單調無母數迴歸在廣義線性模型上之研究

Abstract

本篇文章裡，為解決WAT(Wafer Acceptance Test)-EC(Engineering Control)的問題，我們發展了單調無母數迴歸。藉由Gu(2002)，Zhang(2004)所提出的方法加以結合及修正，將反應變數拓展至整個指數族上，與此相關的演算法也會在本文中提出。我們利用Natural Cubic Splines的性質發展出有效率的計算法，並用模擬資料來探討其效率。當反應變數為Bernoulli或Poisson分部時，其模擬的結果都有不錯的表現。此外，在“真實函數”具有單調性的情形下，有單調限制估計量之ASE(Averages Square Error)與無單調限制並沒有明顯差異。然而，當無單調限制之估計量呈現出非單調時，則單調限制估計量在ASE上之表現會明顯優於前者。最後，我們將說明如何利用此方法來篩選EC中的WAT測試項目並且建立適當的管制上下限。

In this study, motivated by the WAT-EC problem, we develop a nonparametric monotone smoothing spline smoother for analyzing responses from exponential families by combining the methodologies provided in Gu (2002) and Zhang (2004) along with our modification. An algorithm with implementation details is provided. Computation is efficient because we utilize the characteristics of the natural cubic splines. The effectiveness of the proposed method is studied by simulation. The simulation results demonstrate that the proposed method performs well in the regression models with both the Bernoulli and Poisson responses. When the “true” function is monotonic,the proposed monotone estimator performs about the same as the unconstrained smoother in terms of the averaged squared error for the cases when the latter performs well. On the other hand, constrained smoother outperforms the unconstrained smoother when the unconstrained smoother produces non-monotone estimates. As an illustrative example, we demonstrate the proposed method can be used in screening WAT test items for more stringent engineering control and in setting appropriate control limits.