指數分散加速衰變模型最適樣本數配置之研究

Optimal Sample Size Allocation for Accelerated Degradation Test (Based on Exponential Dispersion Model)

李宜真(I-Chen Lee)

指導教授 ： 曾勝滄

國立清華大學/理學院/統計學研究所/碩士(2011年)

https://doi.org/10.6843/NTHU.2011.00442

摘要

加速衰變試驗 (accelerated degradation test, ADT) 是工業界最常被用來推估高可靠度產品的可靠度資訊 (如產品壽命第 p 百分位數) 之重要分析工具，因此如何規劃有效率的加速衰變試驗，尤其是決定加速應力所需之最適樣本數配置，是可靠度工程師經常面臨之重要決策問題。目前文獻上有關此問題的研究方法，大多僅針對特定衰變模型 (如 Wiener, Gamma, Inverse Gaussian 衰變模型等) 進行分析。然而，對於一般化的衰變模型，缺乏有系統的探討其樣本數配置問題。為克服前述困難，本論文首先建構指數分散 (exponential dispersion, ED) 衰變模型，其優點是前述之衰變模型皆為其特例。其次，針對兩個應力水準的 ADT 問題，本論文分別在 V-optimality, D-optimality, 及 A-optimality 準則下，推導出最適 (樣本數) 配置比例之解析解。具體而言，在 V-optimality 及 A-optimality 準則下，最適配置比例皆為 ED 模型中未知參數针 及應力水準之函數；然而，在D-optimality 準則下，最適配置為高低應力都分配相同樣本數，亦即它與未知參數及應力水準皆無關。此外，本文亦深入探討此三種準則下最適樣本配置方法之相對效率 (relative efficiency)，以碳薄膜電阻器的第50 百分位數之估計值為例，採用 D-optimal 及Ａ-optimal 配置比例相對於Ｖ-optimal 配置比例之相對效率約
分別為85%及83%。

關鍵字

加速衰變試驗 ； 指數分散模型

並列摘要

Accelerated Degradation tests (ADTs) are widely used to assess the lifetime information of highly reliable products possessing quality characteristics that both degrade over time and can be related to reliability. Hence, how to design an efficient ADT plan for assessing product’s lifetime information at normal-use stress (especially for the optimal sample-size allocation to higher test-stress levels) turns out to be a challenging issue for reliability analysts. In the literature, several papers had addressed this decision problem. However, the results are only based on a specific degradation model (such as Wiener, Gamma, inverse Gaussian models, etc.) and it lacks of a uniform approach for a general degradation model. To overcome this difficulty, we first propose an exponential dispersion (ED) degradation model which covers all mentioned-above degradation models. Next, by using V-optimality, D-optimality, and A-optimality criterion, we derive the analytical expression of the optimal sample-size allocation for a 2-stress ADT when the underlying degradation model follows an ED degradation model. The results demonstrate that the V-optimal and A-optimal allocations are the functions of unknown parameters and life-stress function, while D-optimal allocation turns out to be an equal sample-size allocation. Furthermore, we also discuss the relative efficiency of the D-optimal and A-optimal allocations with respect to V-optimal allocation and it demonstrates that the relative efficiency of D-optimal and A-optimal allocations with respect to V-optimal allocation are around 85% and 83%, respectively. Key words and phrases: Accelerated Degradation Tests; Exponential Dispersion Model; V-optimality; D-optimality; A-optimality.

並列關鍵字

Accelerated Degradation Tests ； Exponential Dispersion Model

參考文獻

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