機(jī)械設(shè)計(jì)制造及其自動化中英文翻譯--逐次定數(shù)截尾數(shù)據(jù)下對一個簡單的步進(jìn)應(yīng)力模型的估計(jì)

1、<p><b>　　中文3350字</b></p><p>　　本科畢業(yè)設(shè)計(jì)（英文翻譯）</p><p>　　英文原文： Estimations For A Simple Step-Stress Model </p><p>　　With Progressively Type-II Censored Data </

2、p><p>　　院　　系：　 能源與環(huán)境工程學(xué)院　 　 </p><p>　　專業(yè)年級：　 機(jī)械設(shè)計(jì)制造及其自動化2008級 </p><p>　　學(xué)生姓名：　 　 學(xué)號：　 </p><p>　　2012年5月12日 </p

3、><p>　　International Journal of Reliability, Quality and Safety Engineering</p><p>　　Vol.12, No.5(2005) 385–395 </p><p>　　©World Scienti?c Publishing

4、Company</p><p>　　ESTIMATIONS FOR A SIMPLE STEP-STRESS MODEL</p><p>　　WITH PROGRESSIVELY TYPE-II CENSORED DATA</p><p>　　SHUO-JYE WU? and HSIU-MEI LEE</p><p>　　Department

5、 of Statistics, Tamkang University</p><p>　　Tamsui, Taipei 251 Taiwan</p><p>　　?shuo@stat.tku.edu.tw</p><p>　　DAR-HSIN CHEN</p><p>　　Graduate Institute of Finance</p

6、><p>　　National Chiao Tung University</p><p>　　Hsinchu City 300, Taiwan</p><p>　　Received 1 January 2005</p><p>　　Revised 23 May 2005</p><p>　　With today’s hi

7、gh technology, some life tests result in no or very few failures by the end of test. In such cases, an approach is to do life test at higher-than-usual stress conditions in order to obtain failures quickly. This study di

8、scusses the point and interval estimations of parameters on the simple step-stress model in accelerated life testing with progressive type II censoring. An exponential failure time distribution with mean life that is a l

9、og-linear function of stress and a cumulative</p><p>　　Keywords: Accelerated life test; confidence interval; exponential distribution; maximum likelihood method; pivotal quantity; progressive type II censori

10、ng.</p><p>　　Introduction</p><p>　　Accelerated life test (ALT) is often used for reliability analysis. Test units are run at higher-than-usual stress conditions in order to obtain failures quick

11、ly. A model relating life length to stress is fitted to the accelerated failure times and then extrapolated to estimate the failure time distribution under usual conditions. The stress loading in an ALT can be applied va

12、rious ways. They include constant stress, step stress, and random stress. Nelson (Ref. 10, Chap. 1) discussed their advan</p><p>　　In step-stress scheme, a test unit is subjected to successively higher level

13、s of stress. A test unit starts at a specified low stress for a specified length of time. If it does not fail, stress on it is raised and held a specified time. The stress is thus increased step by step until the test un

14、it fails. Generally all test units go through the same specified pattern of stress levels and test times. The simplest step-stress ALT uses only two stress levels and we call simple step-stress ALT. The s</p><

15、p>　　In ALT, tests are often stopped before all units fail. The estimate from the censored data are less accurate than those from complete data. However, this is more than o?set by the reduced test time and expense. On

16、e of the most common censoring schemes is type II censoring. A type II censored sample has observed only the m(1≤m≤n) smallest observations in a random sample of n units. If an experimenter desires to remove live units a

17、t points other than the ?nal termination point of the life test, th</p><p>　　Consider an experiment in which n independent units are placed on a test at time zero, and the failure times of these units are re

18、corded. Suppose that m failures are going to be observed. When the ?rst failure is observed, of the surviving units are randomly selected and removed. At the second observed failure, of the surviving units are randomly

19、 selected and removed. This experiment stops at the time when the m-th failure is observed and the remaining… surviving units are all removed. The m o</p><p>　　In this study, we consider point and interval e

20、stimations for the simple stepstress ALT with (1) progressive type II censoring, (2) an exponential failure time distribution at a constant stress, and (3) the cumulative exposure model. In Sec.2, we describe the model a

21、nd some necessary assumptions. We use the maximum</p><p>　　likelihood method to obtain the point estimators of the model parameters in Sec. 3. The confidence intervals for the model parameters are derived in

