A proposed approach for approximating lower truncated normal cumulative distribution: application to reliability of used devices

References

• Bouzas, P. R., Aguilera, A. M., & Valderrama, M. J. (2002). Forecasting a class of doubly stochastic poisson processes. Statistical Papers, 43(4), 507. https://doi.org/10.1007/s00362-002-0120-0
   Web of Science ®Google Scholar
• Bowling, S. R., Khasawneh, M. T., Kaewkuekool, S., & Cho, B. R. (2009). A logistic approximation to the cumulative normal distribution. Journal of Industrial Engineering and Management, 2, 114–11. https://doi.org/10.3926/jiem.2009.v2n1.p114-127
   Google Scholar
• Cadwell, J. H. (1951). The bivariate normal integral. Biometrika, 38(3–4), 475–479. https://doi.org/10.1080/00949659008811236
   Web of Science ®Google Scholar
• Gupta, A. K., & Tracy, D. S. (1976). Recurrence relations for the moments of truncated multinormal distribution. Communications in Statistics - Theory and Methods, 5(9), 855–865. https://doi.org/10.1080/03610927608827402
   Web of Science ®Google Scholar
• Hamaker, H. C. (1978). Approximating the cumulative normal distribution and its inverse. Applied Statistics, 27(1), 76–77. https://doi.org/10.2307/2346231
   Google Scholar
• Hamasha, M. M. (2017). Practitioner advice: Approximation of the cumulative density of left-sided truncated normal distribution using logistic function and its implementation in microsoft excel. Quality Engineering, 29(2), 322–328. https://doi.org/10.1080/08982112.2016.1196373
   Web of Science ®Google Scholar
• Hamasha, M. M. (2018). Generate random variates using a newly introduced approximation to cumulative density of lower truncated normal distribution for simulation applications. International Journal of Mathematics in Operational Research, 13(3), 265–279. https://doi.org/10.1504/IJMOR.2018.094852
   Google Scholar
• Hamasha, M. M. (2019). Mathematical approximation of single and double-sided truncated normal distribution using logistic function. International Journal of Industrial Engineering: Theory, Practice and Applications, 26(6), 934–944. https://doi.org/10.23055/ijietap.2019.26.6.3344
   Web of Science ®Google Scholar
• Hamasha, M. M., Ali, H., Hamasha, S., & Ahmed, A. Ultra-fine transformation of data for normality. Heliyon, 8(5), e09370. 2022. https://doi.org/10.1016/j.heliyon.2022.e09370.
   Google Scholar
• Hamasha, M. M., Ali, H., Hamasha, S., & Ahmed, A. (2021). A mathematical approximation to left-sided truncated normal distribution based on hart’s model. Journal of Applied Engineering Science, 19(4), 1093–1099. https://doi.org/10.5937/jaes0-29895
   Google Scholar
• Hamasha, M. M., Ali, H., Hamasha, S., & Ahmed, A (2022). An approximation to the inverse of left-sided truncated gaussian cumulative normal density function using to generate random variates for simulation applications. Accepted, Journal of Applied Engineering Science, 20(2). https://doi.org/10.5937/jaes0-35413
   Google Scholar
• Hart, R. G. A. (1963). Close approximation related to the error function. Mathematics of Computation, 20(96), 600–602. https://doi.org/10.1090/S0025-5718-1966-0203907-1
   Google Scholar
• Horrace, W. C. (2005). Some results on the multivariate truncated normal distribution. Journal of Multivariate Analysis, 94(1), 209–221. https://doi.org/10.1016/j.jmva.2004.10.007
   Web of Science ®Google Scholar
• Horrace, W. C. (2015). Moments of the truncated normal distribution. Journal of Productivity Analysis, 43(2), 133–138. https://doi.org/10.1007/s11123-013-0381-8
   Web of Science ®Google Scholar
• Hoyt, J. P. A. (1968). Simple approximation to the standard normal probability density function. The American Statistician, 22(3), 25–26. https://doi.org/10.1080/00031305.1968.10480455
   Google Scholar
• Johnson, N. L., & Kotz, S. (1975). A vector multivariate hazard rate. Journal of Multivariate Analysis, 5(1), 53–66. https://doi.org/10.1016/0047-259X(75)90055-X
   Google Scholar
• Jolani, S. (2014). An analysis of longitudinal data with nonignorable dropout using the truncated multivariate normal distribution. Journal of Multivariate Analysis, 131, 163–173. https://doi.org/10.1016/j.jmva.2014.06.016
   Web of Science ®Google Scholar
• Khasawneh, M. T., Bowling, S. R., Kaewkuekool, S., & Cho, B. R. (2005a). Tables of a truncated standard normal distribution: A singly truncated case. Quality Engineering, 17(1), 33–50. https://doi.org/10.1081/QEN-200028681
   Google Scholar
• Khasawneh, M. T., Bowling, S. R., Kaewkuekool, S., & Cho, B. R. (2005b). Tables of a truncated standard normal distribution: A doubly truncated case. Quality Engineering, 17(2), 227–241. https://doi.org/10.1081/QEN-200057321
   Google Scholar
• McGill, J. I. (1992). The multivariate hazard gradient and moments of the truncated multinormal distribution. Comm Statist Theory Methods, 213053–213060.
   Web of Science ®Google Scholar
• Pearn, W. L., Hung, H. N., Peng, N. F., & Huang, C. Y. (2007). Testing process precision for truncated normal distributions. Microelectronics Reliability, 47(12), 2275–2281. https://doi.org/10.1016/j.microrel.2006.12.001
   Web of Science ®Google Scholar
• Pender, J. (2015). The truncated normal distribution: Applications to queues with impatient customers. Operations Research Letters, 43(1), 40–45. https://doi.org/10.1016/j.orl.2014.10.008
   Web of Science ®Google Scholar
• Rai, B., & Singh, N. (2003). Hazard rate estimation from incomplete and unclean warranty data. Reliability Engineering & System Safety, 81(81), 79–92. https://doi.org/10.1016/S0951-8320(03)00083-8
   Google Scholar
• Robert, C. P. (1995). Simulation of truncated normal variables. Statistics and Computing, 5(2), 121–125. https://doi.org/10.1007/BF00143942
   Web of Science ®Google Scholar
• Sherif, Y. S. (1982). Inverse truncated normal distribution as a failure model. Reliability Engineering, 3(3), 209–211. https://doi.org/10.1016/0143-8174(82)90030-0
   Google Scholar
• Smith, M. J. A. (2008). Probabilistic abstract interpretation of imperative programs using truncated normal distributions. Electronic Notes in Theoretical Computer Science, 220(3), 43–59. https://doi.org/10.1016/j.entcs.2008.11.018
   Google Scholar
• Stein, W. E., Pfaffenberger, R. C., & Mizzi, P. J. (1993). A stochastic dominance comparison of truncated normal distributions. European Journal of Operational Research, 67(2), 259–266. https://doi.org/10.1016/0377-2217(93)90066-V
   Web of Science ®Google Scholar
• Wei, M., Jin, W., & Shen, L. (2012). A platoon dispersion model based on a truncated normal distribution of speed. Journal of Applied Mathematics, 1–13. https://doi.org/10.1155/2012/727839
   Google Scholar