On the estimation of the Wald distribution parameters with diverse applications
DOI :
https://doi.org/10.5269/bspm.79632Résumé
This paper evaluates various parameter estimation methods for the Wald distribution using simulations and real-world datasets. Simulation results confirm that increased sample sizes improve estimates, reducing bias and Root Mean-Squared Error (RMSE). The Maximum Likelihood Estimator (MLE) is generally the most robust method for large samples but unstable and biased for smaller ones, particularly in estimating $\lambda$. The Maximum product of spacing estimation (MPSE) method performs well asymptotically, with bias and RMSE decreasing as sample size increases. Least Squares (LSE) and Weighted Least Squares (WLSE) are suitable alternatives to MLE for small-to-moderate samples, showing similar estimates and reduced bias with larger samples. The Cramer-von Mises Estimator (CvME) displayed the worst efficiency due to high RMSE. Bayesian Estimators (BE) showed greater bias and lower efficiency than their frequentist counterparts, especially for $\lambda$, with performance strongly dependent on prior selection. Application to real datasets (HIV/AIDS mortality, COVID-19 death rates, and industrial gauge measurements) demonstrates the Wald distribution's feasibility in diverse health data analysis and reliability studies. The study concludes that MLE and MPSE are efficient estimation methods, suggesting the Wald distribution is a strong candidate for applied parameter estimation. Future work could focus on improving Bayesian methods via better prior selection.
Références
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8. Cordeiro, G. M., Pescim, R. R., and Ortega, E. M. M., The Kumaraswamy generalized half-normal distribution for skewed positive data, Journal of Data Science, 10, 2, 195–224, (2012).
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11. Heathcote, A., Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package, Behavior Research Methods, Instruments, & Computers, 36, 678–694, (2004). DOI: https://doi.org/10.3758/BF03206550
12. Jamal, F., Chesneau, C., and Elgarhy, M., Type II general inverse exponential family of distributions, Journal of Statistics and Management Systems, 23, 3, 617–641, (2020). DOI: https://doi.org/10.1080/09720510.2019.1668159
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15. Nwankwo, B. C., Obiora-Ilouno, H. O., Almulhim, F. A., SidAhmed Mustafa, M., and Obulezi, O. J., Group acceptance sampling plans for type-I heavy-tailed exponential distribution based on truncated life tests, AIP Advances, 14, 3, (2024). DOI: https://doi.org/10.1063/5.0194258
16. Nwankwo, M. P., Alsadat, N., Kumar, A., Bahloul, M. M., and Obulezi, O. J., Group acceptance sampling plan based on truncated life tests for Type-I heavy-tailed Rayleigh distribution, Heliyon, 10, 19, (2024). DOI: https://doi.org/10.1016/j.heliyon.2024.e38150
17. Obulezi, O. J., Obiora-Ilouno, H. O., Osuji, G. A., Kayid, M., and Balogun, O. S., Weibull Sine Generalized Distribution Family: Fundamental Properties, Sub-model, Simulations, with Biomedical Applications, Electronic Journal of Applied Statistical Analysis, 18, 01, 183–212, (2025). DOI: https://doi.org/10.1285/i20705948v18n1p183
18. Obulezi, O. J., Obiora-Ilouno, H. O., Osuji, G. A., Kayid, M., and Balogun, O. S., A new family of generalized distributions based on logistic-x transformation: sub-model, properties and useful applications, Research in Statistics, 3, 1, 2477232, (2025). DOI: https://doi.org/10.1080/27684520.2025.2477232
19. Rao, C. R., A natural example of a weighted binomial distribution, The American Statistician, 31, 1, 24–26, (1977). DOI: https://doi.org/10.1080/00031305.1977.10479187
20. Seshadri, V., and Seshadri, V., Reliability and Survival Analysis, The Inverse Gaussian Distribution: Statistical Theory and Applications, 92–113, (1999). DOI: https://doi.org/10.1007/978-1-4612-1456-4_5
21. Srisuradetchai, P., Niyomdecha, A., and Phaphan, W., Wald intervals via profile likelihood for the mean of the inverse gaussian distribution, Symmetry, 16, 1, 93, (2024). DOI: https://doi.org/10.3390/sym16010093