22、 Sec.4. A numerical data set is studied to illustrate the inferential procedure in Sec.5.</p><p>　　Model and Assumptions</p><p>　　Let us consider the following simple step-stress accelerated lif

23、e-testing scheme with progressive type II censoring: Suppose n randomly selected units are simultaneously placed on a life test at stress setting ; the failure times of those that fail in a time interval are observed an

24、d some surviving units are removed when a failure occurs; starting from time , the non-removed surviving units are put to a different stress setting ; the failure times of those that fail are observed and some surviv<

25、/p><p>　　， (1)</p><p>　　Here the and are unknown parameters. The log-linear function is a common choice for the life-stress relationship since it includes both the power-law a

26、nd the Arrhenius-relation as special cases. Furthermore, failures occur according to a cumulative exposure model. That is, the remaining life of a unit depends only on the exposure it has seen, and the unit does not reme

27、mber how the exposure was accumulated. (see Miller and Nelson9)</p><p>　　From previous assumptions, the cumulative distribution function of a test unit under simple step-stress test is:</p><p>　

28、　whereandis the solution of. Hence, the probability density function of a test unit is</p><p><b>　　(2)</b></p><p>　　Maximum Likelihood Estimation</p><p>　　Let……be a prog

29、ressively type II censored sample with censoring schemerom a simple step-stress ALT. That is, failure times of the test units are observed while testing at stress,,andis the total number of failures. Thus, the likelihood

30、 function is given by</p><p><b>　　,</b></p><p>　　where……and. By (1),the log-likelihood function for unknown parameters β0 and β1 may then be written as, for ,</p><p>　　,

31、 (3)</p><p><b>　　where</b></p><p><b>　　,</b></p><p><b>　　and</b></p><p><b>　　.</b></p><p>　

32、　letand.We then find that the maximum likelihood estimators (MLEs) for and are</p><p>　　, (4)</p><p><b>　　and</b></p><p>　　. (5)<

33、;/p><p>　　respectively.</p><p>　　Under mild regularity conditions, any of several maximum likelihood largesample procedures might be used to make inferences about and . One possibility is to emplo

34、y the asymptotic normal approximation to obtain confidence intervals for and . We now derive the Fisher information matrix. From (3), we have</p><p>　　, (6)</p><p>　　,

35、 (7)</p><p><b>　　and</b></p><p>　　. (8)</p><p>　　To obtain the Fisher information, we need the expectations of (6), (7), and (8). To get these, let

36、us first consider the following transformation. If a random variable X has the probability density function as in (2), then Xiong12 showed that the random variable</p><p>　　is exponentially distributed with

37、mean 1. Convert all into through (9). Then …… is a progressively type II censored sample from the standard exponential distribution. From Balakrishnan and Aggarwala (Ref. 2, p. 19), we have the expectation of is </

38、p><p>　　whereThus, the Fisher information is</p><p><b>　　where</b></p><p><b>　　and</b></p><p><b>　　.</b></p><p>　　The Fish

39、er information can be inverted to get the asymptotic variance-covariance matrix of the MLEs as</p><p>　　where ，，and .It follows form Bickel and Doksum (Ref. 3, p. 398) that is a consistent estimator of .Ther

40、efore, the approximate confidence intervals for and are</p><p><b>　　and</b></p><p><b>　　,</b></p><p>　　respectively, whereis the upper percentage point of

41、 the standard normal distribution.</p><p>　　Remark 1. In practice, can be 0 or .if and the log-likelihood function becomes</p><p><b>　　.</b></p><p>　　Hence, no MLE for a

42、ndexists. The MLE for is. If and , the log-likelihood function is</p><p><b>　　.</b></p><p>　　Note that, regardless of the values of , the log-likelihood function is an increasingfu

43、nction of . Hence, no MLE for and exists. The MLE for is .</p><p>　　Interval Estimations</p><p>　　In this section, an exact confidence interval for and an at least confidence interval for

44、 are constructed. Consider the following transformation: </p><p><b>　　(10)</b></p><p>　　Balakrishnan and Aggarwala (Ref. 2, p. 18) showed that the generalized spacings , as defined i

45、n (10), are independent and identically distributed as the standard exponential distribution. Hence, by the Theorem 4.5.1 in Lawless,7</p><p>　　has a chi-square distribution with 2 degrees of freedom and<

