Statistics, Testing, and Defense Acquisition Background Papers (1999) / Chapter Skim
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On the Performance of Weibull Life Tests Based on Exponential Life Testing Designs
On the Performance of Weibull Life Tests Based on Exponential Life Testing Designs
Pages 41-123
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From page 41... ...
EXPONENTIAL LIFE TESTING Applications abound in which investigators seek to make inferences about the lifetime characteristics of a "system" of interest from data on the failure times of prototypical systems placed on test. There are a good many different experimental designs that might be considered in planning a given life testing application; often, some form of data censoring (aimed at bounding the experiment's duration)
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From page 42... ...
, rejecting Ho in favor ofH, if the total time on test Tat the time ofthe rth failure is less then the threshold To Among the advantages afforded by an exponential life test plan is the fact the the resources required to perform the test (that is, the number of systems that must be placed on test and the maximum amount of testing time needed to resolve the rest) may be calculated in advance.
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From page 43... ...
For a study which examines similar questions in the contrast of interval estimation, see Woods (1996~. We now turn to a brief description of the mechanics of exponential life testing.
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From page 44... ...
to The fact that the required sample size rO is completely determined by the values of ~x, ,8 and the "discrimination ratio" 6/00 is a special feature of exponential life testing that facilitates the automated application of this methodology. Once a sample size r = rO is obtained through (1 .7)
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From page 45... ...
, represents Ine maximum Total test time that could be required to resolve the test, that is, to be able to accept or reject Ho on the basis of the data. Together, rO and c describe the total resources that must be committed to guarantee successful completion of the life test.
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From page 46... ...
This strategy of course is based on a tacit assumption of the correctness of the exponential model in the application of interest; when exponentiality fails, this practice can yield highly misleading results. There are a host of other experimental designs for exponential life testing, including type I censoring (that is, censoring at a fixed time t)
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From page 47... ...
As a guide for military applications of exponential life testing, DoD Handbook HI OS provides tabled values of the required sample size rO and the constant c/OO through which the total test time required by a particular application can be computed. An excerpt from Table 2B-5 of that Handbook, showing the five tabled values given corresponding to error probabilities cx = .1 and ,8 = .
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From page 48... ...
At 0= 750, for example, the probability of accepting HO: R= 1,000 goes from .615 under exponentialityto .837 under a Weibull distribution with shape parameter equal to 3. In spite of this type of inflation, it is clear that exponential life tests carried out with complete samples offer reasonable performance in that even under rather severe departures of the Weibull type, they deliver error probabilities at selected key parameter values 00 and BI that are smaller than those set at the planning stage.
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From page 49... ...
exponential life testing based on complete samples works fairly well in a Weibull environment, but there should be opportunities for saving resources when that environment is recognized in advance; and (2) exponential life testing based on censored samples works very poorly in a Weibull environment, and alternative procedures should be considered when the exponential assumption is suspect.
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From page 50... ...
Like the gamma model, it contains the exponential distribution as a special case, so that the adoption of a Weibull assumption represents a broadening from the exponential model rather than a rejection of it. Often, statistical extreme value theory forms the basis for the applicability of the Weibull model; when system failure can be attributed to the failure of the weakest of its many components, the Weibull model will tend to describe failure data quite well.
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From page 51... ...
developed tests and confidence intervals for the unknown shape parameter. For the general problem, when both A and B are unknown, there is rather limited guidance on how to proceed.
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From page 52... ...
discusses Weibull life test plans briefly, stating that "life test plans under the Weibull model have not been thoroughly investigated.
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From page 53... ...
For an arbitrary positive random variable X with distribution F and finite mean ,u, the total time on test transform ~ is defined as IF (X)
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From page 54... ...
Plotting a scaled TTT transform for data collected according to an exponential life testing plan is an excellent way to detect possible departures from exponentiality. In Figures 1 to 6 , we display the TTT plots from six consecutive simulated Weibull experiments, each featuring complete samples of size 20 from eight Weibull distributions with varying shape parameters.
