A Bivariate Semiparametric Control Chart Based on Order Statistics

Abstract

In this article, a new bivariate semiparametric Shewhart-type control chart is presented. The proposed chart is based on the bivariate statistic (X (r) , Y (s) ), where X (r) and Y (s) are the order statistics of the respective X and Y test samples. It is created by considering a straightforward generalization of the well-known univariate median control chart and can be easily applied because it calls for the computation of two single order statistics. The false alarm rate and the in-control run length are not affected by the marginal distributions of the monitored characteristics. However, its performance is typically affected by the dependence structure of the bivariate observations under study; therefore, the suggested chart may be characterized as a semiparametric control chart.An explicit expression for the operating characteristic function of the new control chart is obtained. Moreover, exact formulae are provided for the calculation of the alarm rate given that the characteristics under study follow specific bivariate distributions. In addition, tables and graphs are given for the implementation of the chart for some typical average run length values and false alarm rates. The performance of the suggested chart is compared with that of the traditional χ 2 chart as well as to the nonparametric SN 2 and SR 2 charts that are based on the multivariate form of the sign test and the Wilcoxon signed-rank test, respectively. Finally, in order to demonstrate the applicability of our chart, a case study regarding a realworld problem related to winery production is presented.

Bivariate semiparametric control charts for simultaneous monitoring of process mean and variance

“…Recently, Koutras and Sofikitou used the framework of the univariate median chart introduced by Janacek and Meikle and subsequently extended by Chakraborti et al to establish an analogous bivariate chart. The plotting statistic of this chart is a pair of bivariate order statistics obtained from the test sample for 1 ≤ r , s ≤ n , while the CLs make use of four particular order statistics from the reference sample, namely, and , where 1 ≤ a < b ≤ m and 1≤ c < d ≤ m .…”

“…The interested reader may also refer to Koutras and Sofikitou for comparisons of Rule (which is common for both O 2 N 2 and O 4 charts) with existing parametric and nonparametric CCs capable of detecting only mean shifts. Pretty similar results may be reproduced by making use of the new CCs proposed in the present manuscript.…”

“…A detailed proof for a formula providing the joint cdf of the rv's U r:n = F X (X r:n ) and V s:n = F Y (Y s:n ) can be found in the paper of Koutras and Sofikitou, 38 who exploited a formula for the cdf of the pair of bivariate order statistics (X r:n ,Y s:n )…”

“…The purpose of this paper is to introduce new simple and effective bivariate CCs for monitoring the process mean and variance simultaneously. The starting point was a CC recently proposed by Koutras and Sofikitou, whose implementation is relatively easy since the plotting statistic is just a pair of bivariate order statistics. Bringing into play the idea featured in the nonparametric CC suggested by Balakrishnan et al and extending their methodology to the bivariate case, we added in the decision rule of the Koutras and Sofikitou CC a simple extra condition thereof obtaining a monitoring scheme capable of detecting both mean and variance shifts.…”

“…and Sofikitou38 used the framework of the univariate median chart introduced by Janacek and Meikle 24 and subsequently extended by Chakraborti et al25 to establish an analogous bivariate chart. The plotting statistic of this chart is a pair ðX Þ of bivariate order statistics obtained from the test sample for 1 ≤ r, s ≤ n, while the CLs make use of four particular order statistics from the reference sample, namely, LCL ¼ ðX The process is declared in-control if the following conditions hold true:…”