| 摘要: | 推論高可靠度產品壽命資訊是衰變分析的主要目的，其中定義產品的壽命為衰變路徑首次超過特定門檻值的時間 (first-passage-time, FPT)。伽瑪 (gamma) 以及逆高斯 (inverse Gaussian) 過程是衰變分析中經常使用來分析單調的衰變資料，因此本論文考慮更廣義的 Hougaard 過程，涵蓋伽瑪以及逆高斯過程為特例。由於 Hougaard 過程的機率密度函數涉及難以處理的無窮級數，以致於 Hougaard 過程的 FPT 分布常藉由 Birnbaum-Saunders （BS）分布來近似，但缺乏理論基礎來衡量 BS 近似的各階動差。本論文推導出 Hougaard 過程之 FPT 分布的各階動差，藉此計算出 BS 近似的差異與收斂速度。這些收斂條件恰好與 Hougaard 分布的古典收斂結果相吻合。此外亦證明 Hougaard 過程之 FPT 一階動差的 BS 近似恆大於其確切的動差，且為 FPT 分布的各階動差與其 BS 近似之間的差異，提供了以指數積分 (exponential integral) 來表示的上界。
由於 BS近似的 FPT 分布，其動差具有簡潔的表達式，因此 BS 近似廣泛應用於 (加速) 衰變試驗之最佳實驗計畫。在大樣本情況下，一階近似 (亦即 delta) 方法常用來近似產品壽命估計量的精確度 (即目標函數)。然而當衰變試驗的樣本數很少時，採用二階的近似方法可以大幅度地改進其一階近似。根據泰勒展開式及最大概似估計量的漸近常態性，在衰變分析中常用之隨機過程衍生出的各階目標函數，皆可歸納出特定的結構。針對這樣的目標函數，可以理論導出在實驗成本限制下，最佳實驗計畫之測試樣本數、量測次數以及量測頻率。同時以數值結果呈現在實驗成本限制下，以一階及二階近似為目標函數，其最佳實驗計畫存在著顯而易見的差異。
加速衰變試驗常用於評估加速變數對高可靠度產品壽命的影響程度，而直覺上的兩基本性質：應力的變化不改變壽命分布以及壽命分布的隨機次序 (stochastic ordering)，並非文獻上常用的加速衰變模型都能滿足。因此本論文提出加速不變 (accelerated invariance) 原則，以確保加速衰變模型具有此兩基本性質，並應用至 Houggard 過程。除此之外， Houggard 過程之 (加速) 衰變模型的最佳試驗計畫，其需求性在實際應用上日趨重要。本論文通過有理化參數，推導出其費雪訊息矩陣 (Fisher information matrix) 以利未來研究。;Hougaard processes, which include gamma and inverse Gaussian processes as special cases, as well as the moments of the corresponding first-passage-time (FPT) distributions, are commonly used in degradation analysis. Because the probability density function of the Hougaard process involves an intractable infinite series, the Birnbaum-Saunders (BS) distribution is often used to approximate its FPT distribution. This thesis derives the finite moments of FPT distributions based on Hougaard processes, providing a theoretical justification for BS approximation in terms of convergence rates. Further, it can be shown that the first moment of the FPT distribution for the Hougaard process, when approximated by the BS distribution, is always larger. It also provides a sharp upper bound for the difference using an exponential integral. The conditions for convergence coincidentally elucidate the classical convergence results of Hougaard distributions.
Optimal cost-constrained test plan is of practical concern in degradation test for highly reliable products. The first-order (i.e., delta) method is often used to approximate the precision of the estimated quantity of interest for the product′s lifetime (i.e., objective function). When the degradation data are of small size, the approximation can be essentially improved by the second-order approximation. According to Taylor expansion and asymptotic normality of maximum likelihood estimators, the objective function of each order approximation derived from many stochastic processes shares the specific structure in degradation analysis. For such objective functions, the optimal cost-constrained test plan (to determine the measurement frequency, the number of measurements and sample size) can be theoretically obtained under the explicit relationships between experimental costs and model parameters. The numerical results illustrate the dramatic differences of the optimal cost-constrained test plans using the first- and second-order approximations.
Accelerated degradation tests (ADTs) are widely used to assess lifetime information under normal use conditions for highly reliable products. The two fundamental assumptions in ADTs that changing stress levels does not affect the underlying distribution family and that there is stochastic ordering for the life distributions at different stress levels, may not hold for all accelerated degradation models. Therefore, the acceleration invariance principle for ADTs is proposed to ensure the validity of these two assumptions. Furthermore, to find the optimal cost-constrained test plan for the (accelerated) degradation model using the Houggard process, this thesis applies the technique of rationalizing parameters to derive its Fisher information matrix. |
