1、4 3 a EIA JESD37 92 3234600 0502922 672- W BY GWL ERIWEERIWG DOCUMENTS Il#h Tlm panitrbn of EIA unda RWY Snnnd JEDEC STANDARD Standard for Lognormal Analysis f Uncensored Data, and of Singly Right-Censored Data Utilizing the o Persson and Rootzen Method JESD37 OCTOBER 1992 ELECTRONIC INDUSTRIES ASSO
2、CIATION ENGINEERING DEPARTMENT EIA JESD37 92 = 3234600 0502923 509 I NOTICE JEDEC Standards and Publications contain material that has been prepared, progressively 0 reviewed, and approved through the JEDEC Council level and subsequently reviewed and approved by the EIA General Counsel. JEDEC Standa
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7、tion there are procedures whereby a JEDEC Standard or Publication may be further processed and ultimately became an EIA Standard. a Inquiries, comments, and suggestions relative to the content of this JEDEC Standard should be addressed to the JEDEC Executive Secretary at EIA Headquarters, 2001 Penns
8、ylvania Ave., N.W., Washington, D.C. 20006. Published by ELECTRONIC INDUSTRIES ASSOCIATION Engineering Department 2001 Pennsylvania Ave., N.W. Washington, D.C. 20006 PRICE: Please refer to the current Catalog of EIA errors in usage and interpretation are abundant. Frequently, the number of data poin
9、ts in a distribution is small (i.e., less than 20) Statistical tools for analyzing small samples of data are not available to all. who need them. The result is that less than optimum techniques are being used on calculators. This document was written to provide a very basic set of tools for determin
10、ing the parameters of the lognormal distribution for cases where sophisticated tools or expertise do not exist. The techniques handle a majority of the experimental cases experienced by the contributors. The step-by-step standard and appendixes are intended to instruct and to help ensure proper anal
11、ysis. 3, DEFINITIONS 3.1 Terms 3.1.1 Unit I A unit is a single test structure from which one failure time observation is possible. EIA JESD37 72 m 323LIbOO 0502933 b85 m JEDEC Standard No. 37 Page 3 3.1.2 Sample A sample is a subset of units from a homogeneous parent population that has undergone st
12、ress testing. This subset of the population is assumed to retain the characteristics of the parent population from which it was taken. Failure data from the units in this subset are used to determine distribution parameter estimates of the parent population. 3.1.3 Failure set The failure set is the
13、subset of the sample that fails the defined test criterion during the stress time. 3.1.4 Complete Data Complete data indicates that data from all units in the stress test is available; i.e., all units in the sample are also within the failure set. 3.1.5 Censored Data All nonfailhg units removed from
14、 the stress test at the same time are considered censored data; i.e., the failure times are known to be greater than the time units were on stress test and the censor time is the removal time. This is also known as singly right- censored data. 3.1.6 Lognormal Distribution The lognormal distribution
15、is the assumed distribution for the parent population of failure times from which samples are taken. The logarithms of the failure times are assumed to follow a normal distribution. 3.1.7 Bias The bias of an estimator T for a parameter 8 is E(T) - 8; i.e., the difference between the mean (or expecta
16、tion) of T and the true value of the parameter 8. 3.1.8 Confidence Interval An interval of the form (A,B) where A and B are the confidence limits calculated from sample statistics such that P(A I 8 I B) = 1 - a (where a is the probability of error) is a confidence interval . With repeated sampling,
17、at least 100(1-a)% of the similarly constructed intervals will contain the true population parameter 8. JEDEC Standard No. 37 Page 4 3.2 Symbols N: the number of units in the sample. tf: the failure time observation of a unit from the sample. where : tlaet good is the last time duration the unit was
18、 known to be good and tfirst fail is the time duration the unit was known to have failed. tf+=*: the censor time of a unit from the population. The censor time may or may not be the time that another part is known to have failed. o: the real (unknown) standard deviation or shape parameter of the log
19、normal parent distribution. S: sample estimate of the standard deviation, o, of the lognormal parent distribution, calculated from logarithms of the observed failure times. . tSo: the real (unknown) median-time-to-fail (MTF) of the parent distribution. t50s:sample estimate of the MTF of the parent d
