Please see the attached file.

 

Many thanks,

 

T.

 

question_max_functional.pdf

!

Moderator Note

 

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

 

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.

Please see the updated version:

 

Under what conditions can we state the following?

 

[latex]\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[/latex]

 

where,

 

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

 

and

 

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

 

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

 

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

 

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