The maximimum of a Functional

[Tassus]

Please see the attached file.

 

Many thanks,

 

T.

 

question_max_functional.pdfUnavailable

[imatfaal]

!

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Please put the question here on the forum. Lots of members will not read/download a file from the Net.

 

[Tassus]

Ok thank you, I hope now is fine.

 

Please, tell me if the following statement is true or false and if it is possible give me some reference.

 

Many Thanks.

 

T.

[Tassus]

Please see the updated version:

 

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

 

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