20、istribution, calculated from the logarithms of observed failure times. a: The maximum probability of error that is acceptable when making decisions. 4. SUMMARY OF TECHNIQUES 4.1 Complete Data For complete sample data sets (i.e. , sets consisting of the complete population of the sample), S of the pa
21、rent population distribution is the standard deviation of the ln(t,! values, corrected for bias. t50, is determined from the exponential of the mean of the in(t,) values. The confidence intervals for the mean and the standard deviation of the population are obtained throu h distribution, respectivel
22、y. the use of Students t-distribution and the chi-square (x 4 ) I 4.2 Singly Censored Data JEDEC Standard No. 37 Page 5 For censored sample data sets, S and tSoc are the Persson and Rootzen Estimators corrected for bias. These are easily calculated estimators; however, confidence intervals are not a
23、vailable. The censoring time is the time the test was stopped (a unit may or may not have failed at this time). 4.3 Information Required Parties to the test must have t, , tf-=,., (if used), and desired 1-cr confidence level (a error) values at hand to calculate parameter estimates. 5 o INTERFERENCE
24、8 5.1 Distribution If the parent population does not follow a lognormal distribution, this standard should not be used. The distribution fit can be evaluated as described in section 8. If many of the raw t, values do not lie along the prediction line determined by the parameters estimated, this may
25、suggest that the parent population was not lognormal. Other techniques are also available 4,5. Care must be taken with analysis under these circumstances. 5.2 Parent Populations Lack of homogeneity can occur when there are two or more parent populations contained in the sample, or the units in the s
26、ample are not treated alike before or during the stress. As above, a plot (see section 8) of failure times may provide evidence of more than one parent population. 5.3 Frequency of Readout If the failure time is not accurately known, the parameter estimates will be in error. Parties to the test shou
27、ld agree on the frequency of readout for the test, understanding that more frequent readouts will provide more accurate failure times. 5.4 Bias Correction for Censored Data The factors used to correct bias in parameter estimates from censored data are determined assuming that a failure occurred at t
28、he censor time. If units are removed some time after the last failure occurred, the corrections for bias are less accurate. The significance of any error depends upon the time difference (tdiff) between the last failure and the censor time. The magnitude of the error may be investigated as described
29、 in the following sections. EIA JESD37 92 W 3234b00 0502934 394 JEDEC Standard No. 37 Page 6 5.4.1 Small tdiff Where tdiff is small relative to the time when the next failure is anticipated, compare calculated estimates to estimates determined with censoring time set to the last failure time. 5.4.2
30、Large tdiff Where tdif is large relative to the time the next unit is anticipated to fail, compare the calculated estimates to estimates determined assuming a unit failed at the censoring time. 5.5 Independence Lack of independence can occur when two or more failure times are related or are linked b
31、y some effect or event. An example is a power surge causing several parts to fail simultaneously. 5.6 Greater Than 90% censoring with Right Censoring The bias removal techniques for singly right-censored data used in section 7 have only been checked for accuracy with 590% censoring 3. It has not bee
32、n proved that the bias removal technique is valid or more censoring and should be used with caution. 6. PROCEDURE FOR COMPLETE SAMPLE DATA 6.1 Mean and Median Calculate the unbiased sample estirnate, In ( tsOs) , of the mean of the log failure distribution, and the sample estimate, t, , of the media
33、n of the failure time distribution. then , 6.2 Standard Deviation Calculate the biased and unbiased estimates of the standard deviation of the lognormal failure set. then, EIA JESD37 92 323qbOO 0502935 220 JEDEC Standard No. 37 Page 7 N ( in t, - (in t, 12 N-1 1 S = Sunbiased = 1+ 4 (N- 1) biased (4
34、) (5) 6.3 Confidence Interval For The Mean Calculate the 100(1-a)% confidence interval for the median of the failure set t50. 6.3.1 t-Value Find the t-value corresponding to an upper tail probability of a/2 for Students t-distribution with N-1 degrees of freedom, ta/2;N-l, from Table 1 or equivalent