22. Von Alven, W. H., et al., Reliability engineering, (No Title), (1964).
23. Wald, A., On cumulative sums of random variables, The Annals of Mathematical Statistics, 15, 3, 283–296, (1944). URL: https://www.jstor.org/stable/2236250
24. Yang, H., Huang, M., Chen, X., He, Z., and Pu, S., Enhanced Real-Life Data Modeling with the Modified Burr III Odds Ratio–G Distribution, Axioms, 13, 6, 401, (2024). DOI: https://doi.org/10.3390/axioms13060401
25. Zigangirov, K. S., Expression for the Wald distribution in terms of normal distribution, Radio Engineering and Electronic Physics, 7, 145–48, (1962).
26. Obulezi, O. J., Obulezi distribution: a novel one-parameter distribution for lifetime data modeling, Modern Journal of Statistics, 2, 1, 32–74, (2026). DOI: https://doi.org/10.64389/mjs.2026.02140
27. Chesneau, C., Theory on a new bivariate trigonometric Gaussian distribution, Innovation in Statistics and Probability, 1, 2, 1–17, (2025). DOI: https://doi.org/10.64389/isp.2025.01223
28. Gemeay, A. M., Moakofi, T., Balogun, O. S., Ozkan, E., and Hossain, M. M., Analyzing real data by a new heavy-tailed statistical model, Modern Journal of Statistics, 1, 1, 1–24, (2025). DOI: https://doi.org/10.64389/mjs.2025.01108
29. Mousa, M. N., Moshref, M. E., Youns, N., and Mansour, M. M. M., Inference under Hybrid Censoring for the Quadratic Hazard Rate Model: Simulation and Applications to COVID-19 Mortality, Modern Journal of Statistics, 2, 1, 1–31, (2026). DOI: https://doi.org/10.64389/mjs.2026.02113
30. Noori, N. A., Abdullah, K. N., et al., Development and applications of a new hybrid Weibull-inverse Weibull distribution, Modern Journal of Statistics, 1, 1, 80–103, (2025). DOI: https://doi.org/10.64389/mjs.2025.01112
31. Onyekwere, C. K., Aguwa, O. C., and Obulezi, O. J., An updated lindley distribution: Properties, estimation, acceptance sampling, actuarial risk assessment and applications, Innovation in Statistics and Probability, 1, 1, 1–27, (2025). DOI: https://doi.org/10.64389/isp.2025.01103
32. Orji, G. O., Etaga, H. O., Almetwally, E. M., Igbokwe, C. P., Aguwa, O. C., and Obulezi, O. J., A new odd reparameterized exponential transformed-x family of distributions with applications to public health data, Innovation in Statistics and Probability, 1, 1, 88–118, (2025). DOI: https://doi.org/10.64389/isp.2025.01107
33. Rather, A. A., Subramanian, C., Al-Omari, A. I., and Alanzi, A. R. A., Exponentiated Ailamujia distribution with statistical inference and applications of medical data, Journal of Statistics and Management Systems, 25, 4, 907–925, (2022). DOI: https://doi.org/10.1080/09720510.2021.1966206
2. Alzaatreh, A., Lee, C., and Famoye, F., A new method for generating families of continuous distributions, Metron, 71, 1, 63–79, (2013). DOI: https://doi.org/10.1007/s40300-013-0007-y
3. Alzaatreh, A., Lee, C., and Famoye, F., T-normal family of distributions: a new approach to generalize the normal distribution, Journal of Statistical Distributions and Applications, 1, 1–18, (2014). DOI: https://doi.org/10.1186/2195-5832-1-16
4. Anders, R., Alario, F., Van Maanen, L., et al., The shifted Wald distribution for response time data analysis, Psychological methods, 21, 3, 309, (2016).