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From page 55... ...
Through use of the graphical methods described above, or otherwise, assume that the statistician, after gathering data according to an exponential life test plan, determines that the data are more appropriately modeled as a nonexponential Weibull. It will then be necessary to proceed with an analysis appropriate for these broadened assumptions.
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From page 56... ...
Let us, then, assume that a random sample X~ ...,Xr is available from what was originally thought to be an exponential distribution, and that the sample size r was determined on the basis of an exponential life test plan for testing Ho : ~ = To vs HI : ~ = 6, where 00 > RI, at fixed predetermined values of the error probabilities cc and hi. Assume further that, once the data was collected, the assumption (3.~)
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From page 57... ...
Fortunately, in many engineering applications of the Weibull distribution, the shape parameter A turns out to be substantially larger than 1, and the opportunity exists for the execution of tests with smaller error probabilities or tests requiring less in the way of resources for their implementation. When infant mortality (and an initial decreasing failure rate)
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From page 58... ...
, and the realized value of,C when the exponential life test is executed at the indicated nominal significance level a. Our expansion of DoD Handbook HI O8's Table 2B-5 is restricted to four typical choices of ~x, ,8: ct=,8= .01 (Table 2)
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From page 59... ...
, TTTR (for "total time on test ratio") BR (for the ratio of error probabilities ,Cofthe Weibull test and the planned exponential test at the same fixed values of ~ and r)
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From page 60... ...
(3.10) Proof: The upper bound in (3.9)
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From page 61... ...
Am = (S/n, S/n...,Sln)
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From page 62... ...
If A ~ I, then the total time on test required to resolve the Weibull test based on the transformed data will generally be smaller than the maximal TTT called for in the exponential life test plan. In this case, the upper bound in (3.9)
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From page 63... ...
As an aside, we note that the upper bound provided in (3.10) indicates that the total time on test might be as much as 30.4 times as large as that required by an exponential life test plan; thus, while 1,744 hrs of testing are required by the original plan, the TTT in the Weibull environment with A = I/2 will fall between 3,540 and 53,018 furs.
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From page 64... ...
Thus, while one can increase n with impunity in exponential life tests, one must be very careful in using censored life test designs when the underlying distribution is Weibull. We will motivate below a guideline for identifying what might be considered a reasonable upper bound for the amount of censoring one should entertain in a particular application.
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From page 65... ...
samples of size r are available. If, however, the life test plan calls for type TI censoring, and is terminated, at the latest, when the rth failure occurs among the ~ systems on test, then the TTT statistic becomes 6s
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From page 66... ...
the extent to which TTT could have been reduced if an appropriate Weibull life test had been conducted. Suppose we take particular values of the discrimination ratio, a, ~ and A > 1 as fixed and given.
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From page 67... ...
In general, these tables confirm that there are potential resource savings available when one recognizes an IFR Weibull environment and carries out a Weibull life test instead of an exponential one. Similarly, more resources are required in carrying on Weibull life tests when the DFR Weibull analysis is carried out instead of an exponential life test.
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From page 68... ...
It is perhaps worth noting here that the measure BR shows quite dramatically the power of Weibull life tests when the shape parameter A is reasonably large; for fixed values of A, BR appears to vary inversely with the discrimination ratio. We also note that, for fixed A > 1, the amount of censoring that can be accommodated per the r/n computation is an increasing function of the total time on test ratio, which in turn tends to increase as a function of the discrimination ratio.
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From page 69... ...
the value of the Weibull shape parameter, and happens to guess it correctly. It is, of course, necessary to move beyond this first step, and to engage seriously the question of how to execute a Weibull analysis in the general, two-parameter problem.
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From page 70... ...
Table 6 represents a numerical compilation from which one can obtain an estimated shape parameter from an estimated cv. We have relied upon this ~ {~- A table for obtaining A = f (cv)
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From page 71... ...
