MLE Analysis of Right Censored Data: Difference between revisions
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{{ | ==== MLE Analysis of Right Censored Data ==== | ||
When performing maximum likelihood analysis on data with suspended items, the likelihood function needs to be expanded to take into account the suspended items. The overall estimation technique does not change, but another term is added to the likelihood function to account for the suspended items. Beyond that, the method of solving for the parameter estimates remains the same. For example, consider a distribution where <span class="texhtml">''x''</span> is a continuous random variable with <span class="texhtml">''pdf''</span> and <span class="texhtml">''cdf''</span>: | |||
::<math>\begin{align} | |||
& f(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ | |||
& F(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) | |||
\end{align} | |||
</math> | |||
where <span class="texhtml">θ<sub>1</sub>,θ<sub>2</sub>,...,θ<sub>''k''</sub></span> are the unknown parameters which need to be estimated from <span class="texhtml">''R''</span> observed failures at <span class="texhtml">''T''<sub>1</sub>,''T''<sub>2</sub>...''T''<sub>''R''</sub>,</span> and <span class="texhtml">''M''</span> observed suspensions at <span class="texhtml">''S''<sub>1</sub>,''S''<sub>2</sub></span> ... <span class="texhtml">''S''<sub>''M''</sub>,</span> then the likelihood function is formulated as follows: | |||
::<math>\begin{align} | |||
L({{\theta }_{1}},...,{{\theta }_{k}}|{{T}_{1}},...,{{T}_{R,}}{{S}_{1}},...,{{S}_{M}})= & \underset{i=1}{\overset{R}{\mathop \prod }}\,f({{T}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ | |||
& \cdot \underset{j=1}{\overset{M}{\mathop \prod }}\,[1-F({{S}_{j}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})] | |||
\end{align}</math> | |||
The parameters are solved by maximizing this equation. In most cases, no closed-form solution exists for this maximum or for the parameters. Solutions specific to each distribution utilizing MLE are presented in [[Appendix: Distribution Log-Likelihood Equations]]. | |||
Revision as of 22:41, 16 March 2012
MLE Analysis of Right Censored Data
When performing maximum likelihood analysis on data with suspended items, the likelihood function needs to be expanded to take into account the suspended items. The overall estimation technique does not change, but another term is added to the likelihood function to account for the suspended items. Beyond that, the method of solving for the parameter estimates remains the same. For example, consider a distribution where x is a continuous random variable with pdf and cdf:
- [math]\displaystyle{ \begin{align} & f(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & F(x;{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \end{align} }[/math]
where θ1,θ2,...,θk are the unknown parameters which need to be estimated from R observed failures at T1,T2...TR, and M observed suspensions at S1,S2 ... SM, then the likelihood function is formulated as follows:
- [math]\displaystyle{ \begin{align} L({{\theta }_{1}},...,{{\theta }_{k}}|{{T}_{1}},...,{{T}_{R,}}{{S}_{1}},...,{{S}_{M}})= & \underset{i=1}{\overset{R}{\mathop \prod }}\,f({{T}_{i}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}}) \\ & \cdot \underset{j=1}{\overset{M}{\mathop \prod }}\,[1-F({{S}_{j}};{{\theta }_{1}},{{\theta }_{2}},...,{{\theta }_{k}})] \end{align} }[/math]
The parameters are solved by maximizing this equation. In most cases, no closed-form solution exists for this maximum or for the parameters. Solutions specific to each distribution utilizing MLE are presented in Appendix: Distribution Log-Likelihood Equations.