1、JEDEC STANDARD LOGNORMAL ANALYSIS OF UNCENSORED DATA AND OF SINGLY RIGHT-CENSORED DATA UTILIZING THE PERSSON AND ROOTZEN METHOD JESD37A (Revision of JESD37, October 1992) AUGUST 2017 JEDEC SOLID STATE TECHNOLOGY ASSOCIATION NOTICE JEDEC standards and publications contain material that has been prepa
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9、 information, contact: JEDEC Solid State Technology Association 3103 North 10th Street Suite 240 South Arlington, VA 22201-2107 or refer to www.jedec.org under Standards-Documents/Copyright Information. JEDEC Standard No. 37A -i- LOGNORMAL ANALYSIS OF UNCENSORED DATA AND OF SINGLY RIGHT-CENSORED DAT
10、A UTILIZING THE PERSSON AND ROOTZEN METHOD Contents PageForeword iiIntroduction ii1 Scope 12 Normative references 23 Terms, definitions, and symbols 24 Summary of techniques 45 Inferences 46 Procedure for complete sample data 67 Persson-Rootzen procedure for singly right-censored data 88 Data graphi
11、ng, and presentation 109 Reporting 11Annex A (informative) Non-censored data example 12Annex B (informative) Singly right-censored data example 14Annex C (informative) Data graphing example for singly right-censored data 16Annex D (informative) Bibliography 26Annex E (informative) Differences betwee
12、n JESD37A and its predecessor JESD37 27JEDEC Standard No. 37A -ii- Foreword This standard is intended for use by those involved with reliability, lifetime estimation or failure data analysis. This standard enables the user to estimate the parameters of a two parameter lognormal distribution from com
13、plete or singly right-censored independent failure data. The methods described were chosen because they can be easily done with a calculator or spreadsheet and are as accurate as more complex iterative techniques for the cases described. Interferences or issues that can complicate and/or invalidate
14、the analysis are discussed. One section discusses analysis of failure data where all units from within the sample have a known failure time. Another section discusses analysis of failure data for the case where the life experiment was stopped and the failure times for the surviving unis are not know
15、n. Graphical presentation is discussed and sample calculations are presented in several annexes. Introduction Analysis of reliability experiments depends extensively on failure statistics and commonly used failure distribution is the normal distribution after natural logarithms of the failure times
16、have been calculated. Lognormal distributions and the associated statistics are used by analysts working in semiconductor reliability to interpret and communicate results from failure experiments. Unfortunately, many individuals who lack a sufficient background in statistical methods are using the t
17、echniques; errors in usage and interpretation are abundant. This document was written to provide a very basic set of tools for determining the parameters of the lognormal distribution for cases where sophisticated tools or expertise do not exist. The techniques handle a majority of the experimental
18、cases experienced by contributors. The step-by-step standard and annexes are intended to instruct and to help ensure proper analysis.JEDEC Standard No. 37A Page 1 STANDARD FOR LOGNORMAL ANALYSIS OF UNCENSORED DATA AND OF SINGLY RIGHT-CENSORED DATA UTILIZING THE PERSSON AND ROOTZEN METHOD (From JEDEC
19、 Board Ballot JCB-17-28, formulated under the cognizance of the JC-14.2 Committee on Wafer-Level Reliability) 1 Scope 1.1 Intent This standard enables the user to estimate the parameters of a two-parameter lognormal distribution from complete or singly right-censored independent data samples. Specif
20、ically, this standard is intended for analyzing failure-time (tf) data obtained from a stress test of a sample of units when the natural logarithm of the failure-time (ln tf) follow a normal distribution. This standard is not intended to describe techniques used to determine how well the failure dat
21、a fits a lognormal distribution. However, if points lie along a straight line for plots generated in section 8 the lognormal distribution estimators will describe the points along the line. 1.2 Results The results of the analysis provide bias-corrected sample estimates for the median time-to-failure
22、 (t50), mean of the ln tfvalues (ln t50), and the standard deviation () of the ln tfvalue of the lognormal distribution. Additionally, confidence intervals are provided for complete data samples (no censoring). These are all obtained from the failure time values (tf) 1.3 Complete data case This stan
23、dard may be used to analyze complete data where the failure-time data for the entire sample population is known and used. The analysis uses the most efficient estimators for obtaining estimates of the two primary parameters (t50, ) of the distribution. 1.4 Right-censored case This standard may be us
24、ed to analyze singly right-censored (Type II censored) data where the test has been stopped before all the parts have failed. The analysis uses the Persson and Rootzen Estimators 1 corrected for bias 2, 3. These estimators can be calculated with a hand calculator and are as accurate as more complex
25、estimators. JEDEC Standard No. 37A Page 2 1.5 Other methods This standard does not preclude the use of other estimators for determining a lognormal distribution, as long as the method has been shown to provide results similar to those obtained by this standard. For example, Maximum Likelihood Estima
26、tors are widely used, have desirable properties and are acceptable when bias is removed. However, these estimators are difficult to obtain without computerized iterative techniques, are subject to similar, if not identical bias and many software tools do not remove this bias. 1.6 Normal distribution
27、 This standard can also be used to estimate the parameters of a normal distribution from complete and right-censored data samples. Raw data is used without taking the natural logarithm. The mean and standard deviation can be used without modification. 2 Normative references T. Persson and H. Rootzen