-type test: one could estimate the shape parameter A by the maximum likelihood estimate A, and carry out the test in section 3 with A replaced by that estimate. We have confirmed, via simulation, that the performance of that test, in small and moderate samples, is essentially indistinguishable from the likelihood ratio procedure described above.
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From page 72... ...
This latter work suggests that one might test hypotheses concerning Weibull means by first estimating the shape parameter from the least squares fit to data plotted on Weibull probability paper, and then carrying out the appropriate exponential test based on the transformed data X,A, .
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From page 73... ...
The exact, best test of TO vs ye has suitably small a and ,C values if A is not too small, and requires a somewhat larger value A to ensure such behavior when the sample sizes are small. For sample sizes equal to the required sample size for exponential life tests with a = ,5 = .
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From page 74... ...
We expect,Cto be less than 0.1 here since a shape parameter of 1.2 has served to decrease the effective discrimination ratio, so that 1 5 observations is more than actually required to achieve a = ,8 = .1 in the life test based on transformed data. For the tests which did not benefit from knowledge of the true A, we find that the cv-based test had a = .1 1, ,6 = .02, the likelihood ratio test had ~ = .1 3, ,8 = .02, and the leastsquares-based test had a = .09, JIB= .03.
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From page 75... ...
In Table ~ I, we record the realized error probabilities for two tests, the first being the optimal test (kno) when the Weibull shape parameter A is fixed and known and the second being the ise test in which A has been estimated from the Weibull probability plot.
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From page 76... ...
The results of section 3 show quite graphically that Weibull life tests can, in certain circumstances, provide substantially greater statistical power than exponential life tests based on the same sample size, and can offer substantial savings in both sample size and testing time when the goal is to match the statistical power of a planned exponential life test. While Tables 2 to 5 ostensibly offer guidance only for the very special case in which the Weibull shape parameter A is known, 76
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From page 77... ...
First, they show quite emphatically that exponential life testing can be especially misleading when the underlying distribution is a DFR Weibull; it is clear that a much larger sample size and much greater testing time are needed to achieve any given nominal error probabilities than what an exponential life test plan would prescribe. The good news carried by Tables 2 to 5 is that, when the underlying distribution is an IFR Weibull, considerable savings are possible.
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From page 78... ...
It should be emphasized that Weibull life tests having characteristics such as those above require knowledge of the value of the shape parameter. Recall, however, that under precisely the circumstances with which we are dealing, the general Weibull tests of section 4 may be employed with confidence.
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From page 79... ...
Our investigations regarding Weibull life testing in the censored data case, while not as comprehensive as we'd like, support the general conclusion that it is possible to test competing Weibull means reliably in the presence of censoring. In applications of that type, our simulations indicate that extreme censonug can be dangerous, that is, can lead to quite inflated error probabilities unless the shape parameter A is very large.
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From page 80... ...
In the context of the hybrid statistical problem discussed in the introductory section-where an alternative model is selected after data has been gathered according to an exponential life test plan-it would be important, in practice, to be able to carry out an appropriate analysis for the model identified as most suitable, be it the Weibull or some other failure-time model deemed to be applicable to the life testing experiment of interest.
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From page 81... ...
TABLE 1 Life Test Sampling Plans for ax = .1 = ,8.
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From page 84... ...
TABLE 2 cont'd. o-~iiml ~ - 0.01 A-1.3 A-1.4 A-I |,1d, r C/t, dS SSR mR IIR r/a SSR mR BR rJa SSR n1R 0.01 3 0.14S O 1 1.6t9 · 9.6tl 0.333 Q094 · O Q333 0.118 Qm 3 0.14S O 1 1.689 · 9.6tl 1 I.9D3 · 9.X8 1 2.10S QQ3 3 0.14S O 2 1.6t9 · 9.6S1 1 1.9D3 · 9.SC8 1 2.10S 0.04 3 0.14S 0.001 1 1.689 · 9.6S1 1 I.9D3 · 9.S08 1 2.
