In this paper, we are concerned with pure statistical Shewhart control charts for the scale parameter of the three-parameter Weibull control variable, where, and are the location, the scale and the shape parameters, respectively, with fixed (FSI) and variable (VSI) sampling intervals. The parameters and are assumed to be known. We consider two-sided, and lower and upper one-sided Shewhart control charts and their FSI and VSI versions . They jointly control the mean and the variance of the Weibull control variable X. The pivotal statistic of those control charts is the maximum-likelihood estimator of for the Nth random sample XN=(X1N,X2N,…,XnN) of the Weibull control variable X. The design and performance of these control charts are studied. Two criteria, i.e. 'comparability criterion' (or 'matched criterion') under control and 'primordial criterion', are imposed on their design. The performance of these control charts is measured using the function average time to signal. For the VSI versions, the constant which defines the partition of the 'continuation region' is obtained through the 'comparability criterion' under control. The monotonic behaviour of the function average time to signal in terms of the parameters (magnitude of the shift suff ered by the target value 0), and is studied. We show that the function average time to signal of all the control charts studied in this paper does not depend on the value of the parameter or on 0, and, under control, does not depend on the parameter, when Delta (the probability of a false alarm) and n (sample size) are fixed. All control charts satisfy the 'primordial criterion' and, for fixed, on average, they all (except the two-sided VSI, for which we were not able to ascertain proof) are quicker in detecting the shift as increases. We conjecture - and we are not contradicted by the numerical example considered - that the same is true for the two-sided VSI control chart. We prove that, under the average time to signal criterion, the VSI versions are always preferable to their FSI versions. In the case of one-sided control charts, under the 'comparability criterion', the VSI version is always preferable to the FSI version, and this advantage increases with and the extent of the shift. Our one-sided control charts perform better and have more powerful statistical properties than does our two-sided control chart. The numerical example where n=5,0=1,=0.5, 1.0, 2.0, and Delta=1/370.4 is presented for the two-sided, and the lower and upper one-sided control charts. These numerical results are presented in tables and in figures. The joint influence of the parameters and in the function average time to signal is illustrated.

Key Words and Phrases: two-dimensional Markov chain, average run length, run length percentage points, probability of a misleading signal. ABSTRACTChanges in the process mean µ ( ) or in the process standard deviation σ ( ) ought to be regarded as an indication that a production process is out of control. This paper considers the problem of the joint monitoring of these two parameters -when the quality characteristic follows a normal distribution -, using a combined Exponentially Weighted Moving Average CEWMA ( ) scheme.Three performance measures of this joint control scheme are investigated under shifts in the process mean or inflations of the process standard deviation, and under the adoption of head starts: the average run length, the run length percentage points and the probability of a misleading signal.Approximations to these three performance indicators will be obtained considering a two-dimensional Markov chain. The independence between the horizontal and vertical transitions of this approximating two-dimensional Markov chain plays an important role in providing simple expressions to those performance measures which avoid the computation of a probability transition matrix with unusual dimensions.A numerical comparison between these three performance measures and the corresponding ones of the matched combined Shewhart CShewhart ( ) scheme X, S2 ( ) will be also presented, leading to the conclusion that the substituition of this combined scheme by the CEWMA scheme can improve the joint monitoring of the process mean and standard deviation.2

No abstract

In statistical process control (SPC), it is usual to assume that counts have a Poisson distribution. The non-negative, discrete, and asymmetrical character of a control statistic with such a distribution and the value of its target mean may prevent the quality control practitioner to deal with a c-chart with a pre-specified in-control average run length (ARL) or the ability to control not only increases but also decreases in the mean of those counts in a timely fashion. Furthermore, the c-charts proposed in the SPC literature tend to be ARL-biased, in the sense that some out-of-control ARL values are larger than the in-control ARL. In this paper, we explore the notions of randomized and uniformly most powerful unbiased tests to eliminate the bias of the ARL function of the c-chart.

Manufacturing processes are usually monitored by making use of control charts for variables or attributes. Controlling both increases and decreases in a parameter, by using a control statistic with an asymmetrical distribution, frequently leads to an ARL-biased chart, in the sense that some out-of-control average run length (ARL) values are larger than the in-control ARL, i.e., it takes longer to detect some shifts in the parameter than to trigger a false alarm. In this paper, we are going to: