在存活與可靠性研究中最簡單且最重要的分配為指數分配.而且指數分配在壽命研究中所扮演的角色等同於常態分配在其他統計領域的重要性.它最著名的性質為與以往的記憶無關（lack of memory）,也就是物體或動物過去的壽命並不有影響未來存活的機率;亦即指數分配具有失效率為一常數的特徵.然而對許多存活的資料來說此特性是不合理的.由此一些指數的擴充分配如 Weibull, Extreme-value, Gompertz, Pareto, Rayleigh等分配它們的失效率是變動的(可能為時間的一個函數 )而非一定數.因此這些分配可以更寬廣的運用在壽命與可靠度的問題上,並且可以利用適當的數學轉換將這些分配轉換成指數分配的形式,此仍可以借用指數分配的性質來簡化問題以及估計參數.通常這些分配的參數估計往往採用最大概似估計量(Maximum likelihood estimator: MLE), 動差估計量(Moment estimator: ME),修正的動差估計量(Modified moment estimator: MME),貝氏估計量(Bayesian estimator),最小平方法估計量(Least squares estimator: LSE).動差估計量在計算方面是很容易的,但是在許多情況下常常是並不有效.最大概似估計量符合一致性且漸近有效,然而往往對一些設限資料在計算時較複雜且不易解.本文提供兩個簡易的方法來估計這些參數,首先借用指數的性質,提供這些分配中參數的一些恰當聯合信賴區域與信賴區間, 其次提供加權最小平方法來估計這些參數．我們採用次序統計量與穩定的變異數的轉換之觀念建立新的加權最小平方法．並在適當的評估下,新的加權最小平方法的估計量與最大概似估計量並沒有顯著的差異.但是,新的加權最小平方法提供易瞭解的觀念,在計算上直接並簡單.另外,我們提供與參數無關的統計量(pivotal quantity)在多重設限資料下,指數分配中未觀察到的資料的預測區間.這在實際上,對可靠性和壽命實驗中的整個生存期間與 n 個元件系統中 j 個元件損壞的時間,均可以用所建議的與參數無關的統計量(pivotal quantity)來預估的．

　　The simplest and most important distribution in survival studies is the exponential. The exponential distribution plays a role in lifetime studies analogous to that of the normal distribution in other areas of statistics. It is famous for its unique "lack of memory", high requires that the age of the animals or individual does not affect future survival. The exponential distribution has the character of constant hazard rate, however, that it cannot be reasonable and inadequate to describe many survival data set. Some distributions such as Weibull, Extreme-value,Weibull, Gompertz, Pareto, Rayleigh etc are extensions of the exponential distribution, their hazard rate vary broadly. Hence, these distributions can be more widely used in life-testing and reliability problem.All these distributions are extended by appropriately mathematical transformations from the exponential distribution. In the statistical analysis, the parameters of these distributions may be estimated by MLE(Maximum likelihood estimator), AMLE(asymptotic Maximum likelihood estimator),ME(Moment estimator), MME(Modified moment), Bayesian estimator, and LSE(Least squares estimator) etc. The Me are usually easy to calculate, but they are generally inefficient. The MLE are always consistent and asymptotically efficient, however,they are often intractable and complicative for some censored data or truncated data. Bayesian estimators have the property of robustness, however, they need the prior distribution and is complicated in calculation. In this thesis, we propose two simple methods that they can be easy to estimate for these parameters. At first, some exact joint confidence regions and confidence intervals for parameters are derived by using some basic properties of exponential distribution. Secondly, we provide a weighted least squares method to estimate these parameters.We adopt the notion of order statistics and variance stabilizing transforms to set up the weighted least squares method(WLS). In some simulation study, the WLS method and the maximum likelihood estimate do not differ significantly in terms of precision and bias for moderate and large sample size. Finally, we also provide a suitable pivotal quantity to present the prediction interval of the jth future ordered-observation in a sample of size n from one-parameter exponential distribution where a multiple type II censored sample. As applications, the total duration time in a life test and the failure time of a j-out-of-n system may be predicted by some pivotal quantities.

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