Name: Isabel Cristina Ramírez
Institution: Escuela de Estadística, Universidad Nacional de Colombia, Medellín, Colombia
E-mail: iscramirezgu@unal.edu.co
Co-authors: Nelfi González
Abstract:
In this work, control charts for the mean are constructed using a Bayesian approach. It is as- sumed that the quality characteristic to be controlled can be modeled by a normal distribution. Through simulation studies, the performance of the control chart is analyzed assuming two cases: known and unknown variance. A Bayesian conjugate model is established, therefore the prior distri- bution for the mean is normal and in the case where the variance is unknown, the prior distribution for the variance is defined as the Inverse- Gamma (ν, ν) . The posterior predictive distribution, which is also normal, is used to establish the control limits of the chart. For both cases, the effect of the prior distributions for the unknown parameters and the performance of the control charts in phase II is evaluated by the signal probability. In the simulation study we evaluate the effects of sample sizes, distance of the prior mean in relation to the mean of the calibration sample, and an indicator of how informative is the prior distribution of the population mean. Additionally, the effect of the ν parameter was studied in the case of unknown variance. We found that the false alarm rate can be exaggeratedly large if a prior distribution is very informative, which in turn leads to a ARL-biased chart, that is, the maximum of the ARL it is not given when the process is in control. In addition, when the size of the calibration samples and the future sample are small, there is great influence of the specification of the prior distribution on the power of the control chart, especially when the prior is very informative. Regarding the effect of the definition of the parameter ν, it is found that the smaller the value, which means having a less informative prior distribution, the lower the power of the control chart.