If no failure is observed during a test on n units over a duration of T, then:

Because we may have had a small or restricted test, this cannot realistically be true. Moreover, this estimate does not take into account the number on test and the test duration. Had the test been continued, it is highly likely that a failure would eventually have occurred.

An upper level estimate for both one-sided and two-sided confidence limits can be obtained with r = 0. However, the lower limit for the two-sided confidence limit cannot be obtained with r = 0 (because it is defined only for r > 0).

It is possible to relax this limitation by conservatively assuming that a failure occurs in the very next instant. Then, r = 1 can be used to evaluate the lower two-sided confidence limit. This conservative modification, although sometimes used to allow a complete statistical analysis, lacks firm statistical basis.

Therefore, when there are no failures, we can find only an upper 100 (1 − α / 2) confidence limit for λ. Alternatively, we can find only a lower 100 (α / 2) confidence limit for MTTF. The probability statement corresponding to the upper confidence limit on λ is as follows (based on the above approximation):

However, as mentioned in Chi-Square Fundamentals , a Chi-square (χ2) distribution with two degrees of freedom is equivalent to the exponential distribution with a mean of 2. Therefore:

If there is only one unit being tested, then the upper confidence limit on λ is:

Similarly, the lower 100(1 - α) confidence limit for MTTF is given by:

If there is only one unit being tested, then the lower confidence limit on MTTF is:

As shown in the above equations, Chi-square tables are not needed for evaluating these confidence limits. The simple formula obtained using the exponential distribution can be used instead.

The 50% zero failure estimate is often used as a point estimate for λ. This should be interpreted very carefully. It is a value of λ that makes the likelihood of obtaining zero failures in the given test similar to the chance of having a coin that has been flipped landing “heads” side up. We are not really 50% confident of anything; we have just picked a λ that will produce zero failures 50% of the time.

The following table presents the lower bounds for MTTF at various confidence levels for a one-unit test when no failures are observed during the testing period (T).