利用多重型II設限樣本對Burr Type XII 及Lognormal 分配的形狀參數做統計推論

Statistical inference about the shape parameter of the Burr type XII and Lognormal distributions based on multiply type II censored sample

林雅莉(Ya-Li Lin)

指導教授 ： 吳忠武

淡江大學/管理學院/統計學系碩士班/碩士(2005年)

https://doi.org/10.6846/TKU.2005.00417

摘要

在現實生活中，我們常會遇到無法取得完整樣本的情況，例如因為時間、成本的限制或人為疏忽而不能獲得所有的觀察值，而在實際應用上，可靠度分析或存活分析的壽命資料就常有這種問題產生，面對此種不完整資料，我們無法利用一般應用於完整樣本的統計方法來做推論分析。因此，我們所要討論的即為多重型II設限樣本的統計分析方法。

許多產品的故障率函數會呈現浴缸型（bathtub shape）的曲線模式，例如電子、電鍍及機械等的產品。因此，我們在描述這種情況時，如果利用具有浴缸型故障率函數之雙參數分配會比Weibull分配，Extreme value分配及Normal分配等等來得適合。然而，Jiang, Ji和Xiao (2003)曾經提到並非所有產品的故障率函數都呈現浴缸型的曲線模式，例如當產品因疲勞或劣化而故障時，其壽命模式會呈現單峰型(或稱反向的浴缸型)(unimodal或 reversed bathtub shaped)的故障率曲線模式。因此，本文的主要研究目的為探討利用產品壽命來自Burr type XII及Lognormal分配之多重型II設限樣本對其形狀參數提出最適當的檢定及區間估計之方法。

本文主要共分為四章，第一章為緒論，第二章是討論在多重型II設限下有關Burr type XII分配的形狀參數之統計推論，第三章則是討論在多重型II設限下有關Lognormal分配的形狀參數之統計推論。最後，在第四章中我們將會針對二、三章歸納出幾個結果並提出相關建議與未來研究方向。

關鍵字

多重型II設限 ； 形狀參數 ； 單峰型 ； Burr type XII分配 ； Lognormal分配 ； 假設檢定 ； 樞紐量 ； 蒙地卡羅模擬 ； 信賴區間

並列摘要

In human life, we often encounter the situation of getting censored sample, for example, because of the restriction of time and cost or human mistake, we can’t get all observations. In practice, there are often these issues concerned in the data of the reliability or survival analysis. In face of this censored data, we can’t apply traditional statistical inference to perform our analysis on it. Therefore, what we discuss in this paper is statistical inference with multiply type II censored sample.

In this paper, we discuss the lifetime distribution with the unimodal shape or reversed bathtub shape failure rate function under the multiply type II censored sample. First, we provide 18 pivotal quantities to test the shape parameter of the two lifetime distributions and establish confidence interval of the shape parameter under the multiply type II censored sample. Secondly, we also find the best test statistic based on their most power of test among all test statistics. In addition, we obtain the best pivotal quantities with the shortest tolerance length. Finally, we give two examples and the Monte Carlo simulation to assess the behavior (including higher power and more shorter length of confidence interval) of these pivotal quantities for testing null hypotheses under given significance level and establishing confidence interval of the shape parameter under the given confidence coefficient.

並列關鍵字

Multiply Type II Censored Samples ； Shape parameter ； Unimodal shaped ； Burr type XII Distribution ； Lognormal Distribution ； Testing hypotheses ； Pivotal Quantity ； Monte Carlo Simulation ； Confidence interval.

參考文獻

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