28、, “Simple and highly efficient estimators for a Type-I censored normal sample”, Biometrika, Vol. 64, 1977, pp.123. 3 Terms, definitions, and symbols 3.1 Terms and definitions For the purposes of this standard, the following terms and definitions apply. bias (statistical): The difference between the
29、mean (or expectation) of an estimator, T, and the true value, , of a parameter: E(T) . censored data: The set of data for the portion of the stressed samples that had stressing discontinued prior to the originally intended end of the test stress. NOTE This is also known as singly right censored data
30、. complete data: All available data from all units in the stress test, including those in the failure set. 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 B) = 1 (where is the probability of error) is a confi
31、dence interval. NOTE With repeated sampling, at least 100(1 )% of the similarly constructed intervals will contain the true population parameter . failure set: The subset of the sample that fails the defined test criterion during the stress time. JEDEC Standard No. 37A Page 3 3 Terms, definitions, a
32、nd symbols (contd) lognormal distribution: A distribution in which for some value a, ln (X a) has a normal (Gaussian) distribution with a mean and variance 2. NOTE If the parent population does not follow a lognormal distribution, this standard should not be used. sample: A subset of units from a ho
33、mogeneous parent population that has undergone stress testing. NOTE 1 This subset of the population is assumed to retain the characteristics of the parent population from which it was taken. NOTE 2 Failure data from the units in this subset are used to determine distribution parameter estimates of t
34、he parent population. unit: A single test structure from which one failure time observation is possible. 3.2 Symbols N: The number of units in the sample. tf: The failure time observation of a unit from the sample. Fail time nullf=expnulllnnull nulllast goodnull+ln( nullfirst fail)2null OR (1) = nul
35、lnulllast goodnullfirst failwhere: tlast good: The last time duration the unit was known to be good tfirst fail: The time duration the unit was known to have failed tf-cen: 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 fa
36、iled. : The real (unknown) standard deviation or shape parameter of the lognormal parent distribution. s: The sample estimate of the standard deviation of the lognormal parent distribution, calculated from logarithms of the observed failure times. t50:The real (unknown) median-time-to-fail (MTTF) of
37、 the parent distribution. t50s:The sample estimate of the MTTF of the parent distribution, calculated from the logarithms of observed failure times. : The maximum probability of error that is acceptable when making decisions. JEDEC Standard No. 37A Page 4 4 Summary of Techniques 4.1 Complete data Fo
38、r complete sample data sets (i.e., sets consisting of the complete population of the sample) the standard deviation of the parent population distribution is based on the standard deviation of the ln(tf) values, corrected for bias. The t50s is determined from the exponential of the mean of the ln(tf)
39、 values. The confidence intervals for the mean and the standard deviation of the population are obtained through the use of Students t-distribution and the chi-square (2) distribution, respectively. 4.2 Singly censored data For censored sample data sets, s and t50sare the Persson and Rootzen estimat
40、ors corrected for bias. These are easily calculated estimators; however, confidence intervals are not available. 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 tf, tf-cen(if used), and desir
41、ed 1 confidence level ( error) values at hand to calculate parameter estimates. 5 Inferences 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 tfval
42、ues do not lie along the prediction line determined by the parameters estimated, this may 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 wh
43、en there are two or more parent populations contained in the sample, or the units in the sample are not treated alike before or during the stress. As above, a plot (see clause 8) of failure times may provide evidence of more than on parent population. JEDEC Standard No. 37A Page 5 5.3 Frequency of r
44、eadout If the failure time is not accurately known, the parameter estimates will be in error. Parties to the test should 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 facto
45、r used to correct bias in parameter estimates from censored data are determined assuming that a failure occurred at the censor time. If units are removed some time after the last failure occurred, the corrections for bias are less accurate. The significant of any error depends upon the time differen
46、ce (tdiff) between the last failure and the censor time. The magnitude of the error may be investigated as described in the following sections. 5.4.1 Small tdiff When tdiffis small relative to the time when the next failure is anticipated, compare calculated estimates to estimates determined with ce
47、nsoring time se to the last failure time. 5.4.2 Large tdiff When tdiffis large relative to the time when the next failure is anticipated, 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
48、 more failure times are related or are linked by 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 use in section 7 have only been checked for
49、accuracy with less than 90% censoring 3. It has not been proven that the technique is valid for more censoring and should be used with caution. JEDEC Standard No. 37A Page 6 6 Procedure for complete sample data 6.1 Mean and median Calculate the unbiased sample estimate, ln(t50s), of the mean of the log failure distribution and the sample estimate, t50s, of the median of the failure time distribution. ln(null50s) = nullf =1nullnullln(nullf)nullnull=1(2)then, null50null=explnnull50nullnull (3)6.2 Standard deviation Calculate
