Logistic Confidence Bounds Example

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This example appears in the Reliability growth reference.

For the data given in the Reliability Data - Logistic Model example, calculate the 2-sided 90% confidence bounds under the Logistic model for the following:

1. The parameters [math]\displaystyle{ b\,\! }[/math] and [math]\displaystyle{ k\,\! }[/math].
2. Reliability at month 5.

Solution

1. The values of [math]\displaystyle{ \hat{b}\,\! }[/math] and [math]\displaystyle{ \hat{k}\,\! }[/math] that were estimated from the least squares analysis in the reliability data example are:

   [math]\displaystyle{ \begin{align} \widehat{b}= & 3.3991 \\ \widehat{\alpha }= & 0.7398 \end{align}\,\! }[/math]

   Thus, the 2-sided 90% confidence bounds on parameter [math]\displaystyle{ b\,\! }[/math] are:

   [math]\displaystyle{ \begin{align} {{b}_{lower}}= & 2.5547 \\ {{b}_{upper}}= & 4.5225 \end{align}\,\! }[/math]

   The 2-sided 90% confidence bounds on parameter [math]\displaystyle{ k\,\! }[/math] are:

   [math]\displaystyle{ \begin{align} {{k}_{lower}}= & 0.6798 \\ {{k}_{upper}}= & 0.7997 \end{align}\,\! }[/math]

2. First, calculate the reliability estimation at month 5:

   [math]\displaystyle{ \begin{align} {{R}_{5}}= & \frac{1}{1+b{{e}^{-5k}}} \\ = & 0.9224 \end{align}\,\! }[/math]

   Thus, the 2-sided 90% confidence bounds on reliability at month 5 are:

   [math]\displaystyle{ \begin{align} {{[{{R}_{5}}]}_{lower}}= & 0.8493 \\ {{[{{R}_{5}}]}_{upper}}= & 0.9955 \end{align}\,\! }[/math]

   The next figure shows a graph of the reliability plotted with 2-sided 90% confidence bounds.