5. Aslam, M., Jun, C. H., Lio, Y. L., Ahmad, M., and Rasool, M., Group acceptance sampling plans for resubmitted lots under Burr-type XII distributions, Journal of the Chinese institute of Industrial Engineers, 28, 8, 606–615, (2011). DOI: https://doi.org/10.1080/10170669.2011.651165
6. Cheng, R., and Amin, N., Maximum product of spacings estimation with application to the lognormal distribution (Mathematical Report 79-1), Cardiff: University of Wales IST, (1979).
7. Chhikara, R., and Folks, J. L., The inverse Gaussian distribution: theory: methodology, and applications, CRC Press, (2024).
8. Cordeiro, G. M., Pescim, R. R., and Ortega, E. M. M., The Kumaraswamy generalized half-normal distribution for skewed positive data, Journal of Data Science, 10, 2, 195–224, (2012).
9. Ducharme, G. R., Goodness-of-fit tests for the inverse Gaussian and related distributions, Test, 10, 271–290, (2001). DOI: https://doi.org/10.1007/BF02595697
10. Eugene, N., Lee, C., and Famoye, F., Beta-normal distribution and its applications, Communications in StatisticsTheory and methods, 31, 4, 497–512, (2002). DOI: https://doi.org/10.1081/STA-120003130
11. Heathcote, A., Fitting Wald and ex-Wald distributions to response time data: An example using functions for the S-PLUS package, Behavior Research Methods, Instruments, & Computers, 36, 678–694, (2004). DOI: https://doi.org/10.3758/BF03206550
12. Jamal, F., Chesneau, C., and Elgarhy, M., Type II general inverse exponential family of distributions, Journal of Statistics and Management Systems, 23, 3, 617–641, (2020). DOI: https://doi.org/10.1080/09720510.2019.1668159
13. Kundu, D., and Raqab, M. Z., Estimation of R= P (Y¡ X) for three-parameter Weibull distribution, Statistics & Probability Letters, 79, 17, 1839–1846, (2009). DOI: https://doi.org/10.1016/j.spl.2009.05.026
14. Meyer-Grant, C. G., Conjugate Bayesian analysis of the Wald model: On an exact drift-rate posterior, Journal of Mathematical Psychology, 124, 102904, (2025). DOI: https://doi.org/10.1016/j.jmp.2025.102904
15. Nwankwo, B. C., Obiora-Ilouno, H. O., Almulhim, F. A., SidAhmed Mustafa, M., and Obulezi, O. J., Group acceptance sampling plans for type-I heavy-tailed exponential distribution based on truncated life tests, AIP Advances, 14, 3, (2024). DOI: https://doi.org/10.1063/5.0194258
16. Nwankwo, M. P., Alsadat, N., Kumar, A., Bahloul, M. M., and Obulezi, O. J., Group acceptance sampling plan based on truncated life tests for Type-I heavy-tailed Rayleigh distribution, Heliyon, 10, 19, (2024). DOI: https://doi.org/10.1016/j.heliyon.2024.e38150
17. Obulezi, O. J., Obiora-Ilouno, H. O., Osuji, G. A., Kayid, M., and Balogun, O. S., Weibull Sine Generalized Distribution Family: Fundamental Properties, Sub-model, Simulations, with Biomedical Applications, Electronic Journal of Applied Statistical Analysis, 18, 01, 183–212, (2025). DOI: https://doi.org/10.1285/i20705948v18n1p183
18. Obulezi, O. J., Obiora-Ilouno, H. O., Osuji, G. A., Kayid, M., and Balogun, O. S., A new family of generalized distributions based on logistic-x transformation: sub-model, properties and useful applications, Research in Statistics, 3, 1, 2477232, (2025). DOI: https://doi.org/10.1080/27684520.2025.2477232
19. Rao, C. R., A natural example of a weighted binomial distribution, The American Statistician, 31, 1, 24–26, (1977). DOI: https://doi.org/10.1080/00031305.1977.10479187
20. Seshadri, V., and Seshadri, V., Reliability and Survival Analysis, The Inverse Gaussian Distribution: Statistical Theory and Applications, 92–113, (1999). DOI: https://doi.org/10.1007/978-1-4612-1456-4_5
21. Srisuradetchai, P., Niyomdecha, A., and Phaphan, W., Wald intervals via profile likelihood for the mean of the inverse gaussian distribution, Symmetry, 16, 1, 93, (2024). DOI: https://doi.org/10.3390/sym16010093
