Some Monitoring Procedures Related to Asymmetry Parameter of Azzalini’s Skew-Normal Model

Abstract

In the real world, we often observe that the underlying distribution of some Gaussian processes tends to become skewed, when some undesirable assignable cause takes place in the process. Such phenomena are common in the field of manufacturing and in chemical industries, among others, where a process deviates from a normal model and becomes a skew-normal. The Azzalini’s skew-normal (hereafter ASN) distribution is a well-known model for such processes. In other words, we assume that the in-control (hereafter IC) distribution of the process under consideration is normal, that is a special case of the ASN model with asymmetry parameter zero, whereas the out-of-control (hereafter OOC) process distribution is ASN with any nonzero asymmetry parameter. In the ASN model, a change in asymmetry parameter also induces shifts in both the mean and variance, even if, both the location and scale parameters remain invariant. Traditionally, researchers consider a shift either in the mean or in variance or in both the parameters of the normal distribution. Some inference and monitoring issues related to deviation from symmetry are essential problems that are largely overlooked in literature. To this end, we propose various test statistics and design for sequential monitoring schemes for the asymmetry parameter of the ASN model. We examine and compare the performance of various procedures based on an extensive Monte-Carlo experiment. We provide an illustration based on an interesting manufacturing case study. We also offer some concluding remarks and future research problems.