Saddlepoint approximation for the bivariate two-sample tests under the random allocation design

Ibrahim A. A. Shanan; Mona Hosny; Abd El-Raheem M. Abd El-Raheem; Ibrahim A. A. Shanan; Mona Hosny; Abd El-Raheem M. Abd El-Raheem

doi:10.3934/nhm.2025051

Networks and Heterogeneous Media

2025, Volume 20, Issue 4: 1190-1200. doi: 10.3934/nhm.2025051

Previous Article Next Article

Research article Topical Sections

Saddlepoint approximation for the bivariate two-sample tests under the random allocation design

1.
Department of Information Technologies, Management Technical College, Southern Technical University, Basra, Iraq
2.
Department of Mathematics, College of Science, King Khalid University, Abha, 61413, Saudi Arabia
3.
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt

Received: 18 June 2025 Revised: 05 October 2025 Accepted: 21 October 2025 Published: 03 November 2025

This article addresses the challenge of accurately estimating p-values in bivariate two-sample tests under random allocation designs, a common setting in clinical and reliability studies. Existing normal approximations often perform poorly in small samples and in the distribution tails, leading to unreliable inference. To overcome this limitation, we propose the use of the saddlepoint approximation as a highly accurate alternative. Through simulation studies and real-data analyses, we demonstrate that the proposed method consistently yields p-values that are closer to the exact permutation values than those obtained from traditional normal approximations, particularly in small-sample settings.
- bivariate two-sample tests,
- saddlepoint approximation,
- random allocation design,
- nonparametric inference,
- P-value accuracy.
Citation: Ibrahim A. A. Shanan, Mona Hosny, Abd El-Raheem M. Abd El-Raheem. Saddlepoint approximation for the bivariate two-sample tests under the random allocation design[J]. Networks and Heterogeneous Media, 2025, 20(4): 1190-1200. doi: 10.3934/nhm.2025051

Related Papers:

Abstract

This article addresses the challenge of accurately estimating p-values in bivariate two-sample tests under random allocation designs, a common setting in clinical and reliability studies. Existing normal approximations often perform poorly in small samples and in the distribution tails, leading to unreliable inference. To overcome this limitation, we propose the use of the saddlepoint approximation as a highly accurate alternative. Through simulation studies and real-data analyses, we demonstrate that the proposed method consistently yields p-values that are closer to the exact permutation values than those obtained from traditional normal approximations, particularly in small-sample settings.

