A mathematical approximation to left-sided truncated normal distribution based on Hart's model

Abstract

Left-sided truncated distributions (LSTD) have been found in different situations in the industry. For example, the life distribution of used devices is left-sided truncated distribution. Moreover, if a lower specification exists without the upper specification limit, the product distribution is truncated from the left side. Left-sided truncated normal distributions (LSTND) is a special case where the original distribution is normal. LSTND characteristics, as well as cumulative densities and probabilities can be difficult to employ manually, with most practitioners relying largely on specialized (and expensive) software. In many cases, practitioners are against purchasing software, as they are often limited in the number of estimations. The paper will provide an accurate and straightforward approximation to the cumulative density of LSTND. Hart's normal distribution is simplified and used as a foundation of this model. The maximum absolute error for the curve at different truncation points (i.e., ZL) over the definition range (i.e., [zL: ∞]) is as follows: 0.004303 for ZL = -4, 0.00432 for ZL = -3, 0.00449 for ZL = -2, 0.005727 for ZL =-1, and 0.0106 for ZL = 0. Even the maximum errors are very ignorable in probability applications. Further, it is rare to find a truncation point of higher than -2 in the industry.