Suppose one wants to monitor a certain characteristic, say the mean, of the outcomes of a manufacturing process. To this end, at regular time points a produced item is sampled and its outcome is recorded (or, quite often, a few such items are sampled at each point, and e.g. the average of the observed values is used). These values are then depicted as a function of time, thus giving an impression of how the process evolves. However, to transform such a picture into a control chart, an additional ingredient is needed. The idea is that a process which is in control produces outcomes which are between a given upper and lower control limit, except for a very small probability p (e.g. p = 0.001). Hence the picture containing the observed values is adorned by adding three horizontal lines. One in the middle, as a central line representing the actual process mean, around which the observed values fluctuate. This central line is flanked at a safe distance by two parallel lines, representing the upper and lower control limit. As long as the process is in control, i.e. its mean stays at its intended value, the observed values will typically fall within the band thus defined. If the process goes out of control, however, usually quite soon outcomes will arise which fall outside the interval spanned by these two limits. To see this, suppose for example that a shift of the actual process mean has occurred at a certain point in time. From then on, the observed values will again fluctuate around a horizontal line. However, this line no longer lies in the center of the band, but it will have moved towards one of its sides, thus making it (much) easier to jump outside the band. Leaving the band is recorded as a signal, which suggests intervention.
If the distribution leading to the outcomes of the process is given, it is in principle rather straightforward to determine such control limits: one just selects the appropriate quantiles of this distribution in relation to the specified p. (The p-quantile or p-percentile xp of a given distribution can be defined as follows: if the random variable X has this distribution, then xp is such that P(X < xp) = p.) Unfortunately, this assumption is not very realistic, as in practice the underlying distribution is (or at least its parameters are) typically unknown. An obvious and universally popular solution for this seemingly minor obstacle is to plug in estimates for the unknown parameters (or for the distribution as a whole in the nonparametric case) and to proceed as before. The rationale behind this attitude presumably is that exact results are neither attainable nor desired and that all one needs is reasonable guidance for sensible behavior in practice.
Unfortunately again, however, this appealing no-nonsense approach can lead to grossly wrong results which are entirely misleading and on which no sensible behavior can be based. The reason for this debacle is that in view of the small p involved, one is dealing with rather extreme quantiles. Estimating these with relative errors of acceptable size will require (very) large sample sizes, which often are not available in control chart practice. The aim of the present project is to come up with explicit and simple proposals to deal with this largely ignored problem. The idea roughly is to develop a variety of control charts, together with an adequate strategy for choosing one of these on the basis of the available data. In this way a balance can be achieved between accuracy on the one hand and reliability on the other. If the data are "well behaved" in the sense that they are more or less normally distributed, one can stay close to the classical plug-in approach, but if a preliminary check suggests otherwise, one is guided towards more robust alternatives. Suitable software will be produced to facilitate implementation in practice.
As outlined above, the intended products of the project will be simple and readily applicable in control chart practice. The mathematical methods required for their derivation, however, are quite complicated. Neither exact methods nor numerical evaluation can provide a clear insight into what is really going on. Needed are asymptotic techniques, quite often of higher order type. Fortunately, ample experience in this area is available in the project team, extending beyond the area of SPC to rank tests, goodness-of-fit tests, adaptive procedures, data-driven methods, intermediate efficiencies and deficiencies, etc.