Abstract

In case of an ordinal process ($p_1,$ $p_2,$ $p_3$) involving three ordered categories, always the following condition holds: $p_3\le (1-p_1),$ where $p_1$ and $p_3$ are the proportion of items in the first and third categories respectively. Using this fact, Pal and Gauri (2019 Pal, S., and S. K. Gauri. 2019. Evaluating capability of a process with ordinal responses. Communication in Statistics – Simulation and Computation. Advance online publication. doi:10.1080/03610918.2019.1704417.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) defined a measure called process area of proportions ($A_p$) aiming to evaluate capability of a process with ordinal responses. In this article, the statistical distribution of the sample estimate of process area of proportion (${\widehat{A}}_p$) is studied in details using simulations. It is found that ${\widehat{A}}_p$ approximately follows Gamma ($k,$ $\theta$) distribution, where $k$ and $\theta$ are functions of $p_1,$ $p_3$ and $n$ (sample size). Control charts are designed based on the Gamma approximation (GA) as well as empirical distribution (ED) of ${\widehat{A}}_p,$ and performances of these charts are evaluated based on simulation studies. The monitoring statistic (${\widehat{A}}_p$) takes care of the fact that an ordinal process cannot be considered out-of-control if $p_1$ and/or $p_3$ shifts in favorable directions, and therefore, the proposed GA-based chart does not give false out–out-control signal when $p_1$ and/or $p_3$ shifts in favorable directions.

摘要

如果是有序过程 ($p_{\mathrm{1个}},$ $p_{\mathrm{2个}},$ $p_{\mathrm{3个}}$) 涉及三个有序类别，始终满足以下条件：$p_{\mathrm{3个}}\le (\mathrm{1个}-p_{\mathrm{1个}}),$在哪里$p_{\mathrm{1个}}$和$p_{\mathrm{3个}}$分别是第一类和第三类项目的比例。利用这一事实，Pal 和 Gauri（2019 Pal, S.和SK Gauri。2019 年。评估具有有序响应的过程的能力。统计交流——模拟与计算。提前在线发布。内政部：10.1080/03610918.2019.1704417。[Taylor & Francis Online]、[Web of Science®]  、[Google Scholar]）定义了一个称为过程区域比例的度量（$A_p$) 旨在评估具有有序响应的过程的能力。在本文中，样本估计的统计分布过程面积的比例（${\widehat{A}}_p$) 使用模拟进行了详细研究。结果发现${\widehat{A}}_p$大约遵循 Gamma ($k,$ $\theta$) 分布，其中$k$和$\theta$是函数$p_{\mathrm{1个}},$ $p_{\mathrm{3个}}$和$n$（样本量）。控制图的设计基于 Gamma 近似 (GA) 以及经验分布 (ED)${\widehat{A}}_p,$这些图表的性能是根据模拟研究进行评估的。监测统计量（${\widehat{A}}_p$) 考虑到一个有序过程不能被认为是失控的事实，如果$p_{\mathrm{1个}}$和/或$p_{\mathrm{3个}}$向有利的方向移动，因此，所提出的基于 GA 的图表在以下情况下不会给出错误的失控信号$p_{\mathrm{1个}}$和/或$p_{\mathrm{3个}}$向有利的方向转变。