I have a function , for instance $$ \lambda(S, B)=\left(\frac{n-(S+B)}{\sigma_{\text {stat }}}\right)^{2}+\left(\frac{B^{\text {obs }}-B}{\sigma_{\text {syst }}}\right)^{2} $$ maximum likelihood estimator $$ \hat{S}=n-\theta^{\text {obs }} ,~~~\hat{B}=B^{\text {obs }} $$ can be probably taken as $$ \partial\lambda/\partial S= 0, ~~~ \partial\lambda/\partial B= 0 $$ Then, how can I get conditional maximum likelihood estimator $$ \hat{\hat{B}}(S) $$


1 Answer 1


Just treat $S$ as a known constant and maximise with respect to $B$. So you have:

$$\begin{align} \frac{\partial \lambda}{\partial B}(S,B) &= -2 \Bigg[ \frac{n-S-B}{\sigma_\text{stat}} + \frac{B^\text{obs}-B}{\sigma_\text{syst}} \Bigg], \\[12pt] \frac{\partial^2 \lambda}{\partial B^2}(S,B) &= 2 \Bigg[ \frac{1}{\sigma_\text{stat}} + \frac{1}{\sigma_\text{syst}} \Bigg] > 0. \\[12pt] \end{align}$$

This shows that $\lambda$ is strictly convex in $B$, so its maximum occurs at the unique critical point that solves:

$$\frac{n-S-\hat{B}(S)}{\sigma_\text{stat}} + \frac{B^\text{obs}-\hat{B}(S)}{\sigma_\text{syst}} = 0,$$

which gives the conditional MLE:

$$\hat{B}(S) = \frac{(n-S) \sigma_\text{syst} + B^\text{obs} \sigma_\text{stat}}{\sigma_\text{syst} - \sigma_\text{stat}}.$$

(Your value $\theta^\text{obs}$ does not appear in the original objective function and is not defined, so it is not clear where this came from in your question.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.