適應估計誤差之 Hotelling T2 管制圖

Abstract

管制圖的平均連串長度（ARL）績效表現會隨著不同應用中第一階段分析所估計的製程參數不同而有所變異，稱作“從業人員之間”的變異。因此，可將 ARL 視為隨機變數。近來，除了 ARL 的平均數之外，其變異數（SDARL）也已經被許多研究用於探討不同的單變量製程。本論文中，我們研究多變量製程 Hotelling T2 管制圖其 ARL 的性質。研究結果發現，在使用常用的第一階段樣本數時， ARL 的分配是偏斜的，且其 SDARL 相當大。我們需要很多的第一階段數據才能將 SDARL 減少至可接受的數值。此外，當製程在管制狀態下時，存在很高的機率其 ARL 將低於期望的數值。為了解決此問題，我們發展了一個演算法運用拔靴法來建構一個適應估計誤差的管制界限，使從業人員有很高的“信心”實際的 ARL 會高於期望的數值。我們透過模擬研究來評估及比較使用傳統的管制界限與本文所提的調整管制界限其 ARL 績效表現。最後，我們給一個例子來說明的此調整管制界限的應用。

The average run length (ARL) performance of a control chart varies as the process parameter estimates from the Phase I analysis vary in each application. Consequently, the ARL becomes a random variable due to the so-called “practitioner-to-practitioner” variation. In addition to the mean, the standard deviation of the ARL (SDARL) has been considered in several recent studies for univariate processes. In this paper, we study the ARL behavior of the Hotelling T2 control chart for multivariate processes. The results show that the distribution is skewed and SDARL is surprisingly large in common practice and it would require a large amount of Phase I data to reduce SDARL to an acceptable level. Moreover, there is a high probability that the actual in-control ARL would be less than the nominal level. To overcome this problem, we develop an algorithm to construct an adaptive control limit adaptive via bootstrapping with which practitioners would have a high “confidence” that the actual in-control ARL will exceed the nominal in-control ARL level. A simulation study was conducted to evaluate and compare the ARL performance of the conventional and the proposed bootstrap-based adjusted control limits. An example is given to illustrate the application of our adjusted control limit.