Four-parameter beta distribution estimation and skewness test

Sequential tolerance control (STC) is a tolerance control methodology used in discrete parts manufacturing. Recently, an adaptive sphere-fitting method for STC (ASF-STC) was developed to account for potential skewness in manufacturing operations' distributions, a factor not considered in conventional STC. ASF-STC offers significant improvements over conventional STC when such skewness exists. The direction of skewness of an operations' distribution is a necessary input to ASF-STC. Thus, a novel approach to determining the skewness of a distribution for small sample sizes is presented here. ASF-STC has an additional requirement of distribution information for each operation. The beta distribution is an ideal candidate here, as it is very flexible in shape. The literature on four-parameter beta estimation is very limited, and their performance for small sample sizes is poor. STC was designed for low-volume production, thus the estimation for small sample sizes is necessary here. This study presents a heuristic, based on the method of moments estimates for a beta distribution, that estimates the four parameters for a beta distribution with small sample size. Several computational results are provided to compare this heuristic to the best-known procedure, with the heuristic found to perform better for the test problems considered.