線性失敗率( Linear failure rate, LFR )分配在存活分析和可靠度理論的研究中頗適合用來描述整個系統的使用情況或壽命。近年來由於記錄值( Records or record values )的應用範圍相當廣泛，已經逐漸地受到統計學界的重視。由於文獻上並未發現有任何論文曾對 LFR 分配的記錄值多所著墨，而 LFR 分配在實際應用上的重要性又與日俱增，因此我們想針對在 LFR 分配下的記錄值作詳盡的統計估計推論。本篇論文將探討 LFR 分配在記錄統計下的參數估計推論以及 LFR 分配的位置參數和尺度參數之估計推論。在估計 LFR 分配的參數方面，我們分別用傳統最大概似法和EM 演算法來計算 LFR 分配中參數的最大概似估計值；在估計 LFR 分配的位置參數和尺度參數方面，則以傳統最大概似法和最佳線性不偏估計法二種方法。在使用最佳線性不偏估計法之前，我們將先討論 LFR 分配在記錄統計的基本性質，動差及衍生的遞迴式，再利用所獲得的結果計算最佳線性不偏估計值並討論最佳線性不偏預測值( Best linear unbiased prediction )及預測區間( Prediction interval )。然後，以蒙地卡羅( Monde Carlo )模擬方法計算所估計參數之誤差( Bias )及最小平方開平方根( RMSE )分別比較傳統最大概似法和 EM 演算法及傳統最大概似法和最佳線性不偏估計法的差異，最後我們以二個例子說明整個估計推論的過程。

In life testing and reliability studies, linear failure rate (LFR) distributions are often applicable in modeling the life length of a system or component when failures occur at random and also from aging or wear-out. Most of the previous work related to the LFR distributions is devoted to developing the inference procedures. See, e.g., Bain (1974), Salvia (1980), Ashour and Youssef (1991), and Sen and Bhattacharyya (1995).
The field of record values and associated inferences has broadened its appeal in recent years. Balakrishnan and Chan (1994) and Sultan and Balakrishnan (1997) have established recurrence relations for moments of the Rayleigh distribution as well as moments of record values. The similar discussions related to the exponential distribution can be found in Ahsanullah (1995), Balakrishnan and Arnold (1995), Basak (1996), Arnold et al. (1998), and Balakrishnan and Rao (1998). However, the inference for the LFR distributions with record values has not been undertaken. In this thesis we first derive the maximum likelihood estimates (MLEs) of the LFR distributions for record values by two methods. The basic idea underlying the first method is a reduction of the two-dimensional numerical search for the zeros of the LFR log-likelihood gradient vector to a one-dimensional numerical search. The second method presents the MLEs with application of the EM algorithm.
We establish some recurrence relationships for the single and product moments in order to compute the means, variances, and covariances for the record values and then obtain the best linear unbiased estimators (BLUEs) for the location and scale parameters. The prediction of a future record value and the test for spuriosity of the current record values are also evaluated. Simulation results indicate that the MLEs derived by the EM algorithm are more efficient than the traditional maximum likelihood estimation, and the BLUEs are more reliable against those of maximum likelihood estimation. We conclude with two illustrative examples.

1.前言
2.參數λ和ν的估計推論
2.1 傳統的最大概似估計法
2.2 EM 演算法
3.位置參數μ和尺度參數σ的估計推論
3.1 最大概似估計法
3.2 最佳線性不偏估計法
4.數值分析
參考資料
附錄. LFR分配資料的模擬方法

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