MLE Analysis of Right Censored Data - Revision history

Lisa Hacker: Replaced content with 'Category: For Deletion'

2012-08-03T08:39:50Z

Replaced content with 'Category: For Deletion'

← Older revision		Revision as of 08:39, 3 August 2012
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	~~==== MLE Analysis of Right Censored Data ====~~		[[Category: For Deletion]]

	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~~]].

Harry Guo at 22:41, 16 March 2012

2012-03-16T22:41:58Z

← Older revision		Revision as of 22:41, 16 March 2012
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	{{~~MLE Parameter Est 4 Complete Data~~ (~~section~~)}}		==== 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]].

Harry Guo at 22:40, 16 March 2012

2012-03-16T22:40:42Z

← Older revision		Revision as of 22:40, 16 March 2012
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	{{MLE ~~Analysis of Right Censored~~ Data}}		{{MLE Parameter Est 4 Complete Data (section)}}

Harry Guo: Created page with '{{MLE Analysis of Right Censored Data}}'

2012-03-16T22:39:58Z

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