鑑於傳統管制圖高度依賴嚴格分布假設與大量Phase I 初期樣本，導致在實務應用上因樣本不足常面臨缺乏可靠性的問題。為了解決這個問題本論文以超越機率準則（Exceedance Probability Criterion, EPC）為核心，建立一套完整且具理論基礎的非參數統計製程監控的方法，該方法除了具有高度穩健性且無須分布假設優點外，且其適用於小至中等樣本情境下之製程監控。

為此，本研究提出一類新型EPC 相容之管制圖設計，利用分數順序統計量（Fractional Order Statistics, FOS）進行外推分位數估計，並具備理論基礎。所開發之三種方法分別為：經典FOS、具自適性激活函數的ad-FOS，以及三項式FOS（3-term FOS）方法，均可透過解析式推導出滿足指定涵蓋機率之管制界限，且能廣泛適用於多種分布特性下，大幅提升相較於傳統非參數與參數方法之優勢。

本論文進一步將該方法論擴展至多變量製程監控領域，透過創新性地將空間深度（Spatial Depth）轉換為連續離群程度（Outlyingness Measure），克服傳統深度排序值方法中離散性問題。並提出一套二階段分位數估計流程，以有效建構高維空間下之EPC 型上界管制界限。

透過大量模擬實驗及實際半導體製程案例驗證，本研究所提出方法於涵蓋準確度、模型誤設適應性及複雜數據結構的彈性表現上，均展現顯著優勢。整體而言，本論文推進了非參數製程監控領域之發展，建立了一套兼具統一性、彈性與理論嚴謹性的完整方法論，特別適用於樣本數有限、分布形式未知之工業應用場域。

This dissertation establishes a comprehensive and analytically grounded methodology for nonparametric statistical process monitoring under the Exceedance Probability Criterion (EPC). Motivated by the practical limitations of traditional control charts—particularly their heavy reliance on strong distributional assumptions and large Phase I sample sizes—this research addresses the pressing need for robust, distribution-free approaches capable of delivering reliable control limits even under small to moderate sample conditions.

To achieve this, a new class of EPC-compliant control charts is introduced, leveraging fractional order statistics (FOS) to develop extrapolated quantile estimators with theoretical properties. Three variants are systematically developed: the classic FOS, the adaptive FOS (ad-FOS), and the 3-term FOS methods. These estimators provide analytic solutions for constructing control limits that satisfy pre-specified coverage guarantees across a wide range of distributional characteristics, offering significant advantages over conventional nonparametric and parametric techniques.

The methodology is further extended to multivariate process monitoring through a novel transformation of spatial depth into continuous outlyingness measures, overcoming the discreteness inherent in traditional depth-based approaches. A two-stage quantile estimation framework is proposed to enable the construction of multivariate EPC-based control limits, ensuring both scalability and accuracy in high-dimensional settings.

Extensive simulation studies and real-world applications, particularly involving semiconductor manufacturing processes, validate the superior performance of the proposed methods in terms of coverage accuracy, robustness against model misspecification, and adaptability to complex data structures. Collectively, this research advances the state of nonparametric process monitoring by providing a unified, flexible, and theoretically rigorous framework that is especially well-suited for industrial environments characterized by limited sample sizes and unknown distributional forms.

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