A New Distribution‐free Control Chart for Joint Monitoring of Unknown Location and Scale Parameters of Continuous Distributions

Abstract

While the assumption of normality is required for the validity of most of the available control charts for joint monitoring of unknown location and scale parameters, we propose and study a distribution‐free Shewhart‐type chart based on the Cucconi statistic, called the Shewhart‐Cucconi (SC) chart. We also propose a follow‐up diagnostic procedure useful to determine the type of shift the process may have undergone when the chart signals an out‐of‐control process. Control limits for the SC chart are tabulated for some typical nominal in‐control (IC) average run length (ARL) values; a large sample approximation to the control limit is provided which can be useful in practice. Performance of the SC chart is examined in a simulation study on the basis of the ARL, the standard deviation, the median and some percentiles of the run length distribution. Detailed comparisons with a competing distribution‐free chart, known as the Shewhart‐Lepage chart (see Mukherjee and Chakraborti) show that the SC chart performs just as well or better. The effect of estimation of parameters on the IC performance of the SC chart is studied by examining the influence of the size of the reference (Phase‐I) sample. A numerical example is given for illustration. Summary and conclusions are offered. Copyright © 2013 John Wiley & Sons, Ltd.

Performance evaluation of joint monitoring control charts

“…The lower control limits of both charts is zero as both statistics can never be negative (cf. [5,7] and the upper control limits of both charts, say H, are used to make decision. The values of H are provided in [5,7] for some selective values of n and m. We have covered more combinations of n and m to nd the upper control limit, say H, for both charts, using a simulation study with 100,000 replicates (in R 3.…”

“…The values of H are provided in [5,7] for some selective values of n and m. We have covered more combinations of n and m to nd the upper control limit, say H, for both charts, using a simulation study with 100,000 replicates (in R 3. [5]). The shift in location is noted when p 3 is very low except p 4 ; if p 3 is relatively high except p 4 , then there is the indication of a shift in scale.…”

“…Moreover, a traditional approach used in SPC is to monitor each parameter separately; however, simultaneous monitoring of more than one parameter is also becoming popular in industry. Chowdhury et al [5], McCracken and Chakraborti [6], Mukherjee and Chakraborti [7], and Mukherjee et al [8] and the references therein may be seen in literature on simultaneous charts.…”

“…Lepage [9]), and this chart is hereafter named as Shewhart-Lepage (SL) chart. On the same lines, Chowdhury et al [5] developed a distribution-free Shewhart chart for joint monitoring that utilizes Cucconi test proposed by Cucconi [10], hereafter referred to as Shewhart Cucconi (SC) chart. Marozzi [11] provided a comparative analysis of Cucconi test versus Lepage test under some distributional setups and favored Cucconi test over Lepage test. This study intends to investigate the impact of light and heavy-tailed distributions on the performance of SL and SC charts.…”

Multinomial interpoint distances

“…In some applications such as two-phase control charting (Chowdhury et al 2014) the sample sizes are very unbalanced. Hence, we also study cases where N x = N y .…”