References

[1]	L. Guo, R. Modarres, Testing the equality of matrix distributions, Stat. Methods Appl., 29 (2020), 289–307. https://doi.org/10.1007/s10260-019-00477-7 doi: 10.1007/s10260-019-00477-7
[2]	B. Sürücü, Testing for censored bivariate distributions with applications to environmental data, Environ. Ecol. Stat., 22 (2015), 637–649. https://doi.org/10.1007/s10651-015-0323-x doi: 10.1007/s10651-015-0323-x
[3]	H. Wendt, P. Abry, G. Didier, Wavelet domain bootstrap for testing the equality of bivariate self-similarity exponents, IEEE Stat. Signal Process. Workshop (SSP), (2018), 563–567.https://doi.org/10.1109/SSP.2018.8450710
[4]	R. Modarres, Testing the equality and exchangeability of bivariate distributions, Commun. Stat. Theory Methods, 41 (2012), 3750–3758. https://doi.org/10.1080/03610926.2011.566973 doi: 10.1080/03610926.2011.566973
[5]	G. K. Bhattacharyya, R. A. Johnson, A layer rank test for ordered bivariate alternatives, Ann. Math. Stat., 41 (1970), 1296–1310. https://doi.org/10.1214/aoms/1177696904 doi: 10.1214/aoms/1177696904
[6]	R. A. Johnson, K. G. Mehrotra, Nonparametric tests for ordered alternatives in the bivariate case, J. Multivar. Anal., 2 (1972), 219–229. https://doi.org/10.1016/0047-259x(72)90029-2 doi: 10.1016/0047-259x(72)90029-2
[7]	E. J. Dietz, Multivariate generalization of Johckheere's test for ordered alternatives, Commun. Stat. Theory Methods, 18 (1989), 3763–3783. https://doi.org/10.1080/03610928908830122 doi: 10.1080/03610928908830122
[8]	K. V. Mardia, A non-parametric test for the bivariate two-sample location problem, J. R. Stat. Soc. Ser. B (Methodol.), 29 (1967), 320–342. https://doi.org/10.1111/j.2517-6161.1967.tb00699.x doi: 10.1111/j.2517-6161.1967.tb00699.x
[9]	J. S. Williams, Quick test for comparing two populations with bivariate data, Biometrics, 30 (1977), 215–219. https://doi.org/10.2307/2529314 doi: 10.2307/2529314
[10]	D. Peters, R. H. Randles, A bivariate signed rank test for the two-sample location problem, J. R. Stat. Soc. Ser. B (Methodol.), 53 (1991), 493–504. https://doi.org/10.1111/j.2517-6161.1991.tb01841.x doi: 10.1111/j.2517-6161.1991.tb01841.x
[11]	T. P. Hettmansperger, H. Oja, Affine invariant multivariate multisample sign tests, J. R. Stat. Soc. Ser. B (Stat. Methodol.), 56 (1994), 235–249. https://doi.org/10.1111/j.2517-6161.1994.tb01974.x doi: 10.1111/j.2517-6161.1994.tb01974.x
[12]	K. Sen, S. K. Mathur, A test for bivariate two-sample location problem, Commun. Stat. Theory Methods, 29 (2000), 417–436. https://doi.org/10.1080/03610920008832492 doi: 10.1080/03610920008832492
[13]	H. Samawi, M. F. Al-Saleh, O. Al-Saidy, The matched pair sign test using bivariate ranked set sampling for different ranking-based schemes, Stat. Methodol., 6 (2009), 397–407. https://doi.org/10.1016/j.stamet.2009.02.002 doi: 10.1016/j.stamet.2009.02.002
[14]	S. K. Mathur, A new nonparametric bivariate test for two-sample location problem, Stat. Methods Appl., 18 (2009), 375–388. https://doi.org/10.1007/s10260-008-0095-7 doi: 10.1007/s10260-008-0095-7
[15]	F. Roland, D. Herold, Robust nonparametric tests for the two-sample location problem, Stat. Methods Appl., 20 (2011), 409–422. https://doi.org/10.1007/s10260-011-0164-1 doi: 10.1007/s10260-011-0164-1
[16]	H. E. Daniels, Saddlepoint approximation in statistics, Ann. Math. Stat., 25 (1954), 631–650.
[17]	H. E. Daniels, Tail probability approximations, Int. Stat. Rev., 55 (1987), 37–48. https://doi.org/10.1515/9783112319086-033 doi: 10.1515/9783112319086-033
[18]	I. M. Skovgaard, Saddlepoint expansions for conditional distributions, J. Appl. Probab., 24 (1987), 875–887. https://doi.org/10.1017/s0021900200116754 doi: 10.1017/s0021900200116754
[19]	J. E. Kolassa, Series Approximation Methods in Statistics, New York: Springer, 2006. https://doi.org/10.1007/0-387-32227-2
[20]	R. W. Butler, Saddlepoint Approximations with Applications, Cambridge: Cambridge Univ. Press, 2007.
[21]	A. M. Abd El-Raheem, M. Hosny, E. F. Abd-Elfattah, Statistical inference of the class of nonparametric tests for the panel count and current status data from the perspective of the saddlepoint approximation, J. Math., 2023 (2022), 9111653. https://doi.org/10.1155/2023/9111653 doi: 10.1155/2023/9111653
[22]	A. M. Abd El-Raheem, M. Hosny, Saddlepoint p-values for a class of nonparametric tests for the current status and panel count data under generalized permuted block design, AIMS Math., 8 (2023), 18866–18880. https://doi.org/10.3934/math.2023960 doi: 10.3934/math.2023960
[23]	I. A. Shanan, E. F. Abd-Elfattah, A. M. Abd El-Raheem, A new approach for approximating the p-value of a class of bivariate sign tests, Sci. Rep., 13 (2023), 19133. https://doi.org/10.1038/s41598-023-45975-7 doi: 10.1038/s41598-023-45975-7
[24]	A. M. Abd El-Raheem, I. A. Shanan, M. Hosny, Saddlepoint approximation of the p-values for the multivariate one-sample sign and signed-rank tests, AIMS Math., 9 (2024), 25482–25493. https://doi.org/10.3934/math.20241244 doi: 10.3934/math.20241244
[25]	A. M. Abd El-Raheem, H. N. Mohamed, E. F. Abd-Elfattah, Saddlepoint p-values for a class of location-scale tests, J. Biopharm. Stat., 35 (2025), 641–660. https://doi.org/10.1038/s41598-024-53451-z doi: 10.1038/s41598-024-53451-z
[26]	A. D. Alhejaili, A. A. AlGhamedi, A review of saddle-point approximation: Theory and applications, Adv. Appl. Stat., 92 (2025), 343–390. https://doi.org/10.17654/0972361725015 doi: 10.17654/0972361725015
[27]	A. M. Abd El-Raheem, M. Hosny, On the distribution of the random sum and linear combination of independent exponentiated exponential random variables, Symmetry, 17 (2025), 200. https://doi.org/10.3390/sym17020200 doi: 10.3390/sym17020200
[28]	W. F. Rosenberger, J. M. Lachin, Randomization in Clinical Trials: Theory and Practice, Hoboken: John Wiley & Sons, 2015.
[29]	E. F. Abd-Elfattah, Saddlepoint p-values for two-sample bivariate tests, J. Stat. Plann. Inference, 171 (2016), 92–98. https://doi.org/10.1016/j.jspi.2015.11.002 doi: 10.1016/j.jspi.2015.11.002
[30]	D. J. Hand, F. Daly, K. McConway, D. Lunn, E. Ostrowski, A Handbook of Small Data Sets, Boca Raton: CRC Press, 1993.
[31]	H. M. Samawi, M. Pararai, An optimal bivariate ranked set sample design for the matched pairs sign test, J. Stat. Theory Pract., 3 (2009), 393–406. https://doi.org/10.1080/15598608.2009.10411932 doi: 10.1080/15598608.2009.10411932

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)