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The maximimum of a Functional


Tassus
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  • 4 months later...

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]

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