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中文論文名稱 利用排序集合樣本對柏拉圖分配作貝氏預測區間
英文論文名稱 Bayesian predictive interval for the Pareto distribution based on ranked set sample
校院名稱 淡江大學
系所名稱(中) 統計學系碩士班
系所名稱(英) Department of Statistics
學年度 93
學期 2
出版年 94
研究生中文姓名 王顗熒
研究生英文姓名 Yi-Ying Wang
學號 692460420
學位類別 碩士
語文別 中文
口試日期 2005-05-27
論文頁數 47頁
口試委員 指導教授-吳忠武
委員-吳錦松
委員-李汶娟
委員-李秀美
中文關鍵字 排序集合樣本  柏拉圖分配  貝氏預測  蒙地卡羅模擬  預測區間 
英文關鍵字 Ranked Set Sample  Pareto Distribution  Bayesian prediction  Monte Carlo Simulation  Prediction Interval 
學科別分類 學科別自然科學統計
中文摘要 在研究有關產品可靠度方面的問題時,通常需要進行壽命試驗,而在試驗進行當中,常常希望能預測部份尚未發生故障的樣本壽命,以決定是否變更生產計畫或採行其他決策的參考。本文即希望能利用所取得之產品發生故障之排序集合樣本壽命觀測值來預測未來型II 設限樣本產品發生故障之壽命觀測值的貝氏預測區間和平均覆蓋機率,做為評估及改善產品可靠度的依據。
研究有關產品可靠度方面的文獻,大部分選擇產品壽命分配為指數分配。指數壽命分配適用於失敗率穩定的產品;但是,如果假設每條生產線之產品壽命皆為指數分配具有失敗率 ,而經其生產線產出的產品之失敗率 為隨機變數服從Gamma分配,那麼由此混合母體隨機抽取的生產線之產品,其產品壽命便服從第二類型的柏拉圖分配。
本文主要探討的分配為柏拉圖分配,以求取未來型II 設限樣本之貝氏預測區間和平均覆蓋機率。其中,第二章是討論其樣本壽命觀測值服從指數分配的排序集合樣本,則樣本壽命觀測值為 的情況之下,分別針對尺度參數已知、尺度和形狀參數皆未知及一般化無資訊的事前分配,求取未來型 設限樣本壽命觀測值的貝氏預測區間和平均覆蓋機率。第三章舉出數值範例,以及利用蒙地卡羅(Monte Carlo)模擬方法以建立給定在 下,求取未來型 設限樣本壽命觀測值的貝氏預測區間和平均覆蓋機率。最後第四章則是結論。
英文摘要 In the researching of Products’ reliability, the result of life testing is used as the basis for the evaluation and improvement of reliability. During life testing, however, the future observation in an ordered sample is often expected to be predicted so as to determine whether the life testing experiment be redesigned or used as the reference for other decisions.
In most literatures, the exponential distribution is widely used as a model of lifetime data. The distribution is characterized by a constant failure rate. But in a population of component there could be a ubiquitous variation in failure rate because of small fluctuations in manufacturing tolerances so that a component selected at random can be regarded as belonging to a random subpopulation. Let the lifetime of a particular component have an exponential distribution with failure rate and let the failure rate follow a Gamma distribution, then the failure time of a component selected at random from such a mixed population has a Pareto distribution of the second kind.
This paper presents that under a ranked set sample from a Pareto distribution, we adopted Bayesian method only based on the only to obtain the prediction intervals of the future Type II censored lifetime observations.
論文目次 目錄
表目錄 …………..………………………….……………………… III
第一章 緒論 …..………………….………………………………….. 1
1.1 研究動機與目的 ……………………………………………. 1
1.2 文獻探討 ……………………………………………………. 3
1.3 本文架構 ……………………………………………………. 5
第二章 未來樣本壽命觀測值之貝氏預測區間 ……..………………7
2.1 排序集合樣本 ………………………………………………. 9
2.2 未來型 設限樣本壽命觀測值之貝氏預測區間 ………… 10
2.2.1 當分配的尺度參數(Scale Parameter)已知時之貝氏預測區間 …...……………………………………………... 11
2.2.2 當分配的尺度參數(Scale Parameter)和形狀參數(Shape Parameter)皆未知時之貝氏預測區間 ….………….. 19
2.2.3一般化無資訊事前分配之貝氏預測區間 …………. 25
第三章 數值模擬 ……………………………………………..……. 32
3.1 數值範例 ………………………..……………..………..… 32
3.1.1 尺度參數已知的未來型II 設限樣本壽命觀測值之貝氏預測區間 …………………………………………… 32


3.1.2 尺度和形狀參數皆未知的未來型II 設限樣本壽命觀測值之貝氏預測區間 ………………………………… 34
3.2統計模擬 ………………..………………………………..…. 36
第四章 結論 ……………..…………………………………….….... 42
參考文獻 …………………………...……………………………….. 45