46、/p><p>　　has a chi-square distribution with2m-2 degrees of freedom. We can also show that U and V are independent.</p><p>　　In the following discussion, let be the upper a percentage point of the

47、F distribution with and degrees of freedom and let be the upper a percentage point of the chi-square distribution with degrees of freedom. </p><p>　　4.1. Confidence Intervals for </p><p>　　C

48、onsider the case that .Let ,where</p><p><b>　　,</b></p><p><b>　　and</b></p><p><b>　　.</b></p><p>　　It is easy to show that and, hen

49、ce has a chi-square distribution with 2m-2 degrees of freedom. It is also easy to see that has a chi-square distribution with 2 degrees of freedom, and and are independent.</p><p>　　Now, we are going to de

50、rive a confidence interval for . Consider the pivotal quantity</p><p><b>　　.</b></p><p>　　Let and .For ,we have</p><p>　　Hence, if ,a confidence interval for is ,</

51、p><p><b>　　where</b></p><p>　　. (11)</p><p>　　If ，a confidence interval for is .</p><p>　　Note that the previous confidence interval for is v

52、alid under the condition However, in practice, can be 0 or m. Thus, we have the following two remarks. </p><p>　　Remark 2. When,consider the pivotal quantity</p><p><b>　　.</b></p

53、><p>　　where .Then the confidence interval for is ,for ，an empty set elsewhere.</p><p>　　Remark 3. when and ,consider the pivotal quantity</p><p><b>　　.</b></p>&l

54、t;p><b>　　Let</b></p><p><b>　　.</b></p><p>　　Then, the confidence interval for is ,for , elsewhere.</p><p>　　4.2. Confidence Interval for </p><

55、;p>　　Suppose that .We have is distributed as , is distributed as , and and are independent. Let , , and . For , we have</p><p>　　Hence, if , an at least confidence interval for is ,where</p>

56、<p>　　, (12)</p><p><b>　　and</b></p><p>　　. (13)</p><p>　　If , an at least confidence interval for is .</p><p>　　Note that

57、the above discussion is under the condition . However, inpractice, can be 0 or m. Therefore, we have the following two remarks.</p><p>　　Remark4. When, we have</p><p>　　Hence, an at least conf

58、idence interval for is , for , an empty set elsewhere.</p><p>　　Remark5. Whenand, we have</p><p>　　Thus, an at least confidence interval for is , for , where</p><p><b>　　，

59、</b></p><p><b>　　and</b></p><p><b>　　.</b></p><p>　　If , the confidence interval is .</p><p>　　An Illustrative Example</p><p>

60、;　　To illustrate the use of the methods given in this paper, Table 1 presents the simulated data from a simple step-stress ALT model with progressive type II censoring. These data were simulated by generating a progressi

61、vely type II censored sample from the standard exponential distribution using the algorithm presented in Balakrishnan and Aggarwala (Ref. 2, p. 32), and then the transformation (9) is used to get the sample from model (2

62、). We choose , , , , , , , and censoring scheme listed in Tab</p><p>　　Table1. Simulated failure time data.</p><p>　　The numbers of failures observed at stress and at stress are and , respe

63、ctively. Using (4) and (5), we are able to calculate the MLEs of and, and therefore, the estimates are</p><p><b>　　and</b></p><p>　　In addition to point estimation, we may also consi

64、der interval estimations of and. A 90% confidence interval for is obtained as</p><p>　　by using (11). We also obtain from (12) and (13) that an at least 90% confidence interval for is</p><p>&

65、lt;b>　　.</b></p><p>　　Acknowledgements</p><p>　　The authors would like to thank the Editor and referee for providing helpful comments. The work of the first author was partially support

66、ed by the National Science Council of ROC grant NSC 89-2118-M-032-016. </p><p>　　References</p><p>　　1. A. A. Alhadeed and S.-S. Yang, Optimal simple step-stress plan for cumulative exposure mod

67、el using log-normal distribution, IEEE Transactions on Reliability 54 (2005)64–68.</p><p>　　2. N. Balakrishnan and R. Aggarwala, Progressive Censoring — Theory, Methods, and Applications (Birkh¨auser, B

68、oston, 2000).</p><p>　　3. P. J. Bickel and K. A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Vol. 1, 2nd edn. (Prentice Hall, Upper Saddle River, NJ, 2001).</p><p>　　4. A.