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From page 86... ...
TABLE 2 cont'd.
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From page 90... ...
TABLE 3 cont'd.
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From page 94... ...
TABLE 4 cont'd.
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From page 97... ...
TABLE 5 ',IS, r 0.01 1 o.m I 0.0 1 0.04 1 d.o5 1 0.06 1 Q07 1 0.08 1 QQ9 1 0.1 S 0.11 1 0.12 1 0.13 1 0.14 1 02S 1 0.16 1 0.17 1 nl~ 1 Q19 1 0.2 1 0.21 2 0.22 2 0.23 2 0.24 2 Q25 2 0.26 2 0.27 2 0.2S 2 0.29 2 0.3 2 Q.31 2 0.32 2 0.33 2 0.34 2 0.3S 2 0.36 3 0.37 3 0.38 3 0.39 3 0.4 3 0.41 3 0.42 3 0.43 3 0.44 3 0.4S 4 0.46 4 0.47 4 Q48 4 0.49 4 O.S 5 O.SI 5 O.S2 5 0.53 S O.S4 6 O.SS 6 O.S6 6 O.S7 7 O.S8 7 O.S9 7 06 S 0.61 S 0.62 9 0.63 9 0.64 10 0.6S 11 0.66 11 Q6?
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From page 101... ...
TABLE 5 cont'd.
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From page 102... ...
TABLE 6 Shape Parameter A Corresponding to the Coefficient of Variation cv cv A cv A cv A 428.8314 0.10 0.7238 1.40 0.3994 2.70 47.0366 0.15 0.7006 1.45 0.3929 2.75 15.8430 0.20 0.6790 1.50 0.3866 2.80 8.3066 0.25 0.6588 1.55 0.3805 2.85 5.4077 0.30 0.6399 1.60 0.3747 2.90 3.9721 0.35 0.6222 1.65 0.3690 2.95 3.1409 0.40 0.6055 1.70 0.3634 3.00 2.6064 0.45 0.5897 1.75 0.3581 3.05 2.2361 0.50 0.5749 1.80 0.3529 3.10 1.9650 0.55 0.5608 1.85 0.3479 3.15 1.7581 0.60 0.5474 1.90 0.3430 3.20 1.5948 0.65 0.5348 1.95 0.3383 3.25 1.4624 0.70 0.5227 2.00 0.3336 3.30 1.3529 0.75 0.5112 2.05 0.3292 3.35 1.2605 0.80 0.5003 2.10 0.3248 3.40 1.1815 1.1130 1.0530 0.90 0.95 0.4898 0.4798 0.4703 2.15 0.3206 3.45 2.20 0.3165 3.50 2.25 0.3124 3.55 1.0000 1.00 0.4611 2.30 0.3085 3.60 0.9527 1.05 0.4523 2.35 0.3047 3.65 0.9102 1.10 0.4438 2.40 0.3010 3.70 0.8718 1.15 0.4341 2.45 0.2974 3.75 0.8369 1.20 0.4279 2.50 0.2938 3.80 102
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From page 104... ...
TABLE 7 Olll~eo.2s r-22 r-8 r-4 r-3 r-2 l~no c ~lr ke kno cv lr b~c Icso cv k Isc lu ~c ~lr Isc Icoo c ~lr kc 0.1 ~ 0.13 0.98 O.S9 0.3S 0.12 0.99 0.72 0.36 0.08 0.97 0.7S 0.40 0.10 0.91 0.78 0.43 0.09 Q96 0.81 O.S2 B 0.82 0.00 O.S6 O.S6 0.7S 0.00 0.26 O.S7 0.85 0.02 Q20 0.61 0.85 0.01 0.17 O.SI 0.91 0.01 0.14 0.48 0.2`r 0.06 0.8S 0.77 0.23 0.10 0.81 0.72 0.29 0.12 0.81 0.74 0.39 0.10 0.79 0.74 0.39 0.13 0.77 0.68 0.44 B O
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From page 105... ...
