Economic statistical design of X bar control chart for non-normal symmetric distribution of quality characteristic

Abstract

In economic statistical design of a control chart, the economic-loss function is minimized subject to a constrained minimum value of power, maximum value of probability of false alarms and average time to signal an expected shift. This paper is concerned with the optimum economic statistical design of the X bar chart when quality characteristic has non-normal symmetric distribution. We considered three types of distributions: Student distribution, standard Laplace distribution and logistic distribution. For each of these distributions, we calculated theoretical distribution of standardized sample mean (or its best approximation) and approximated it with normal, Pearson VII and Johnson SU distributions. For considered example, constrained minimization of expected loss function was done using genetic algorithm in statistical software R. We compared results of economic statistical design of X-bar chart for theoretical distribution of standardized sample mean with the results for normal, Pearson and Johnson distributions. We found that, for all chosen distributions of quality characteristic, Pearson VII distribution and Johnson SU distribution give
results very close to results based on theoretical distribution of standardized sample mean, while normal distribution gives much worse fit.