表目錄

表 2.1.1 n組樣本大小為n之集合 ….……………………………… 9
表 2.1.2 n組樣本大小為n之排序集合 ………………..…………. 10
表3.1-1 尺度參數 已知下的n組樣本大小為n之樣本集合 ……………………………………………………….… 33
表 3.1-2對於 , 和 的未來型II 設限樣本壽命觀測值 之95%貝氏預測區間 …..……………… 33
表 3.1-3 尺度和形狀參數皆未知下的n組樣本大小為n之樣本集合 ………….………………………………………… 34
表 3.1-4 對於 、 和 的未來型II 設限樣本壽命觀測值 之95%貝氏預測區間 …..…………… 35
表3.2.1 對於 , , 和 之
和 的95%貝氏預測水準下之平均覆蓋機率及其均方
誤 …….…………………………………………………… 39
表3.2.2 對於 , , 和 之
和 的95%貝氏預測水準下之平均區間長度 …..… 40
表3.2.3 對於 , , 和 之
和 的99%貝氏預測水準下之平均覆蓋機率及其均方
誤 ………………………………………………………… 40

表3.2.4 對於 , , 和 之
和 的99%貝氏預測水準下之平均區間長度 ……... 41
參考文獻 [中文部份]
王盟發,曾玉玲,採集合排序樣本時常態平均值之較佳檢定,中國統計學報,九十一年,391-418頁

[英文部份]
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[2] Ali Mousa, M. A. M. (2001), Inference and Prediction for Pareto
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[3] Ali Mousa, M. A. M. (2003), Bayesian Predcition based on Pareto Dounly Censored data. Statistics., 37(1), 65-72.

[4] Al-Hussaini, E. K., Nigm, A. M. and Jaheen, Z. F. (2001), Bayesian
prediction based on finite mixtures of Lomax components model and type I censoring, Statistics, 35(3), 259-268.

[5] Arnold, B. C. and Press, S. J. (1989), Bayesian estimation and prediction for Pareto data, Journal of the American Statistical
Association., 84, 1079-1084.

[6] Compaq Visual Fortran, Professional Edition V6.6 Intel Version and IMSL (2000), Compaq Computer Corporation.

[7] David, H. A. (1981), Order Statistics, 2nd ed., John Wiley and Sons, Inc., New York.

[8] Dell, T. R. (1969), The theory and some applications of ranked set sampling. Ph.D. Dissertation, University of Georgia.

[9] Dell, T. R., and Clutter, J. L. (1972), Ranked set sampling theory with order statistics background. Biometrics, 28, 545-555.

[10] Engelhardt, M., Bain, L. J. and Shiue, W. K. (1986), Statistical analysis of a compound exponential failure model, Journal of Statistical Computation and Simulation, 23, 229-315.
[11] Fei. H., Sinha, B. K., and Wu, Z. (1994), Estimation of parameter Weibull and extreme-value distribution using ranked set sampling. Journal of Statistical Research, 28, 149-161.

[12] Johnson, N. L., Kotz, S., and Balakrishnan, N. (1994), Continous
Univariate Distribution, (2nd ed.), Vol. 1., Wiley, New York.

[13] Lawless, J. F. (1971), A prediction problem concerning samples form the exponential distribution with application in life testing, Technometrics, 13, 725-730.

[14] Likeš J. (1974), Prediction of s-th ordered observation for the two-parameter exponential distribution, Technometrics, 16, 241-244.

[15] M. A. M. Ali Mousa and Jaheen Z. F. (1997), Bayesian Prediction For The Burr Type XII Model Based on Doubly Censored Data, Statistics 29 , 285-294.

[16] Martz, H. F. and Waller, R. A. (1982), Bayesian Reliability Analysis, Wiley, New York.

[17] McNolty, F., Doyle, J. and Hansen, E. (1980), Properties of the Mixed Exponential Failure Process. Technometrics, 22, 241-244.

[18] McIntyre, G. A. (1952). A method for unbiased selective sampling, using ranked sets. Australian J. Agricultural Research 3, 385-390.

[19] Al-Saleh M. Fraiwan, Al-Shrafat Khalaf and Muttlak H. (2000), Bayesian Estimation Using Ranked Set Sampling, Biometrical Journal 42 4, 489-500.

[20] Nigm, A. M., Al-Hussaini, E. K. and Jaheen, Z. F. (2003), Bayesian one-sample prediction of future observations under Pareto distribution, Statistics, 37, 527-536.

[21] Nigm, A. M. and Hamdy H. I. (1987), Bayesian Prediction Bounds for the Pareto Lifetime Model, Communications in Statistics-Theory and Methods, 16, 1761-1772.




[22] Ouyang, L. Y. and Wu, S. J. (1994), Prediction Intervals for an Ordered Observation from a Pareto Distribution, IEEE Transactions on Reliability, 43, 264-269.

[23] Shirahata, S.(1993), Interval estimation in ranked set sampling, Bulletin of the Computational Statistics of Japan, 6, 15-22.

[24] Wu, J. W., Lu, H. L., Chen, C. H. and Yang, C. H. (2002), A note on the prediction intervals for a future ordered observation from a pareto distribution. Accepted by Quality and Quantity.

[25] Wu, J. W. and Yang C. C. (2002), Weighted Moments Estimation of The Scale Parameter of The Exponential Distribution Based on a Multiply Type Censored Sample, Quality and Reliability Engineering International, 18, 149-154.

[26] Chen Zehua, Bai Zhidong and Sinha Bimal K. (2004), Ranked Set Sampling Theory and Applications, Springer , New York.



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