69、D. Dharmadhikari and M. M. Rahman, A model for step-stress accelerated life testing, Naval Research Logistics 50 (2003) 841–868.</p><p>　　5. E. Gouno, An inference method for temperature step-stress accelera

70、ted life testing, Quality and Reliability Engineering International 17 (2001) 11–18.</p><p>　　6. I. H. Khamis and J. J. Higgins, A new model for step-stress testing, IEEE Transactions on Reliability 47 (1998

71、) 131–134.</p><p>　　7. J. F. Lawless, Statistical Models and Methods for Lifetime Data, 2nd edn. (John Wiley & Sons, New York, 2003).</p><p>　　8. E. O. McSorley, J.-C. Lu and C.-S. Li, Perfo

72、rmance of parameter-estimates in step-stress accelerated life-tests with various sample-sizes, IEEE Transactions on Reliability 51 (2002) 271–277.</p><p>　　9. R. Miller and W. B. Nelson, Optimum simple step

73、stress plans for accelerated life testing, IEEE Transactions on Reliability 32 (1983) 59–65.</p><p>　　10. W. Nelson, Accelerated Testing: Statistical Models, Test Plans, and Data Analysis (John Wiley & S

74、ons, New York, 1990).</p><p>　　11. L. C. Tang, Y. S. Sun, T. N. Goh and H. L. Ong, Analysis of step-stress acceleratedlife-test data: A new approach, IEEE Transactions on Reliability 45 (1996) 69–74.</p&g

75、t;<p>　　12. C. Xiong, Inferences on a simple step-stress model with type-II censored exponential data, IEEE Transactions on Reliability 47 (1998) 142–146.</p><p>　　13. K.-P. Yeo and L.-C. Tang, Planni

76、ng step-stress life-test with a target accelerationfactor, IEEE Transactions on Reliability 48 (1999) 61–67.</p><p>　　About the Authors</p><p>　　Shuo-Jye Wu is a Professor in the Department of S

77、tatistics at Tamkang University. He obtained his Ph.D. in Statistics from the University of Wisconsin-Madison. His professional interests are in the development and application of statistical methodology for problems in

78、reliability.</p><p>　　Hsiu-Mei Lee obtained her Ph.D. inManagement Sciences from Tamkang University in 1996. She is currently employed as an Associate Professor in the Department of Statistics at Tamkang Uni

79、versity. Her research interest is applied statistics.</p><p>　　Dar-Hsin Chen is an Associate Professor in the Graduate Institute of Finance at National Chiao Tung University. He obtained his Ph.D. in Finance

80、 from the University of Mississippi. His professional interests are applying statistical and econometric models in modeling financial assets’volatility.</p><p>　　International Journal of Reliability, Quality

81、 and Safety Engineering</p><p>　　Vol.12, No.5(2005) 385–395 </p><p>　　©World Scienti?c Publishing Company</p><p>　　逐次定數(shù)截尾數(shù)據(jù)下對一個簡單的步進(jìn)應(yīng)力模型的估計(jì)</p&g

82、t;<p><b>　　吳碩杰，李秀美</b></p><p><b>　　淡江大學(xué)統(tǒng)計(jì)學(xué)系</b></p><p>　　臺灣 臺北 淡水251號</p><p>　　*shuo@stat.tku.edu.tw</p><p><b>　　陳達(dá)新</b></

83、p><p>　　國立交通大學(xué) 財(cái)務(wù)金融研究所</p><p>　　臺灣 新竹市 300號</p><p>　　2005年1月1日 收到</p><p>　　2005年5月23日 修訂</p><p>　　在今天的高新技術(shù)下，一些壽命試驗(yàn)在測試結(jié)束后沒有或很少會產(chǎn)生失效。在這種情況下，為了獲得迅速地實(shí)現(xiàn)產(chǎn)品失效，一種方法是在

84、高于正常應(yīng)力的條件下進(jìn)行壽命試驗(yàn)。本研究目的在于探討在逐次定數(shù)截尾加速壽命試驗(yàn)中的簡單步進(jìn)應(yīng)力模型的參數(shù)的點(diǎn)估計(jì)和區(qū)間估計(jì)。通過一個應(yīng)力的線性分布和一個累積失效模型下一個有效壽命的平均壽命分布，我們得出模型參數(shù)的極大似然估計(jì)量。模型參數(shù)的置信區(qū)間通過使用樞軸量而被建立，并且可以適用于任何大小的樣本。通過研究了一個數(shù)值例子來說明所提出的方法。</p><p>　　關(guān)鍵詞：加速壽命試驗(yàn)；置信區(qū)間；指數(shù)分布；極大似然估

