## Abstract

The very nature of process control in contract manufacturing is typically confounded by the magnitude of subcontractor's operations. As the number of the company's product groups increases, so does the number of input and quality parameters that need to be monitored and evaluated. Hence, to avoid proliferation of monitored parameters, stratification is done. Parametric monitoring is modeled using a mixture structure (Y, X) where X represents the product group and Y the monitored characteristic. The study explored to establish a single characteristic of a bivariate mixture as given by Olkin and Tate (1961). Based on the given unconditional bivariate mixture, a new Cpk , say , was formulated. It was observed, that was a biased estimator of . Similarly, was also biased estimator of . Since and are both estimators of their respective parameters, the asymptotic variance of is also a biased estimator. However, for large n the bias tends to zero. In comparing the efficiency of and based on asymptotic variances, i.e., Var and Var( ), it is noted that Var is less than Var( ). This implies that the derived variance is most appropriate for data coming from the unconditional bivariate mixture.

## References

Bissel AF. 1986. How Reliable Is Your Capability Index. John Wiley and Sons, Inc. pp. 226 -227, 1968.

Bonzo DC, De Leon, AR. Likelihood-Based Limits and Measure for Stratified Product Groups. Proceedings of MASM.

Chou Y, Owen DB. 1989. On the Distribution of the Estimated Process Capability Indices. Communications in Statistics – Theory and Methods 18. pp. 4549-4560.

Franklin LA, Wasserman GS. 1992. Bootstrap Confidence Limits for Capability Indices. Journal on Quality Technology 24. pp. 196-210.

McCormack DW. 1996. Capability Indices for Nonnormal Data. ASA 1996 Proceedings of the Section on Quality and Productivity. pp. 108-113.

Olkin I, Tate RF. 1961. Multivariate Correlation Models with Mixed Discrete and Continuous Variables. The Annals of Mathematical Statistics. 32(2): 448-465.

Zhang NF, Sternback GA, Wardrop DM. 1990. Interval Estimation of Process Capability Index. Comunications in Statistics – Theory and Methods 19, pp. 4455-4470.