Logistic

The Logistic reliability growth model has an S-shaped curve and is given by Kececioglu [3]:


 * $$R = \frac{1}{1+be^{-kt}}, b > 0, k > 0, T \simeq 0$$

where $$b\,\!$$ and $$k\,\!$$ are parameters. Similar to the analysis given for the Gompertz curve, the following may be concluded:

 The point of inflection is given by:
 * $${{T}_{i}}=\frac{\ln (b)}{k}\,\!$$

 When $$b>1\,\!$$, then $${{T}_{i}}>0\,\!$$ and an S-shaped curve will be generated. However, when $$0 The value of $$R\,\!$$ is equal to 0.5 at the inflection point.  

Parameter Estimation
In this section, we will demonstrate the parameter estimation method for the Logistic model using three examples for different types of data.

Confidence Bounds
Least squares is used to estimate the parameters of the following Logistic model.


 * $$\ln (\frac{1}-1)=\ln (b)-k{{T}_{i}}\,\!$$

Thus, the confidence bounds on the parameter $$b\,\!$$ are given by:


 * $$b=\hat{b}{{e}^{{{t}_{n-2,\alpha /2}}SE(\ln \hat{b})}}\,\!$$

where:


 * $$\begin{align}

SE(\ln \hat{b})&=\sigma \cdot \sqrt{\frac{\underset{i=1}{\overset{n}{\mathop{\sum }}}\,{{({{T}_{i}})}^{2}}}{n\cdot {{S}_{xx}}}},\ \ \ {{S}_{xx}}=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{({{T}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,{{T}_{i}} \right)}^{2}} \\ \\ \sigma &=\sqrt{SSE/(n-2)} \end{align}\,\!$$

and the confidence bounds on the parameter $$k\,\!$$ are:


 * $$k=\hat{k}\pm {{t}_{n-2,\alpha /2}}SE(\hat{k})\,\!$$

where:


 * $$SE(\hat{k})=\frac{\sigma }{\sqrt},\ \ {{S}_{xx}}=\left[ \underset{i=1}{\overset{n}{\mathop \sum }}\,{{({{T}_{i}})}^{2}} \right]-\frac{1}{n}{{\left( \underset{i=1}{\overset{n}{\mathop \sum }}\,{{T}_{i}} \right)}^{2}}\,\!$$

Since the reliability is always between 0 and 1, the logit transformation is used to obtain the confidence bounds on reliability, which is:


 * $$CB=\frac{{{{\hat{R}}}_{i}}+(1-{{{\hat{R}}}_{i}}){{e}^{\pm {{z}_{\alpha }}{{{\hat{\sigma }}}_{R}}/\left[ {{{\hat{R}}}_{i}}(1-{{{\hat{R}}}_{i}}) \right]}}}\,\!$$