Gompertz分配產品之壽命績效的評估

Assessing the lifetime performance for the products with Gompertz distribution

吳亭潔(Ting-Chieh Wu)

指導教授 ： 吳錦全

淡江大學/管理學院/統計學系碩士班/碩士(2009年)

https://doi.org/10.6846/TKU.2009.00702

摘要

近年來，製程能力指標被製造業廣泛的應用在品質監控方面，以評估製程
能力是否合乎水準。大多數的製程能力指標都是假設產品的品質特性在常態分
配下；然而，產品壽命往往是服從非常態分配，例如：指數分配、柏拉圖分配、
韋伯分配等等。對於產品壽命之相關分配，在實務上會利用壽命績效指標L C 來
衡量產品的壽命績效，其中L 是規格下界。在壽命試驗中常因時間限制以及人
力和成本的考量而無法取得完整的樣本資料，使得必須使用設限樣本資料。
本文主要目的是利用產品壽命具有Gompertz 分配之右型Ⅱ設限樣本，
(1) (2) ( ) , , , k X X " X ，來評估壽命績效指標L C ，並且利用L C 的最大概似估計量發展
一個新的檢定程序與信賴區間。此外，也利用產品壽命來自一個具有雙參數指
數分配之右型Ⅱ設限樣本，建構出壽命績效指標L C 的信賴區間。利用此檢定程
序與信賴區間，可以提供廠商評估產品壽命是否達到要求水準。

關鍵字

Gompertz 分配 ； 雙參數指數分配 ； 右型Ⅱ設限樣本 ； 壽命績效指標 ； 最大概似估計量 ； 蒙地卡羅模擬

並列摘要

In recent years, many process capability indices (PCIs) have been widely
used in quality monitoring by many manufacturing industries. Most PCIs assume
that the quality characteristic has a normal distribution. However, the
lifetime of products frequently possesses an exponential distribution, a
Pareto or Weibull distribution etc.. In practice, the lifetime performance
index L C is utilized to measure lifetime performance for products with some
lifetime distributions, where L is the lower specification limit. In
lifetime testing experiments, we may not be able to obtain a complete sample
due to time limitation or other restrictions. Therefore, censored samples
arise in practice.
This research constructs a maximum likelihood estimator (MLE) of L C
based on the right type Ⅱ censored sample from the Gompertz distribution.
The MLE of L C is then utilized to develop a new hypothesis testing procedure
and the confidence interval in the condition of known L . In addition, the
confidence interval of L C is also constructed for the two-parameter
exponential distribution with the right type Ⅱ censored sample. The
purchasers can then employ the new hypothesis and the confidence interval
to determine whether the lifetime performance of products adhere to the
required level.

並列關鍵字

Gompertz distribution ； Two-parameter exponential distribution ； Type Ⅱ censored sample ； Lifetime performance index ； Maximum likelihood estimator ； Monte Carlo simulation

參考文獻

performance assessment system for Weibull lifetime product,

[1] Balakrishnan, N. and Cohen, C. A. (1991), Order Statistics and

Inference, Academic Press, Inc.

[2] Boyles, R. A. (1991), The Taguchi capability index. Journal of

Quality Technology, 23, pp. 17-26.