2.5.10.2 Algorithm: Demonostration Test Plan

1. Accept the assumed distribution (including its shape or scale parameter) and the reliability target, and convert the target into a target parameter \(\theta_0\) for the selected distribution.
2. Form the exact binomial lower-confidence-bound equation:
   \( \sum_{k=0}^{m} \binom{n}{k} \, p(\theta_0)^k \, \big[1 - p(\theta_0)\big]^{n-k} = \alpha \)
   where \(\theta\) is the parameter to be demonstrated (scale, location, or an equivalent derived parameter), and \(\theta_0\) is the target value of \(\theta\); \(m\) is the maximum number of failures allowed for a pass
3. Solve the unknown \(t\) or \(n\) using the reliability function \(R(t|\theta_0)\) of the selected distribution.

This procedure ensures that a passing test demonstrates the one-sided claim \(\theta \geq \theta_0\) with confidence \(1- \alpha \).

Exact Binomial Lower-Confidence-Bound Equation Construction

Converting User Goals to a Common Parameter

The reliability target can be specified in four ways: scale/location, percentile, reliability at a time, or MTTF. Each of these is converted to a common target parameter \(\theta_0\) using the inverse relationships of the chosen distribution:

• A reliability-at-time target \(R\) at time \(t_0\) is converted to the parameter value \(\theta_0\) that satisfies \(R(t_0 \mid \theta_0) = R\).
• A percentile target \(t_p\) at cumulative failure probability \(p\) is converted to the parameter value \(\theta_0\) that satisfies \(F(t_p \mid \theta_0) = p\), where \( F=1-R \).
• An MTT target is converted to the corresponding scale or location parameter via the mean formula of the chosen distribution.

Once \(\theta_0\) is determined, the same binomial design equation is applied.

Failure Count as a Binomial Random Variable

Consider a test in which \(n\) units are each placed on test for a fixed duration \(t\). At the end of the test, each unit has either failed or survived.

The probability that a single unit fails during the test is:

\[ p(\theta) = 1 - R(t \mid \theta) \]

Conditional on the true parameter \(\theta\), the failure indicators for the \(n\) units are independent Bernoulli trials with common failure probability \(p(\theta)\). Therefore, the total number of failures \(X\) follows a binomial distribution:

\[X \sim \text{Binomial}\big(n,\; p(\theta)\big)\]

The test is declared a "pass" if the observed number of failures does not exceed the acceptance number:

\[X \le m\]

If \(X > m\), the demonstration fails and no claim is made.

Confidence-Bound Construction

A one-sided lower confidence bound for the parameter \(\theta\) is constructed. The bound is obtained by inverting the binomial cumulative distribution function: the lower bound \(\theta_L\) is the parameter value such that, when the true parameter equals \(\theta_L\), the probability of observing a passing result is exactly equal to the specified significance level \(\alpha\):

\[P\big(X \le m \mid \theta = \theta_L\big) = \alpha\]

Expanding the binomial cumulative probability:

\[\sum_{k=0}^{m} \binom{n}{k} \, p(\theta_L)^k \, \big[1 - p(\theta_L)\big]^{n-k} = \alpha\]

where \( p(\theta_L) = 1 - R(t \mid \theta_L) \)

For the demonstration plan, the lower bound is set to the target:

\[\theta_L = \theta_0\]

Thus the design equation is:

\[ \sum_{k=0}^{m} \binom{n}{k} \, p(\theta_0)^k \, \big[1 - p(\theta_0)\big]^{n-k} = \alpha \]

If this equation holds, then observing a passing result (\( X \leq m \)) demonstrates that the lower \(1- \alpha \) confidence bound on \(\theta\) is at least \(\theta_0\).

Solving the Design Equation

The design equation is solved for whichever quantity the user did not fix.

Zero-failure plans (m = 0)

When no failures are allowed, the summation collapses to a single term:

\[P(X = 0 \mid \theta_0) = \big[1 - p(\theta_0)\big]^n = \big[R(t \mid \theta_0)\big]^n = \alpha\]

Taking logarithms:

\[n \, \ln\big[R(t \mid \theta_0)\big] = \ln \alpha\]

Given \(n\), solve for \(t\):

\[ R(t \mid \theta_0) = \alpha^{1/n}\]

The required test time is obtained by inverting the reliability function of the chosen distribution.

Given \(t\), solve for \(n\):

\[ n = \frac{\ln \alpha}{\ln\big[R(t \mid \theta_0)\big]}\]

Because \(n\) must be an integer, the tool rounds up to the next integer. This upward rounding makes the actual confidence slightly higher than the target because \(P(X=0 \mid \theta_0) = R(t \mid \theta_0)^n\) decreases as \(n\) increases.

Plans allowing failures (m >= 1)

When \(m \geq 1\), the full binomial sum must be used:

\[\sum_{k=0}^{m} \binom{n}{k} \, p(\theta_0)^k \, \big[1 - p(\theta_0)\big]^{n-k} = \alpha\]

When a sample size is calculated, the result must be an integer. The theoretical \(n\) is therefore rounded upward. Because the pass probability at \( \theta_0 \) decreases as \(n</math increases, with <math>t\) and \(m\) fixed, rounding \(n\) upward makes the actual confidence level slightly greater than the target \(1- \alpha \).

When a test time \(t\) is calculated given an integer sample size \(n\), no rounding is needed, and the actual confidence level equals the target exactly.

Probability of Passing Curve

The Probability of Passing (POP) curve plots the probability of a pass as a function of the true parameter value. For a true parameter \( \theta \):

\[P_{\text{pass}}(\theta) = \sum_{k=0}^{m} \binom{n}{k} \, p(\theta)^k \, \big[1 - p(\theta)\big]^{n-k}\]

where \( p(\theta) = 1 - R(t \mid \theta)\).

Note that \(P_pass(\theta_0) = \alpha\) by construction, so the curve passes through the significance level at the target.