I'm facing this statistical data analysis problem, where I have to maximize a certain statistic in order to find the optimal filtering function. I'm a little bit out of practice with the mathematics needed for solving it...
Well, the statistic I got from the Neyman-Pearson Lemma is a function of the data set $\{s(t)\}$ and of some ($6$) unknown functions $S^{AB}(t)$ (symmetric under $A \leftrightarrow B$), with $A,B=1,2,3$, of the form: $$\Lambda\big(s(t)\big)=\sum_{A,B}\int_{-\infty}^{+\infty} \mathrm{d}t\ M^{AB}\big(s(t)\big)\,S^{AB}(t).$$ $M^{AB}\big(s(t)\big)$ is a known function of the data set. For the sake of simplicity of the notation, I considered continuous times and an infinite interval of integration. Of course this is not true in practice but I think it won't matter for this analysis problem. In order to evaluate the $6$ unknown functions $S^{AB}(f)$ through their maximum likelihood estimators, I'm asked to maximize $\Lambda$: $$\widehat{S^{AB}}:\qquad \max_{S^{AB}}\Lambda\big(s(t)\big).$$
Well, tese are my attempts: in order to find this maximum, I thought to find the derivative of $\Lambda\big(s(t)\big)$ with respect to $S^{AB}$ and impose it equals to zero for every $A,B$: $$ {\partial\over\partial S^{A'B'}}\Lambda\big(s(t)\big)=0.$$How could I perform this derivative? Can I consider the unknown function as an independent variable and put the derivative inside the integrand, writing: $$\sum_{A,B}\int_{-\infty}^{+\infty}\mathrm{d}t\,M^{AB}(t)\delta^{AA'}\delta^{BB'}=0$$ which does not depends on the filtering functions I'm looking for... Is it correct? How can I interpret this result?
Also, I thought to try to solve the problem with this change of variables: $$ {\partial\over\partial S^{A'B'}}\Lambda\big(s(t)\big)= \left|{\partial S^{A'B'}\over\partial t}\right|^{-1}{\partial\over\partial t}\Lambda=\left|{\partial S^{A'B'}\over\partial t}\right|^{-1}\sum_{A,B}M^{AB}(t)\,S^{AB}(t)=0$$ where I need to "guess" some templates for the filtering functions $S^{AB}$ and calculate their derivatives.
So, I'm stuck on this problem... Can someone suggest me a hint or correct what I did? Thank you, guys!
PS: I'm thinking about moving (or copying) this question on dsp.stackexchange.com but since my problems are mostly related on mathematical aspects, I think it's best suited here. Make me know what should I do! Thanks!