Logistic Confidence Bounds Example

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

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


 * 1)	The parameters $$b\,\!$$ and $$k\,\!$$.


 * 2)	Reliability at month 5.

Solution


 * 1)	The values of $$\hat{b}\,\!$$ and $$\hat{k}\,\!$$ that were estimated from the least squares analysis in the reliability data example are:


 * $$\begin{align}

\widehat{b}= & 3.3991 \\ \widehat{\alpha }= & 0.7398 \end{align}\,\!$$


 * Thus, the 2-sided 90% confidence bounds on parameter $$b\,\!$$ are:


 * $$\begin{align}

{{b}_{lower}}= & 2.5547 \\ {{b}_{upper}}= & 4.5225 \end{align}\,\!$$


 * The 2-sided 90% confidence bounds on parameter $$k\,\!$$ are:


 * $$\begin{align}

{{k}_{lower}}= & 0.6798 \\ {{k}_{upper}}= & 0.7997 \end{align}\,\!$$


 * 2) First, calculate the reliability estimation at month 5:


 * $$\begin{align}

{{R}_{5}}= & \frac{1}{1+b{{e}^{-5k}}} \\ = & 0.9224 	\end{align}\,\!$$
 * Thus, the 2-sided 90% confidence bounds on reliability at month 5 iare:


 * $$\begin{align}

{{[{{R}_{5}}]}_{lower}}= & 0.8493 \\ {{[{{R}_{5}}]}_{upper}}= & 0.9955 \end{align}\,\!$$

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