Suppose that $X$ is a random variable with distribution $G$. Let $H(X;\theta)$ be a parametric function with $\theta \in \Theta \subset {\mathbb R}^p$. I want to maximize the function

$$\varphi(\theta) = \int H(x;\theta) dG(x).$$ I know that this has some relationship with the Kullback Liebler divergence and the Entropy, but I don't know how to proceed to maximize this function. Any hints would be appreciated.

  • $\begingroup$ For me the question is unclear? What is the expected result? If the functions are not given how can you take an integral? But maybe I'm missing something $\endgroup$ – keiv.fly Jul 18 at 17:33
  • $\begingroup$ @keiv.fly I just want to find some general procedure as I do not want to fix H and G for the moment. Hopefully related to maximizing the Entropy. Of course, if $\varphi$, then that's a Calculus I matter. $\endgroup$ – Lex Jul 18 at 17:36

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