Template:MLE lloyd-l

Maximum Likelihood Estimators
For the $${{k}^{th}}$$  stage:


 * $${{L}_{k}}=const.\text{ }R_{k}^{{(1-{{R}_{k}})}^{{{n}_{k}}-{{S}_{k}}}}$$

And assuming that the results are independent between stages:


 * $$L=\underset{k=1}{\overset{N}{\mathop \prod }}\,R_{k}^{{(1-{{R}_{k}})}^{{{n}_{k}}-{{S}_{k}}}}$$

Then taking the natural log gives:


 * $$\Lambda =\underset{k=1}{\overset{N}{\mathop \sum }}\,{{S}_{k}}\ln \left( {{R}_{\infty }}-\frac{\alpha }{k} \right)+\underset{k=1}{\overset{N}{\mathop \sum }}\,({{n}_{k}}-{{S}_{k}})\ln \left( 1-{{R}_{\infty }}+\frac{\alpha }{k} \right)$$

Differentiating with respect to $${{R}_{\infty }}$$  and  $$\alpha ,$$  yields:


 * $$\frac{\partial \Lambda }{\partial {{R}_{\infty }}}=\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{{{R}_{\infty }}-\tfrac{\alpha }{k}}-\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{{{n}_{k}}-{{S}_{k}}}{1-{{R}_{\infty }}+\tfrac{\alpha }{k}}$$


 * $$\frac{\partial \Lambda }{\partial \alpha }=-\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{\tfrac{k}}{{{R}_{\infty }}-\tfrac{\alpha }{k}}+\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{\tfrac{{{n}_{k}}-{{S}_{k}}}{k}}{1-{{R}_{\infty }}+\tfrac{\alpha }{k}}$$

Rearranging Eqns. (R1) and (alpha1) and setting equal to zero gives:


 * $$\frac{\partial \Lambda }{\partial {{R}_{\infty }}}=\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{\tfrac-\left( {{R}_{\infty }}-\tfrac{\alpha }{k} \right)}{\tfrac{1}\left( {{R}_{\infty }}-\tfrac{\alpha }{k} \right)\left( 1-{{R}_{\infty }}+\tfrac{\alpha }{k} \right)}=0$$


 * $$\frac{\partial \Lambda }{\partial \alpha }=-\underset{k=1}{\overset{N}{\mathop \sum }}\,\frac{\tfrac{1}{k}\tfrac-\left( {{R}_{\infty }}-\tfrac{\alpha }{k} \right)\tfrac{1}{k}}{\tfrac{1}\left( {{R}_{\infty }}-\tfrac{\alpha }{k} \right)\left( 1-{{R}_{\infty }}+\tfrac{\alpha }{k} \right)}=0$$

Eqns. (R2) and (alpha2) can be solved simultaneously for $$\widehat{\alpha }$$  and  $${{\hat{R}}_{\infty }}$$. It should be noted that a closed form solution does not exist for either of the parameters; thus they must be estimated numerically.