85、計(jì)法；樞軸量；逐次定數(shù)截尾。</p><p><b>　　介紹</b></p><p>　　加速壽命試驗(yàn)(ALT)經(jīng)常被用于可靠性分析。為了使衰減更快速，試驗(yàn)裝置運(yùn)行在一個高于正常壓力的條件下。一個有關(guān)使用壽命的應(yīng)力模型被用來加速衰減時間，然后推算估計(jì)在正常條件下的失效時間分布。ALT可以應(yīng)用與多種方式應(yīng)力加載。它們包括恒定應(yīng)力，步進(jìn)應(yīng)力、和隨機(jī)應(yīng)力。Nelson(參

86、考文獻(xiàn)10，第一章)討論了它們的優(yōu)點(diǎn)和缺點(diǎn)。</p><p>　　在步進(jìn)應(yīng)力方案中，試驗(yàn)裝置逐次在更高的應(yīng)力下進(jìn)行試驗(yàn)。試驗(yàn)裝置開始在一個特定的低應(yīng)力下進(jìn)行試驗(yàn)，持續(xù)一定的時間。如果它不失效，加在裝置上的應(yīng)力升高并維持一定時間。應(yīng)力從而逐次增加，一直到試驗(yàn)裝置的失效。一般所有的試驗(yàn)裝置都經(jīng)歷同樣的指定模式的應(yīng)力水平和測試時間。最簡單的步進(jìn)應(yīng)力ALT僅僅使用兩個應(yīng)力水平，我們稱之為簡單步進(jìn)應(yīng)力ALT。對這個簡單步進(jìn)

87、應(yīng)力ALT的統(tǒng)計(jì)推斷已被如下幾位作者調(diào)查，諸如Tang 等，11Khamis 和 Higgins，6Xiong，12 Yeo和Tang，13Gouno，5McSorley et al.，8 Dharmadhikari 和 Rahman，4和Alhadeed 和 Yang.1。</p><p>　　在ALT中，試測試往往在所有裝置都已失效前就停止了。相比于完整數(shù)據(jù)，從截尾數(shù)據(jù)中得到的估計(jì)是不精確的。然而，這難以抵消

88、減少的測試時間和費(fèi)用。最常見的一種截尾方案是定數(shù)型截尾方案。一個定數(shù)型截尾樣品只觀測記錄隨機(jī)抽樣n個單位的m(1≤m≤n)個最小的觀察值。如果試驗(yàn)者希望移除壽命試驗(yàn)的最后終止點(diǎn)和其他的正在進(jìn)行試驗(yàn)的裝置，上述計(jì)劃將不能再被試驗(yàn)者使用。定數(shù)型截尾不允許從測試中刪除最后終止點(diǎn)以外的點(diǎn)單位。然而，這還算可取的，就像在意外破損情況下的試驗(yàn)裝置，在其中損失的單位中除了終止點(diǎn)外可能都是不可避免的。在消耗的時間和觀察一些極端值之間尋求妥協(xié)時，中間去除

89、，也可能是可取的。這些原因?qū)е铝宋覀冞M(jìn)入逐次截尾的領(lǐng)域。</p><p>　　考慮一個在時間零點(diǎn)的時候?qū)個獨(dú)立單位被放置在一個測試的試驗(yàn)，且這些失效次數(shù)都被記錄。假設(shè)m的失效要觀察。當(dāng)觀察第一次失效時，的存活單位隨機(jī)選擇并刪除。在觀察第二個失效時，的存活單位隨機(jī)選擇并刪除。這個試驗(yàn)在時間停止時觀察m次失效，其余…存活單位都刪除。在m次觀察失效時間，被稱為是從大小為n的樣本用截尾方案（）逐次定數(shù)截尾大小為m的有序

90、統(tǒng)計(jì)量。注意，如果…，然后，是對應(yīng)定數(shù)型截尾，并且如果…，那么，就是對應(yīng)完整的樣本。</p><p>　　在本研究中，我們認(rèn)為點(diǎn)估計(jì)和區(qū)間靠(1)逐次定數(shù)截尾，(2)正常盈利下恒定應(yīng)力指數(shù)的失效時間分布，以及(3)累積失效模型來估計(jì)簡單步進(jìn)應(yīng)力ALT。在第2節(jié)，我們描述了模型以及一些必要的假設(shè)。在第3節(jié)，我們使用極大似然法獲得了模型參數(shù)的點(diǎn)估計(jì)。在第4節(jié)，推導(dǎo)出了置信區(qū)間的模型參數(shù)。在第5節(jié)，對一個具體數(shù)值的數(shù)據(jù)

