On the asymptotic correlationship for some process capability indices (C)over-cap(p), (C)over-cap(pk) and (C)over-cap(pm) under bivariate normal distribution

The process capability index is used to determine whether a production process is capable of producing items within a specified tolerance. Some process capability indices C-p , C-pk and C-pm have been of particular interest as useful management tools for tracking process performance. Most evaluations on process capability indices focus on statistical estimation and test of hypothesis. It is necessary to investigate their asymptotic correlationship among basic estimators (C) over cap (p) , (C) over cap (pk) and (C) over cap (pm) of process capability indices C-p , C-pk and C-pm. , In this paper, we study their asymptotic correlationship for three process capability indices (C) over cap (p) , (C) over cap (pk) and (C) over cap (pm) under bivariate normal distribution BN(mu(x), mu(y), sigma(2)(x), sigma(2)(y), rho). With some nonnormal processes, the asymptotic correlation coefficient of any two respective process capability index estimators could be established.