Tolerance Interval for the Mixture Normal Distribution

Tolerance intervals (TIs) are widely used in numerous industries, ranging from engineering to pharmaceuticals. In these applications, it is commonly assumed that data are normally distributed. However, the normality assumption may not apply in many situations, such as in the case of multiple production lines. As a result, the mixture normal distribution may be a more applicable model than the normal distribution to fit real data. Although the conventional distribution-free TI can be adopted for the mixture normal distribution, it leads to an unsatisfactory coverage probability when the sample size is not sufficiently large. In this study, we propose two Tls for the mixture normal distribution. The first is based the expectation-maximization (EM) algorithm combined with the bootstrap method and the second is based on the asymptotic property of sample quantiles. The simulation results show that the proposed TIs have coverage probability closer to the nominal level than the distribution-free interval. A real engineering data example is used to illustrate the methods.