Template:Confidence limits for the MCF

Confidence Limits for the MCF
Upper and lower confidence limits for $$M({{t}_{i}})$$  are:


 * $$\begin{align}

& {{M}_{U}}({{t}_{i}})= {{M}^{*}}({{t}_{i}}).{{e}^{\tfrac{{{K}_{\alpha }}.\sqrt{Var[{{M}^{*}}({{t}_{i}})]}}{{{M}^{*}}({{t}_{i}})}}} \\ & {{M}_{L}}({{t}_{i}})= \frac{{{M}^{*}}({{t}_{i}})} \end{align}$$

where $$\alpha $$  ( $$50%<\alpha <100%$$ ) is  confidence level,  $${{K}_{\alpha }}$$  is the  $$\alpha $$  standard normal percentile and  $$Var[{{M}^{*}}({{t}_{i}})]$$  is the variance of the MCF estimate at recurrence age  $${{t}_{i}}$$. The variance is calculated as follows:


 * $$Var[{{M}^{*}}({{t}_{i}})]=Var[{{M}^{*}}({{t}_{i-1}})]+\frac{1}{r_{i}^{2}}\left[ \underset{j\in {{R}_{i}}}{\overset{}{\mathop \sum }}\,{{\left( {{d}_{ji}}-\frac{1}{{{r}_{i}}} \right)}^{2}} \right]$$

where $${r}_{i}$$  is defined in the equation of the survivals,  $${{R}_{i}}$$   is the set of the units that have not been suspended by  $$i$$  and  $${{d}_{ji}}$$  is defined as follows:


 * $$\begin{align}

& {{d}_{ji}}= 1\text{ if the }{{j}^{\text{th }}}\text{unit had an event recurrence at age }{{t}_{i}} \\ & {{d}_{ji}}= 0\text{  if the }{{j}^{\text{th }}}\text{unit did not have an event reoccur at age }{{t}_{i}} \end{align}$$

Example 2:

Example 3: