# The maximimum of a Functional

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Many thanks,

T.

question_max_functional.pdf

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• 4 months later...

Under what conditions can we state the following?

$\max_{\theta>0} F \left( \theta \right)= \int_{\rho_{min}}^{\rho_{max}} g \left( \rho \right)\pi\left(\rho,\widehat{\theta \left( \rho \right)} \right)d\rho$

where,

$F\left ( \theta \right )=\int_{\rho_{min}}^{\rho_{max}}g\left(\rho \right )\pi\left(\rho,\theta \right)d\rho$

and

$\widehat{\theta\left( \rho \right)}$ is the argument that maximize $\pi(\rho,\theta)$ with respect to $\theta$

Let $\rho_{min}=0$ and $\rho_{max}=1$. Assume also that $g(\theta)$ and $\pi(\rho,\theta)$ are proper unimodal densities of $\rho$ and the parameter $\theta>0$

Alternatively, we can state the problem in the following way: Determine the conditions that satisfy

$\max_{\theta>0} F \left( \theta \right)= \int_{0}^{1} g \left( \rho \right)\max_{\theta>0}(\pi\left(\rho,\theta \right))d\rho$

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