X charts with estimated control limits are commonly used in practice and treated as if the in-control process parameters were known. However, the former can behave quite differently from the latter. To understand the ...X charts with estimated control limits are commonly used in practice and treated as if the in-control process parameters were known. However, the former can behave quite differently from the latter. To understand the differences, it is necessary to study the run length distribution (RLD), its mean (ARL) and standard deviation (SDRL) of the X charts when the control limits are estimated. However, ARL and SDRL are integrals over an infinite region with a boundless integrand, the finiteness has not been proved in literature. In this paper, we show the finiteness and uniform integrability of ARL and SDRL. Furthermore, we numerically evaluate the ARL, SDRL and the RLD using number theory method. A numerical study is conducted to assess the performance of the proposed method and the results are compared with those given by Quesenberry and Chen.展开更多
基金This research is is partially supported by the National Natural Science Foundation of China.
文摘X charts with estimated control limits are commonly used in practice and treated as if the in-control process parameters were known. However, the former can behave quite differently from the latter. To understand the differences, it is necessary to study the run length distribution (RLD), its mean (ARL) and standard deviation (SDRL) of the X charts when the control limits are estimated. However, ARL and SDRL are integrals over an infinite region with a boundless integrand, the finiteness has not been proved in literature. In this paper, we show the finiteness and uniform integrability of ARL and SDRL. Furthermore, we numerically evaluate the ARL, SDRL and the RLD using number theory method. A numerical study is conducted to assess the performance of the proposed method and the results are compared with those given by Quesenberry and Chen.