22. Von Alven, W. H., et al., Reliability engineering, (No Title), (1964).
23. Wald, A., On cumulative sums of random variables, The Annals of Mathematical Statistics, 15, 3, 283–296, (1944). URL: https://www.jstor.org/stable/2236250
24. Yang, H., Huang, M., Chen, X., He, Z., and Pu, S., Enhanced Real-Life Data Modeling with the Modified Burr III Odds Ratio–G Distribution, Axioms, 13, 6, 401, (2024). DOI: https://doi.org/10.3390/axioms13060401
25. Zigangirov, K. S., Expression for the Wald distribution in terms of normal distribution, Radio Engineering and Electronic Physics, 7, 145–48, (1962).
26. Obulezi, O. J., Obulezi distribution: a novel one-parameter distribution for lifetime data modeling, Modern Journal of Statistics, 2, 1, 32–74, (2026). DOI: https://doi.org/10.64389/mjs.2026.02140
27. Chesneau, C., Theory on a new bivariate trigonometric Gaussian distribution, Innovation in Statistics and Probability, 1, 2, 1–17, (2025). DOI: https://doi.org/10.64389/isp.2025.01223
28. Gemeay, A. M., Moakofi, T., Balogun, O. S., Ozkan, E., and Hossain, M. M., Analyzing real data by a new heavy-tailed statistical model, Modern Journal of Statistics, 1, 1, 1–24, (2025). DOI: https://doi.org/10.64389/mjs.2025.01108
29. Mousa, M. N., Moshref, M. E., Youns, N., and Mansour, M. M. M., Inference under Hybrid Censoring for the Quadratic Hazard Rate Model: Simulation and Applications to COVID-19 Mortality, Modern Journal of Statistics, 2, 1, 1–31, (2026). DOI: https://doi.org/10.64389/mjs.2026.02113
30. Noori, N. A., Abdullah, K. N., et al., Development and applications of a new hybrid Weibull-inverse Weibull distribution, Modern Journal of Statistics, 1, 1, 80–103, (2025). DOI: https://doi.org/10.64389/mjs.2025.01112
31. Onyekwere, C. K., Aguwa, O. C., and Obulezi, O. J., An updated lindley distribution: Properties, estimation, acceptance sampling, actuarial risk assessment and applications, Innovation in Statistics and Probability, 1, 1, 1–27, (2025). DOI: https://doi.org/10.64389/isp.2025.01103
32. Orji, G. O., Etaga, H. O., Almetwally, E. M., Igbokwe, C. P., Aguwa, O. C., and Obulezi, O. J., A new odd reparameterized exponential transformed-x family of distributions with applications to public health data, Innovation in Statistics and Probability, 1, 1, 88–118, (2025). DOI: https://doi.org/10.64389/isp.2025.01107
33. Rather, A. A., Subramanian, C., Al-Omari, A. I., and Alanzi, A. R. A., Exponentiated Ailamujia distribution with statistical inference and applications of medical data, Journal of Statistics and Management Systems, 25, 4, 907–925, (2022). DOI: https://doi.org/10.1080/09720510.2021.1966206
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2026-03-11
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