Logistic Confidence Bounds Example

New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images and more targeted search.

As of January 2024, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest references at RGA examples and RGA reference examples.

---

This example appears in the Reliability Growth and Repairable System Analysis 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.