TABLE 8 O,/8o'O.S r-87 r-29 r~lS r=7 r=4 ~cv Ir Ise b~o cv Ir Isc kno cv Ir Ise Icno cv Ir Ise l ~Ir Ise 0.1 ~ 0.08 l.QO O.S4 0.2S 0.07 I.QO O.S2 0.36 0.09 0.99 0.66 0.42 0.08 O.9S 0.77 0.34 0.06 0.94 0.82 0.37 B 0.68 O.QO 0.7S O.SS 0.8S O.QO O.S4 OS8 0.89 0.00 0.40 0.68 0.92 Q01 0.19 Q60 0.89 0.02 0.22 O.S3 0.2 ~ 0.12 l.QO 0.84 0.23 0.11 0.91 0.84 0.21 0.07 0.84 0.80 0.26 O.IS 0.80 0.74 0.28 0.11 0.7S 0.67 0.30 B O.S6 0.00 0.07 O.SO 0.77 0.03 0.12 0.66 0.79 0.06 0.14 0.70 0.80 0.18 Q2S O.S9 0.81 O.IS 0.30 0.61 0.3 o 0.11 O.S4 0.42 0.24 0.13 O.SS 0.48 0.18 0.13 OSS 0.49 0.2S 0.12 OS9 O.S6 0.33 0.09 O.S2 0.49 0.27 B 0.29 0.10 0.14 0.36 OS6 0.06 0.12 OSS 0.70 0.18 0.23 OS6 0.74 0.23 0.34 0.6S Q82 0.26 0.29 0.67 Q4 ~ 0.06 0.26 0.22 0.12 0.12 0.31 0.27 0.19 O.OS 0.32 0.30 0.16 0.08 0.34 0.33 0.14 0.10 0.43 0.41 Q26 B 0.04 0.06 QO9 O.1S O.SI Q21 O.t7 0.47 0.67 0.24 0.28 O
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From page 106... ...
TABLE 9 dl/B.~0.7S r-498 r- 164 r-81 r-36 r-21 kno c~r lr kc ltno c ~lr Isc 100 cv Ir Ise h~o cv lr Isc h~o cv Ir lse 0.1 ~ 0.08 1.00 0.62 0.30 0.07 1.00 O.S7 Q24 0.08 1.00 0.41 0.29 0.11 1.00 O.S9 0.3S 0.12 1.00 0.62 0.30 ~0.76 0.00 0.42 0.61 0.84 0.00 O.SS Q69 0.84 0.00 0.61 O.S9 Q86 0.00 0.47 O S6 0.82 0.01 0.49 0.6S 0.2 a 0.11 0.93 0.68 0.2S 0.11 O.9S 0.81 0.2S 0.14 0.98 0.89 0.21 0.11 Q89 0.77 0.2S 0.08 0.8S 0.70 0.20 B 0.42 0.01 0.12 0.46 0.67 Q01 0.10 O.S8 0.70 0.00 0.12 0.64 0.86 0.(~5 Q17 Q73 0.8S O.OS 0.14 0.70 0.3 a 0.16 0.71 0.41 0.19 0.12 Q63 0.49 0.24 0.08 0.6S O.S3 0.18 0.06 0.61 OS8 0.17 0.09 O.S9 0.47 0.20 B 0.30 0.19 0.26 0.40 0.66 0.16 0.28 O
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From page 107... ...
TABLE 1 0 Bounds on Shape Parameter Values Achieving Error Probabilities < .1 5 a.
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From page 108... ...
