In an attempt to simultaneously monitor more than one property of an absolutely continuous random variable X, Grimshaw (1991) introduced a control chart which monitors characteristics defined in terms of the quantile function. The test statistic monitors conformance to the target by using a (kx1) vector of the target quantile function Q{dollar}sb{lcub}0{rcub}{dollar} = (Q{dollar}sb0{dollar}(u{dollar}sb{lcub}rm i{rcub}{dollar})), 0 {dollar}leq{dollar} u{dollar}sb{lcub}rm i{rcub}{dollar} {dollar}leq{dollar} 1. Deviation from the target is determined by comparing a vector of sample quantiles Q{dollar}spsim{dollar} with Q{dollar}sb0{dollar}. Q{dollar}spsim{dollar} is asymptotically normal with mean vector Q{dollar}sb0{dollar} and covariance matrix {dollar}Sigmasb0{dollar} which depends on the u{dollar}sb{lcub}rm i{rcub}{dollar}'s and fQ{dollar}sb0rm (usb{lcub}rm i{rcub}{dollar})'s. The performance of the chart is measured by the ARL.; This dissertation addresses the question of whether the ARL can be improved by choosing optimal u{dollar}sb{lcub}rm i{rcub}{dollar}'s. The problem is formulated and the cases of location, scale and simultaneous location-scale shifts in the process distribution are investigated. When a shift occurs, the test statistic of interest has a non-central {dollar}chisp2{dollar} distribution with k d.f and non-centrality parameter {dollar}lambda{dollar}, which we seek to maximize since the power function is an increasing function of {dollar}lambda{dollar}.; Computational efficiency in maximizing {dollar}lambda{dollar} has been obtained by proving that {dollar}Sigmasbsp{lcub}0{rcub}{lcub}-1{rcub}{dollar} is tridiagonal and cross-symmetric. Two algorithms based on known search procedures were used to determine the optimal u{dollar}sb{lcub}rm i{rcub}{dollar}'s that minimize the ARL for different values of k and known shifts. Moreover, we have shown that for location-only and scale-only shifts, the optimal u{dollar}sb{lcub}rm i{rcub}{dollar}'s for minimum ARL's are independent of the size of the shift. This is not true for simultaneous location-scale shifts.; For the optimal u{dollar}sb{lcub}rm i{rcub}{dollar}'s, the ARLs are compared with the ad hoc selection of the u{dollar}sb{lcub}rm i{rcub}{dollar}'s given in Grimshaw's proposed quantile based control chart (QCC). The optimal QCC proves superior in the sense that it provides lower ARLs for detecting all shifts in the process distribution.; Furthermore, comparison of the ARLs of the optimal QCC and the traditional X -and R charts under normality reveals that the optimal QCC is comparable to the X -and R charts and even proves superior for detecting certain simultaneous shifts in the process location and scale parameters.
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