Nonparametric procedures for process control when the control value is not specified
In industrial production processes, control charts have been developed to detect changes in the parameters specifying the quality of the production so that some rectifying action can be taken to restore the parameters to satisfactory values. Examples of the control charts are the Shewhart chart and the cumulative sum control chart (CUSUM chart). In designing a control chart, the exact distribution of the observations, e.g. normal distribution, is usually assumed to be known. But, when there is not sufficient information in determining the distribution, nonparametric procedures are appropriate. In such cases, the control value for the parameter may not be given because of insufficient information.
To construct a control chart when the control value is not given, a standard sample must be obtained when the process is known to be under control so that the quality of the product can be maintained at the same level as that of the standard sample. For this purpose, samples of fixed size are observed sequentially, and at each time a sample is observed a two-sample nonparametric statistic is obtained from the standard sample and the sequentially observed sample. With these sequentially obtained statistics, the usual process control procedure can be done. The truncation point is applied to denote the finite run length or the time at which sufficient information about the distribution of the observations and/or the control value is obtained so that the procedure may be switched to a parametric procedure or a nonparametric procedure with a control value.
To lessen the difficulties in the dependent structure of the statistics we use the fact that conditioned on the standard sample the statistics are i.i.d. random variables. Upper and lower bounds of the run length distribution are obtained for the Shewhart chart. A Brownian motion process is used to approximate the discrete time process of the CUSUM chart. The exact run length distribution of the approximated CUSUM chart is derived by using the inverse Laplace transform. Applying an appropriate correction to the boundary improves the approximation.