Applications

Construction of simultaneous confidence intervals for multinomial proportions is one of the most important statistical procedures in many real-life applications, such as statistical quality control, survey sampling, opinion polls, etc. Basically, there are two popular forms of simultaneous confidence intervals for m-dimension multinomial proportions. The first one is the conventional confidence intervals (CCI) which have lengths proportionate to the estimated standard error of sample proportions. Therefore, the length of each interval depends on how much the corresponding sample proportion deviates from 0.5, longer interval for less deviation and shorter interval for more deviation. Gold (1963), Cochran (1963), Quesenberry and Hurst (1964), Goodman (1965), Tortora (1978), Bailey (1980), Kwong and Iglewicz (1996), focused on the construction of CCI. The second one is called quick simultaneous confidence intervals (QSCI) which have same length for all the intervals. Angers (1974), Thompson(1987), Fitzpatrick and Scott (1987), Sison and Glaz (1995), Kwong (1998) discussed the procedures for the construction of QSCI. A brief comparison between the two forms of confidence intervals can be found in Fitzpatrick and Scott (1987).

After incorporating the singular correlation structure based on Kwong's Inequalities into the evaluation of the critical values for the construction of QSCI, Kwong (1998) improved the performance of QSCI. However, the approach works only for $m \leq 12$ as the computational time increases rapidly for large m. With the application of the new approach of evaluating singular multivariate normal distribution described in Section 2.2, we can construct QSCI (see Kwong (1998) for details) for any m. Analogous to the case of construction of QSCI, we modify the existing approaches for constructing CCI after taking the singular correlation structure of sample proportions into consideration. Besides, we extend the results to the construction of all pairwise simultaneous confidence intervals for multinomial proportions in the forms of CCI and QSCI.