MLE Analysis of Right Censored Data

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,...,θ_k are the unknown parameters which need to be estimated from R observed failures at T₁,T₂...T_R, and M observed suspensions at S₁,S₂ ... S_M, 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.