TABLE 1 1 n= 7 0.90 0.80 n= 96 n= 32 n= 16 n= 7 n=4 n= 108 n= 36 n= 18 n= 8 r = 87 r = 29 r = 15 r = 7 r = 4 r = 87 r = 29 r = 15 r = 7 r = 4 kno Ise kno Ise kno Ise kno lse lmo Ise lcno lse kno Isc Icoo Ise Icno Ise kno Ise 0.1 a 0.05 0.22 0.07 0.32 0.09 0.20 0.09 0.22 0.06 0.27 0.05 0.28 0.11 0.31 0.03 0.32 0.07 0.23 0.1 1 0.44 B 0.72 0.63 0.84 0.66 0.83 0.64 0.80 0.70 0.93 0.69 0.73 0.6S 0.82 0.64 0.86 0.70 0.87 0.65 0.87 0.49 0.2a 0.08 0.24 0.09 0.21 0.10 0.27 0.05 0.2S 0.08 0.27 0.21 0.38 0.11 0.28 0.07 0.22 0.09 0.25 0.08 0.39 B O.S1 0.59 0.76 0.58 0.74 0.66 0.85 0.78 0.84 0.71 0.58 O.S8 0.72 0.61 0.80 0.67 0.79 0.71 0.91 0.67 0.3a 0.07 0.18 0.06 0.17 0.13 0.19 0.13 0.20 O.OS 0.16 0.08 0.24 0.08 0.21 0.06 0.2S 0.11 0.17 0.11 0.32 B 0.24 0.34 0.66 0.63 0.72 0.67 0.82 0.7S 0.74 O.S8 0.23 0.31 O.S8 0.61 0.71 0.6S 0.80 0.65 0.81 0.66 0.4a 0.11 0.18 0.05 0.13 0.09 0.19 0.10 0.20 0.06 0.21 0.11 0.26 0.08 0.19 0.12 0.19 0.08 0.21 O.OS 0.25 B 0.08 0.2S 0.48 0.52 0.61 0.63 0.76 0.69 0.86 0.67 0.09 0.17 O.SO 0.43 O.S2 0.51 0.72 0.56 0.84 0.76 O.Sa 0.10 0.16 0.07 0.17 0.07 0.13 0.120.12 0.070.16 0.11 0.16 0.08 0.18 0.170.19 0.07 0.16 0.10 0.29 B 0.03 0.11 0.33 0.37 0.48 0.48 0.75 0.70 0.81 0.66 0.01 0.10 0.18 0.33 O.S6 0.56 0.69 0.60 0.72 0.56 0.6a 0.13 0.17 0.10 0.13 0.07 0.13 0.09 0.11 0.11 O.1S 0.14 0.16 0.08 0.18 0.13 0.14 0.17 0.16 0.09 0.24 B 0.00 0.03 0.16 0.30 0.32 0.36 O.S6 O.S5 0.78 0.66 0.00 0.06 0.20 0.29 0.37 0.42 0.55 O.S8 0.73 0.58 0.7a 0.09 0.09 0.12 0.07 0.06 0.09 0.08 0.17 0.14 0.24 0.08 0.13 0.06 0.10 0.07 O.1S 0.13 0.17 0.11 0.21 B 0.00 0.00 0.08 0.17 0.36 0.38 0.59 O.S1 0.73 0.68 0.00 0.04 0.07 0.20 0.27 0.29 0.57 O.S4 0.57 0.49 0.8a 0.14 O.1S 0.11 0.09 0.13 0.10 0.11 0.13 0.16 0.14 0.03 0.12 O.OS 0.11 0.04 0.07 0.12 0.15 0.10 0.19 ~0.000.01 0.020.06 0.23 0.25 O.S1 0.61 0.63 O.S4 0.00 0.01 0.06 0.16 0.200.40 O.S1 O.SO 0.69 0.63 O.9a 0.10 0.11 0.14 0.12 0.12 0.14 0.09 0.09 0.12 0.18 -Z71~717- 0.06 0.13 0.09 0.14 0.10 O.IS 0.12 0.22 B 0.00 0.00 0.01 0.08 0.10 0.21 0.34 0.43 O.SO 0.48 0.00 0.41 0.01 0.06 0.09 0.27 0.29 0.42 0.62 0.50 1.0a 0.10 0.08 0.12 -- 0.12 0.20 0.16 0.10 0.10 0.09 0.10 0.10 0.16 0.13 0.13 0.09 0.07 0.13 0.19 0.16 0.19 B 0.00 0.00 0.00 0.03 0.05 0.13 0.40 0.4S 0.42 0.38 0.00 0.00 0.00 0.03 0.09 0.19 0.32 0.37 0.53 O.S9 1.1 a 0.06 O.OS 0.07 0.07 0.12 0.12 0.08 0.11 0.09 O.1S 0.10 0.10 0.06 0.11 0.09 0.13 0.08 0.14 0.14 0.16 B 0.00 0.00 0.01 0.04 0.06 0.11 0.30 0.38 0.47 0.46 0.00 0.00 0.00 0.02 0.04 0.15 0.27 0.33 O.S3 0.47 1.20 0.06 0.07 0.07 0.07 0.12 0.09 0.10 0.12 0.10 0.12 0.09 0.10 0.08 0-.09 0.07 0.11 0.09 0.10 0.13 0.17 ~0.00 0.00 0.00 0.00 0.00 0.04 0.24 0.33 0.48 0.45 