Abstract
The Gaussian or normal distribution is vital in most areas of industrial engineering, including simulation. For example, the inverse of the Gaussian cumulative density function is used in all simulation software (e.g., ARENA, ProModel) to generate a group of random numbers that fit Gaussian distribution. It is also used to estimate the life expectancy of new devices. However, the Gaussian distribution that is truncated from the left side is not defined in any simulation software. Estimation of the expected life of used devices needs left-sided truncated Gaussian distribution. Additionally, very few works examine generating random numbers from left-sided truncated Gaussian distribution. A high accuracy mathematical-based approximation to the left-sided truncated Gaussian cumulative density function is proposed in the current work. Our approximation is built based on Polya’s approximation of the Gaussian cumulative density function. The current model is beneficial to approximate the inverse of the left-sided truncated Gaussian cumulative density function to generate random variates, which is necessary for simulation applications.
Keywords
References
Tunno, B. J., Michanowicz, D. R., Shmool, J. L., Kinnee, E., Cambal, L., Tripathy, S., Gillooly, S., Roper, C., Chubb, L., Clougherty, J. E. (2016). Spatial variation in inversion-focused vs 24-h integrated samples of PM2. 5 and black carbon across Pittsburgh, PA. Journal of exposure science & environmental epidemiology, vol. 26, no. 4, 365-376. https://doi.org/10.1038/jes.2015.14
Wade, T., McKenzie, C. A., Rutt, B. K. (2014). Flip angle mapping with the accelerated 3D look‐locker sequence. Magnetic resonance in medicine, vol. 71, no. 2, 591-598. https://doi.org/10.1002/mrm.24697
Fischer, M., & Jakob, K. (2016). pTAS distributions with application to risk management. Journal of Statistical Distributions and Applications, vol. 3, no. 1, 1-18. https://doi.org/10.1186/s40488-016-0049-9
Cha, J., Cho, B. R. (2015). Classical statistical inference extended to truncated populations for continuous process improvement: test statistics, P‐values, and confidence intervals. Quality and Reliability Engineering International, vol. 31, no. 8, 1807-1824. https://doi.org/10.1002/qre.1719
Ross, S. R. (2014). Introduction to probability models, 11th Ed. Elsevier, Oxford, UK
Zhang, X., Shen, C., Cheng, P., Li, Q. (2017). An image-processing based method for the measurement of the film thickness of a swirl atomizer. Journal of Visualization, vol. 20, no. 1, 1-5. https://doi.org/10.1007/s12650-016-0370-x
Harrison, D., Sutton, D., Carvalho, P., Hobson, M. (2015). Validation of Bayesian posterior distributions using a multidimensional Kolmogorov–Smirnov test. Monthly Notices of the Royal Astronomical Society, vol. 451, no. 3, 2610-2624. https://doi.org/10.1093/mnras/stv1110
Mahajan, A., Tatikonda, S. (2015). An algorithmic approach to identify irrelevant information in sequential teams. Automatica, vol. 61, 178-191. https://doi.org/10.1016/j.automatica.2015.08.002
Davidson, R. (2015). Computing, the bootstrap and economics. Canadian Journal of Economics/Revue canadienne d'économique, vol. 48, no. 4, 1195-1214. https://doi.org/10.1111/caje.12158
Zuverink, A., (2015) Surface roughness scattering of electrons in bulk mosfets. Thesis Submitted to the University of Wisconsin-Madison, Madision, USA
Fang, X., Li, J., Wong, W. K., Fu, B. (2016). Detecting the violation of variance homogeneity in mixed models. Statistical methods in medical research, vol. 25, no. 6, 2506 2520. http://dx.doi.org/10.1177/0962280214526194
Dutta, S., Misra, I. S. (2014). Error Analysis of 2-tierM-ary Star QAM Modulation in Shadowed Fading Channels. International Journal of Computer Applications, vol. 88, no. 1, 9-16.