35、. 6.3.2 Limits Calculate the lower and upper limits of the lOO(l-a)% confidence interval for the median of the time distribution, LCL(t5,) , CL(t,) using the following: 1 . biased JN . biased JN (7) where Sbiased is from (4) above. 6.4 Confidence Interval for The Standard Deviation Calculate the 100
36、(1-a)% confidence interval for CJ of the log failure set. JEDEC Standard No. 37 Page 8 6.4.1 Chi-square Find the 1-a/2 and a/2 percentiles of the chi-squared distribution with N-1 degrees of freedom (x2l-a/2;N-1J , x2a/2;N-i) from Tables 2a,b or equivalent. 6.4.2 Limits Calculate the lower and upper
37、 limits of the 100(1-a)% confidence intervals LCL(a), UCL(a) from the following equations: r I 7. PERSSON-ROOTBEN PROCEDURE FOR SINGLY RIGHT-CENSORED DATA 18283 7.1 Calculate Intermediate Terms 7.1.1 20 Determine the standard normal value zo that corresponds to the (1- K/N) percentile of the standar
38、d normal distribution, where N = the total number of parts on test, and K = number of parts (from sample size N) that failed during the test. Use Table 3 or equivalent. 7.1.2 apR 7.1.3 c, 7.1.4 M JEDEC Standard No. 37 Page 9 7.1.5 StdDev 7.2 Btandard Deviation Calculate the biased and unbiased estim
39、ates of standard deviation. a 7*2*1 PR biased calculate the biased sample estimate, S, biqsed, of the standard deviation of the censored log failure population fromthe following equation: 7.2.2 s Calculate the unbiased estimate, S, of the standard deviation of the censored log failure set from the f
40、ollowing equation: 7.3.2 In t5os Calculate the unbiased estimate, In t50, , of the mean of the log failure set from the following equation: EIA JESD37 92 W 323YbO 0502938 T3T JEDEC Standard No. 37 Page 10 7.3 Mean and Median Calculate the biased and unbiased estimates of the mean of the log failure
41、set. ln (t50S PR biased) Calculate the biased estimate, In (tSoc PR bigsed), of the mean of the log failure set from the following equation: in( 50s PRbiased) * + PR RML (17) y) unbiased 3 0.98 0.068N - -+ K2 Calculate the estimate, t5,os, of the median of the failure time population from the follow
42、ing equation: 8. DATA GRAPBING AND PRESENTATION Use this section to generate graphs of raw fail times and fitted distributions. It is the intent of this section to provide graphs that fully describe the data gathered and any analysis results from the data set. 8.1 Prepare Raw Fail Times For Graphing
43、 8.1.1 Rank Order Fails Rank order log fail times (in tfi), with i=1 being the shortest fail time, i=2 the 2nd fail time, etc. a EIA JESD37 92 m 3234600 0502939 976 W 8.1.2 Probabilities JEDEC Standard No. 37 Page 11 Assign failure probabilities to each tfi 5: 8.1.3 Z-Values Determine the standard n
44、ormal Z-value that corresponds to each failure probability (area). 8.2 Graph Raw Data Use Table 3 or equivalent. Graph Z-value -vs- In tfi for each failure time. used in a key with a label. probability values and one with Z values. time values. Include symbol Label two Z axes, one with failure Label
45、 In tfi axis with 8.3 Censored Data Points At the time point when censoring has occurred, include a marker or symbol that denotes censoring has occurred. Include enough description of censoring in the graph key to explain censoring that has occurred. 8.4 Graph Fitted Lognormal Distribution Graph the
46、 following line using available tSos and S values: In t - In t4 S z=J Include fitted line and description in graph key. If the tfi data belong to a lognormal distribution, they will fall along the line. 9. REPORTING 9.1 Minimum Data Reporting As a minimum, the following information should be reporte
47、d: o estimate for median-time-to-fail (tsOs) o o number of samples, failures, and censored units. estimate for standard deviation of log distribution (S) ETA JESD37 72 W 3234600 0502940 698 W t JEDEC Standard No. 37 Page 12 9.2 optional Reporting Report the following for engineering and/or internal
48、improvement o items listed in 9.1 o raw fail data and censor time (if any) o confidence intervals (if available) o technique used to produce estirnators o state whether estimators are biased or unbiased o provide log-normal graph containing raw data and estimated distribution line as described in se
49、ction 8. 10eREFERENCES T. Persson and H. Rootzen, flSimple and highly efficient estimators for a Type I censored normal sample1, Biometrika, 64, 1977, p. 123. H. Schafft and J. Lechner, ltAnalysis of Time-to-Fail Data,“ Presentation at 1989 Wafer-Level Reliability Workshop, Lake Tahoe, 1989. J. Lechner, IlEstirnators for Type-II Censored (Log) Normal Samples,“ Transactions on Reliability, vol 40, no 5, 1991, pp. 547. M. Stephens, “EDF Statistics for Goodness of Fit and Some Comparisons,I Journal of the American Statistical Association, 69, 197
