Reliability Test Confidence Bounds

Content

Reliability test confidence bounds

Quantify uncertainty in exponential life tests and pass/fail demonstrations, or plan a transparent zero-event test.

Exponential life-test observations

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How to use

  1. Choose Life test for exponentially distributed time-to-failure data, Pass / fail for independent binary trials, or Test planning for a zero-failure and zero-defect demonstration.
  2. For a life test, enter total accumulated unit-hours, the integer failure count, and the stopping rule selected before testing. Failure-truncated analysis requires at least one failure.
  3. For pass/fail data, enter the integer sample size and number of defects or failures; defects cannot exceed the sample size.
  4. For planning, enter target MTBF in hours and target pass probability as a percentage. The plan assumes zero observed events.
  5. Enter confidence as a percentage strictly between 0 and 100, then select Calculate. Select Reset to restore the documented examples and Life test mode.
  6. Interpret the large canonical value as the one-sided lower claim; use the paired metrics for two-sided uncertainty and point estimates. “Unbounded” is mathematically expected for the upper MTBF after zero failures.
  7. Warnings identify invalid counts, ranges, or stopping rules. More confidence, fewer events, and smaller samples produce wider intervals or larger test plans.
  8. Do not use these models when lifetimes are not exponential, failure intensity changes with age, trials are dependent, units have unequal exposure that is not accumulated correctly, stopping rules were changed after seeing results, or competing risks/censoring require a richer survival model.

Equations used

Let C be confidence, α = 1 − C, T be accumulated unit-hours, r be failures, n be binary trials, x be defects, θ be MTBF, λ be failure rate, p be defect probability, and R = 1 − p be reliability.

θ̂ = T/r,   λ̂ = r/T
θL,time = 2T/χ²1−α/2, 2r+2,   θU,time = 2T/χ²α/2, 2r
θL,failure = 2T/χ²1−α/2, 2r,   θU,failure = 2T/χ²α/2, 2r
θL,1-sided = 2T/χ²C,νL;   λ bounds = reciprocals of θ bounds
p̂ = x/n;   pL = Beta⁻¹(α/2; x, n−x+1);   pU = Beta⁻¹(1−α/2; x+1, n−x)
pL,1 = Beta⁻¹(α; x, n−x+1);   pU,1 = Beta⁻¹(C; x+1, n−x)
R̂ = 1 − p̂;   RL = 1 − pU;   RU = 1 − pL
RL,1 = 1 − pU,1;   RU,1 = 1 − pL,1
Treq = −θ₀ ln(1 − C);   Cach,exp = 1 − exp(−Treq/θ₀)
nreq = ceil[ln(1 − C)/ln(R₀)];   Cach,bin = 1 − R₀nreq

Exact endpoint handling: For x = 0, pL,1 = 0 (and pL = 0); for x = n, pU,1 = 1 (and pU = 1). The complementary reliability endpoints follow from R = 1 − p.

Definitions, units, and limits: χ²q,ν is the q quantile of chi-square with ν degrees of freedom; νL = 2r+2 for a time-truncated test and 2r for a failure-truncated test. Beta⁻¹ is the beta-distribution quantile and endpoint cases are evaluated exactly. Hours remain hours; rates are displayed per 10⁶ unit-hours; probabilities are displayed as percentages. Sign convention: failures and defects are non-negative event counts; all time, MTBF, and rates are magnitudes.

FunctionalProduct

Assumptions: exponential tests use constant failure rate, independent identical units, correctly accumulated exposure, and a preselected stopping rule. Clopper–Pearson bounds use independent identical Bernoulli trials and are exact but conservative. Planning equations apply only when zero failures or zero defects will support the claim.

References:exponential confidence-limit equations; NIST binomial confidence intervals; MIL-HDBK-338B.