Büyükkaracığan, N. (2014). Determining the best fitting distributions for minimum flows of streams in Gediz Basin. International Journal of Civil and Environmental Engineering, vol. 8, no. 6, 417-422. https://doi.org/10.5281/zenodo.1093263
Moheghi, H., Niaki, S. T. A., Bootaki, B., Bakhshesh, D. (2017). On the effect of inducted negative correlation rate for beta acceptance–rejection algorithms. Communications in Statistics-Simulation and Computation, vol. 46, no. 3, 2152-2167. https://doi.org/10.1080/03610918.2015.1039128
Hörmann, W., Leydold, J. (2014). Generating generalized inverse Gaussian random variates. Statistics and Computing, vol. 24, no. 4, 547-557. https://doi.org/10.1007/s11222-013-9387-3
Lijoi, A., Prünster, I. (2014). Discussion of “On simulation and properties of the stable law” by L. Devroye and L. James. Statistical methods & applications, vol. 23, no. 3, 371-377. https://doi.org/10.1007/s10260-014-0269-4
Zhu, H., Dick, J. (2014). Discrepancy bounds for deterministic acceptance-rejection samplers. Electronic Journal of Statistics, vol. 8, no. 1, 678-707. https://doi.org/10.1214/14-EJS898
Xi, B., Tan, K. M., Liu, C. (2013). Logarithmic transformation-based gamma random number generators. Journal of Statistical Software, vol. 55, 1-17. https://doi.org/10.18637/jss.v055.i04
Hung, Y. C., Chen, W. C. (2017). Simulation of some multivariate distributions related to the Dirichlet distribution with application to Monte Carlo simulations. Communications in Statistics-Simulation and Computation, vol. 46, no. 6, 4281-4296. https://doi.org/10.1080/03610918.2015.1115066
Romano, P. K. (2015). An algorithm for generating random variates from the Madland–Nix fission energy spectrum. Computer Physics Communications, vol. 187, 152-155. https://doi.org/10.1016/j.cpc.2014.11.001
Favaro, S., Nipoti, B., Teh, Y. W. (2015). Random variate generation for Laguerre-type exponentially tilted $alpha $-stable distributions. Electronic Journal of Statistics, 9, no. 1, 1230-1242. http://doi.org/10.1214/15-EJS1033
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, vol. 2, no. 1, 114-127. http://dx.doi.org/10.3926/jiem..v2n1.p114-127
Jawitz, J. W. (2004). Moments of truncated continuous univariate distributions. Advances in water resources, vol. 27, no. 3, 269-281. https://doi.org/10.1016/j.advwatres.2003.12.002
Khasawneh, M. T., Bowling, S. R., Kaewkuekool, S., Cho, B. R. (2004). Tables of a truncated standard normal distribution: A singly truncated case. Quality Engineering, vol. 17, no. 1, 33-50. https://doi.org/10.1081/QEN-200028681
Khasawneh, M. T., Bowling, S. R., Kaewkuekool, S., Cho, B. R. (2005). Tables of a truncated standard normal distribution: A doubly truncated case. Quality Engineering, vol. 17, no. 2, 227-241. https://doi.org/10.1081/QEN-200057321
Kim, T. M., Takayama, T. (2003). Computational improvement for expected sliding distance of a caisson-type breakwater by introduction of a doubly-truncated normal distribution. Coastal Engineering Journal, vol. 45, no. 3, 387-419. https://doi.org/10.1142/S0578563403000816
Makarov, Y. V., Loutan, C., Ma, J., De Mello, P. (2009). Operational impacts of wind generation on California power systems. IEEE transactions on power systems, vol. 24, no. 2, 1039-1050. https://doi.org/10.1109/TPWRS.2009.2016364
Cha, J., Cho, B. R., Sharp, J. L. (2013). Rethinking the truncated normal distribution. International Journal of Experimental Design and Process Optimisation, vol. 3, no. 4, 327-363.
Hamasha, M., Al, 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, vol. 19, no. 4, 1049-1055. https://doi.org/10.5937/jaes0-29895
Burkardt, J. (2014). The truncated normal distribution. Department of Scientific Computing Website, Florida State University, 1, 35.
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