0.00 0.00 0.00 0.00 0.04 0.11 0.17 0.27 0.40 0.43 1.3a O.Og 0.09 0.10 0.11 0.06 0.08 0.11 0.13 0.09 0.08 0.09 0.07 0.06 0.08 0.09 0.11 0.12 0.13 0.11 0.15 B 0.00 0.00 0.00 0.01 0.01 0.07 0.14 0.18 0.3S 0.39 0.00 0.00 0.00 0.02 0.00 0.04 0.22 0.28 0.44 0.38 1.4a 0.10 0.09 0.06 0.06 0.09 0.10 ~0.09 0.12 0.08 0.08 0.05 0.07 0.08 0.08 0.12 0.14 ~ O.1S O.1S 0.09 0.21 B 0.00 0.00 0.00 0.00 0.00 0.03 ~0.12 0.16 0.19 0.28 0.00 0.00 0.
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From page 109... ...
TILE 1 1 coned.
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From page 110... ...
T1VE3I,E ~ ~ COnt'6.
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From page 111... ...
FIGURE ~ First simulation. 1 ilk ~ ~ / ~ A-1 O · A=0.4 {} A=0.7 o FIGURE 2 Second simulation.
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From page 112... ...
FIGURE 3 Third simulation. Am Zi1 - _ O ~ o -'-IS -- ,-_ FIGURE 4 Fourth simulation.
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From page 113... ...
FIGURE 5 Fifth simulation. 1 FIGURE 6 Sixth simulation.
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From page 114... ...
FIGURE 7 First plot.
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From page 115... ...
FI~9 Third plot. 10 1 g 5 o odor FIN 10 fog plot 10 1 o o 0.1 0.01 I I I I IIIII .
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From page 116... ...
FIGURE 1 1 Fifth plot.
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From page 117... ...
FIGURE 13 Seventh plot. 10 1 o :o ._ 4 0.1 0.01 0.1 FIGURE 14 Eighth plot.
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From page 118... ...
FIGURE ~ 5 Level curves for TTTR when a = ~ = .
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From page 119... ...
Bain, L.~., and M Engelhardt 1 99 1 Statistical Analysis of Reliability and Life Testing Models, Theory and Methods.
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From page 120... ...
Lindley, eds., Accelerated Life Testing and Experts ' Opinions in Reliability. Lindley, Amsterdam: North-Holiand.
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From page 121... ...
Klefsjo, B 1980 Some tests against aging based on the total time on test transform.
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From page 122... ...
1990 Accelerated Testing: Statistical Models, Test Plans and Data Analyses. New York: John Wiley and Sons.
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From page 123... ...
Dannemiller 961 The robustness of life testing procedures derived from the exponential distribution. Technometrics 3:29-49.
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