14-22. https://doi.org/10.11648/j.ijtam.20241002.11
ACS Style
Eidous, O. M.; Daradkeh, S. K. On Inference of Weitzman Overlapping Coefficient ∆(X,Y) in the Case of Two Normal Distributions. Int. J. Theor. Appl. Math. 2024, 10(2), 14-22. doi: 10.11648/j.ijtam.20241002.11
AMA Style
Eidous OM, Daradkeh SK. On Inference of Weitzman Overlapping Coefficient ∆(X,Y) in the Case of Two Normal Distributions. Int J Theor Appl Math. 2024;10(2):14-22. doi: 10.11648/j.ijtam.20241002.11
@article{10.11648/j.ijtam.20241002.11, author = {Omar Mohammad Eidous and Salam Khaled Daradkeh}, title = {On Inference of Weitzman Overlapping Coefficient ∆(X,Y) in the Case of Two Normal Distributions }, journal = {International Journal of Theoretical and Applied Mathematics}, volume = {10}, number = {2}, pages = {14-22}, doi = {10.11648/j.ijtam.20241002.11}, url = {https://doi.org/10.11648/j.ijtam.20241002.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20241002.11}, abstract = {The Weitzman overlapping coefficient ∆(X,Y) is the most important and widely used overlapping coefficient, which represents the intersection area between two probability distributions. This research proposes a new general technique to estimate ∆(X,Y) assuming the existence of two independent random samples following normal distributions. In contrast to some studies in this scope that place some restrictions on the parameters of the two populations such as the equality of their means or the equality of their variances, this study did not assume any restrictions on the parameters of normal distributions. Two new estimators for ∆(X,Y) were derived based on the proposed new technique, and then the properties of the estimator resulting from taking their arithmetic mean was studied and compared with some corresponding estimators available in the literature based on the simulation method. An extensive simulation study was performed by assuming two normal distributions with different parameter values to cover most possible cases in practice. The parameter values were chosen taking into account the exact value of ∆(X,Y), which taken to be small (close to zero), medium (close to 0.5) and large (close to 1). The simulation results showed the effectiveness of the proposed technique in estimating ∆(X,Y). By comparing the proposed estimator of ∆(X,Y) with some existing corresponding estimators, its performance was better than the performances of the other estimators in almost all considered cases. }, year = {2024} }
TY - JOUR T1 - On Inference of Weitzman Overlapping Coefficient ∆(X,Y) in the Case of Two Normal Distributions AU - Omar Mohammad Eidous AU - Salam Khaled Daradkeh Y1 - 2024/08/20 PY - 2024 N1 - https://doi.org/10.11648/j.ijtam.20241002.11 DO - 10.11648/j.ijtam.20241002.11 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 14 EP - 22 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20241002.11 AB - The Weitzman overlapping coefficient ∆(X,Y) is the most important and widely used overlapping coefficient, which represents the intersection area between two probability distributions. This research proposes a new general technique to estimate ∆(X,Y) assuming the existence of two independent random samples following normal distributions. In contrast to some studies in this scope that place some restrictions on the parameters of the two populations such as the equality of their means or the equality of their variances, this study did not assume any restrictions on the parameters of normal distributions. Two new estimators for ∆(X,Y) were derived based on the proposed new technique, and then the properties of the estimator resulting from taking their arithmetic mean was studied and compared with some corresponding estimators available in the literature based on the simulation method. An extensive simulation study was performed by assuming two normal distributions with different parameter values to cover most possible cases in practice. The parameter values were chosen taking into account the exact value of ∆(X,Y), which taken to be small (close to zero), medium (close to 0.5) and large (close to 1). The simulation results showed the effectiveness of the proposed technique in estimating ∆(X,Y). By comparing the proposed estimator of ∆(X,Y) with some existing corresponding estimators, its performance was better than the performances of the other estimators in almost all considered cases. VL - 10 IS - 2 ER -