Control chart improvements for monitoring times between infrequent events

Due to the increasing use of modern technologies in the manufacturing of products in the industries and the interest in the control of the appearance of diseases in public health, the Shewhart control charts for the monitoring of times between events (TEE) infrequent are quite relevant in practice. These graphs have various applications, for example, in industrial processes with a certain level of automation or in the monitoring of public health occurrences, such as rare congenital diseases or pandemics. Frequently, in the construction and evaluation of the performance of control charts, including Shewhart control charts for TEE monitoring, various interactive statistical books, manuals, guides and software are used that include CEP modules (such as Minitab and SPSS ) that do not consider the effects of parameter estimation on the performance and design of control charts. This omission leads to a misinterpretation of the graph's performance measures, which in turn has a negative impact on process quality monitoring. This problem was initially addressed primarily through an unconditional probability perspective. However, this perspective is based on an "average" performance of the graph rather than the actual or achieved performance of any particular graph in a given application using estimators from a given reference sample, an approach known as the conditional probability perspective. which was proposed in mid-2015. For that reason, although some authors in recent years (including members of our research team) have emphasized the importance and have worked on this conditional perspective, more studies are required to fill the theoretical gaps. and gain a comprehensive understanding of the impact of the estimate. On the other hand, the recent literature that considers the effect of parameter estimation on the performance and design of control charts to monitor TEE provides methods whose formulas can be very complicated for users because they require, for example, equation solution techniques. non-linear. With this motivation, we have sought the development of adaptations of traditional methods and, mainly, of new and innovative mathematical and statistical methods that allow to overcome the problem described.