91、集進(jìn)行研究，并說明推理過程。</p><p><b>　　模型和假設(shè)</b></p><p>　　讓我們考慮一下用逐次定數(shù)截尾方法實(shí)施以下簡單步進(jìn)應(yīng)力加速壽命試驗(yàn)方案：假設(shè)隨機(jī)選擇的單位n在應(yīng)力設(shè)定下同時放到壽命試驗(yàn)上；那些失效次數(shù)無法在時間區(qū)間內(nèi)觀察和當(dāng)發(fā)生失效時一些存活單位已經(jīng)被移除；從時間開始，那些沒被刪除的存活單位被放到一個不同的應(yīng)力設(shè)定上；那些失效次數(shù)被觀察

92、和當(dāng)失效發(fā)生時一些存活單位被移除；在第m次失效時，壽命試驗(yàn)將被停止。在任何應(yīng)力下，那些測試單位的失效壽命分布都是一個指數(shù)分布。在應(yīng)力水平下，平均壽命測試裝置是一種應(yīng)力的對數(shù)線性函數(shù)。那是，</p><p>　　， (1)</p><p>　　這里的和是未知參數(shù)。這個線性函數(shù)是一種常見的選擇應(yīng)力壽命關(guān)系，因?yàn)樽鳛樘厥馇闆r它既包括冪規(guī)律和Arrheniu

93、s關(guān)系。此外，按照累積失效模型已經(jīng)發(fā)生失效。換言之，其裝置剩余的壽命的只取決于已發(fā)生的失效，并裝置不記得如何累積失效。(見Miller and Nelson9)</p><p>　　從先前的假設(shè)，測試裝置的累積分布函數(shù)在簡單步進(jìn)應(yīng)力壽命測試是:</p><p>　　這里和是的解。因此，最合適測試裝置的概率密度函數(shù)是</p><p><b>　　(2)<

94、;/b></p><p><b>　　極大似然估計(jì)</b></p><p>　　設(shè)……是用截尾方案從一個簡單步進(jìn)應(yīng)力ALT中得到的逐次定數(shù)截尾樣本。也就是說，當(dāng)測試在應(yīng)力，觀察到測試裝置的失效時間，并且是失效的總數(shù)。因此，似然函數(shù)被如下給出：</p><p><b>　　，</b></p><p&

95、gt;　　這里……和。根據(jù)(1)，對于未知參數(shù)和的對數(shù)似然函數(shù)可以被寫成，對于時，</p><p>　　， (3)</p><p><b>　　其中</b></p><p><b>　　，</b></p><p><b>　　和</b></

96、p><p><b>　　。</b></p><p>　　設(shè)和。然后我們會發(fā)現(xiàn)和的極大似然估計(jì)量（MLEs）分別是</p><p>　　， (4)</p><p><b>　　和</b></p><p>　　。

97、 (5)</p><p>　　在輕微規(guī)律條件下，任何幾個最大似然大樣本程序都可能會被用來推斷和。一個最大的可能就是采用逐步正態(tài)近似獲得和的置信區(qū)間?，F(xiàn)在我們得出了Fisher信息矩陣。從公式（3），有</p><p>　　， (6)</p><p>　　， (7)</p>&l

98、t;p><b>　　和</b></p><p>　　。 (8)</p><p>　　為了獲得Fisher信息量，我們需要(6)(7)(8)的期望。為了得到這些，讓我們首先考慮以下的信息量。如果一個隨機(jī)變量在(2)中有概率密度函數(shù)，那么Xiong12表示這種隨機(jī)變量</p><p>　　是平均為1的指數(shù)分布

99、。通過(9)把所有的轉(zhuǎn)換成。然后……是從標(biāo)準(zhǔn)指數(shù)分布中得到的逐次定數(shù)截尾樣本。從Balakrishman和Aggarwala（參考文獻(xiàn)2，第19頁）中，我們可以得到的期望是</p><p>　　這里。因此，F(xiàn)isher信息量是</p><p><b>　　其中</b></p><p><b>　　和</b></p&g

