The properties of the ATTRIVAR chart based on the sum of truncated normal distributions

Abstract

The attribute-variable sampling procedure has been considered in recent process control studies. The items of the sample are first submitted to a go-no-go gauge test and, according to the result of the test, they are approved or reproved with regard to an X dimension. If the number of reproved items exceeds a specified threshold, all the sample items are inspected again, but now by measuring the X dimension. The control chart with attribute-variable sampling is called ATTRIVAR chart - "ATTRI from Attribute and VAR from variable". With the ATTRIVAR chart in use, the signaling rule is based on the value, the mean of the X observations. This way, the ATTRIVAR chart signals when the go-no-go gauge test reproves more sample items than a pre-specified threshold and, in the second stage, the point falls outside the control limits of the ATTRIVAR chart. With the approved /reproved classification and with the assumption that the X observations are normally distributed, it follows that the is function of truncated normals. Because of that, the falsa alarm risk and the power of the ATTRIVAR chart are not easy to obtain, their computation involve numerical integrations with growing complexity as the sample size increases. The motivation to conduct the proposed research lies on the great applicability of the ATTRIVAR charts combined with the lack of appropriate tools to deal with the sum of truncated distributions.