Let the observation $x$ has the following density:

begin{equation}

p(x|theta)=

begin{cases}

frac{1}{theta},& text{if } 0 < x leq theta\

0, & text{otherwise}

end{cases}

end{equation}

and the r.v $theta$ has:

begin{equation}

p(theta)=

begin{cases}

theta exp(-theta),& text{if } 0 leq theta\

0, & text{otherwise}

end{cases}

end{equation}

I try to find MAP and MMSE estimation of the parameter $theta$.

MAP:

$hat{theta} = argmax_{theta}$ $p(theta|x) = argmax_{theta}$ $p(x|theta)p(theta) = argmax_{theta}$ $exp(-theta)$ which gives $hat{theta} = 0$. But it seems $p(x|theta)$ is not well defined at $theta = 0$. Am I missing something here?

MMSE:

$hat{theta} = E(theta | x) = int_{0}^{infty} theta exp(-theta) ,dtheta = 1$. I want to verify if it’s correct.