100、t;<p><b>　　。</b></p><p>　　Fisher信息量能夠轉(zhuǎn)換成如下MLEs的漸進(jìn)方差-協(xié)方差矩陣</p><p>　　其中，，和。它遵循了Bickel和Doksum（參考文獻(xiàn)3，第398頁）所說的，是的一個相合估計(jì)。因此，和的近似置信區(qū)間分別是</p><p><b>　　和</b>&l

101、t;/p><p><b>　　，</b></p><p>　　其中是標(biāo)準(zhǔn)正態(tài)分布上的個百分點(diǎn)。</p><p>　　備注1.在實(shí)際中，能夠是0或者。如果和，對數(shù)似然函數(shù)變?yōu)?lt;/p><p><b>　　。</b></p><p>　　所以，沒有和存在的MLE。的MLE是。如果和，

102、對數(shù)似然函數(shù)變?yōu)?lt;/p><p><b>　　。</b></p><p>　　值得注意的是，無論的值為多少，對數(shù)似然函數(shù)是一個的增加函數(shù)。所以，沒有和存在的MLE。的MLE是。</p><p><b>　　區(qū)間估計(jì)</b></p><p>　　在這節(jié)中，一個確切的置信區(qū)間和至少的置信區(qū)間被構(gòu)造?？紤]

103、以下的信息量：</p><p><b>　　(10)</b></p><p>　　Balakrishnan and Aggarwala（參考文獻(xiàn)2，第18頁）顯示，廣義的間距，在(10)中定義，獨(dú)立同分布為標(biāo)準(zhǔn)的指數(shù)分布。所以，根據(jù)在Lawless11中4.5.1的定理，</p><p>　　有2個自由度的卡方分布和</p>&l

104、t;p>　　有2m-2自由度的卡方分布。我們可以發(fā)現(xiàn)和是獨(dú)立的。</p><p>　　在下面的討論，設(shè)在有和個自由度的分布的一個百分點(diǎn)上并且設(shè)在有個自由度的卡方分布的一個百分點(diǎn)上。</p><p>　　4.1. 的置信區(qū)間</p><p>　　考慮的情況。設(shè)，其中</p><p><b>　　，</b></p

105、><p><b>　　和</b></p><p><b>　　。</b></p><p>　　簡單表現(xiàn)了，并且，于是有一個具有2m-2自由度的卡方分布。顯而易見，也有一個具有2個自由度的卡方分布，并且和是獨(dú)立的。</p><p>　　現(xiàn)在，我們將要得出的一個置信區(qū)間?？紤]到樞軸量</p>

106、<p><b>　　。</b></p><p><b>　　設(shè)和。在，我們得到</b></p><p>　　故，如果，一個的的置信區(qū)間是，其中</p><p>　　。 (11)</p><p>　　如果，一個的置信區(qū)間是。</p><p>　　注意，先前的

107、置信區(qū)間在情況下是有效的。然而，在實(shí)際中，可能是0或者。因?yàn)?，我們有了以下兩個備注。</p><p>　　備注2.當(dāng)，考慮到樞軸量</p><p><b>　　。</b></p><p>　　其中。然后，的置信區(qū)間是，對于，空集在別處。</p><p>　　備注3. 當(dāng)和，考慮到樞軸量</p><p&

108、gt;<b>　　。</b></p><p><b>　　設(shè)</b></p><p><b>　　。</b></p><p>　　然后，的置信區(qū)間是，對于，在別處。</p><p>　　4.2. 的置信區(qū)間</p><p>　　假設(shè)。我們得到被分布成，被

109、分布成，并且和是獨(dú)立的。設(shè)，，和。對于，我們得到</p><p>　　故，如果，一個的至少置信區(qū)間是，其中</p><p>　　， (12)</p><p><b>　　和</b></p><p>　　。 (13)</p><p>　　如果，一個的至少置信區(qū)間是。</

110、p><p>　　值得注意的是，以上討論是基于情況下的。然而，在實(shí)際中，可能是0或者。因?yàn)?，我們有了以下兩個備注。</p><p>　　備注4. 當(dāng)，我們得到</p><p>　　因此，一個的至少置信區(qū)間是，對于，空集在別處。</p><p>　　備注5. 當(dāng)和，我們得到</p><p>　　由此，一個的至少置信區(qū)間是，對于