Warning
After hours of trying, it has been proven (thanks, @leonbloy) that my attempt at a solution contained lots of mistakes. Maybe the correct answer is that there is no solution, but I don't know!
Homework question
Let $X_1, ..., X_n$ be a random sample from a distribution with pdf:
$f_X(x | \lambda, \theta) = \lambda \, e^{-\lambda(x - \theta)} \, \mathbf{1}_{\{x \geq \theta \}}$
The (independent) prior distributions of $\lambda$ and $\theta$ are:
$ \begin{cases} p(\lambda) = \frac{1}{\lambda^2} \, \mathbf{1}_{\{\lambda > 1\}} \\ p(\theta) = \frac{1}{\pi(1+\theta^2)} \, \mathbf{1}_{\{\theta \in \mathbf{R}\}} \end{cases} $
The goal is to find Bayesian estimates under quadratic loss of $\lambda$ and $\theta$.
First approach
What we want is to find $\mathbf{E}[p(\lambda | \mathbf{x})]$ and $\mathbf{E}[p(\theta | \mathbf{x})]$.
The posterior distribution is:
$ p(\lambda, \theta \, | \, \mathbf{x}) \propto p(\theta,\lambda) f(\mathbf{x} \, | \, \lambda, \theta) $
And then the marginal posterior distributions are:
$ \begin{cases} p(\lambda | \mathbf{x}) =\int_{-\infty}^{x_{(1)}} p(\lambda, \theta \, | \, \mathbf{x}) \, d\theta \\ p(\theta | \mathbf{x}) = \int_1^\infty p(\lambda, \theta \, | \, \mathbf{x}) \, d\lambda \end{cases} $
Trying to answer the question
If I'm not wrong:
$ p(\lambda, \theta \, | \, \mathbf{x}) \propto \underbrace{\frac{1}{\pi\lambda^2(1+\theta)^2}}_{p(\lambda,\theta)} \, \underbrace{\lambda^n e^{-\lambda(S-n\theta)}}_{ f(\mathbf{x} \, | \, \lambda, \theta)} \qquad \left(S = \sum_{i=1}^n x_i\right) $
I think that the normalizing constant is:
$ c_n = \frac{1}{\pi} \int_{-\infty}^{x_{(1)}} \frac{1}{1+\theta^2} \int_{1}^\infty \lambda^{n-2} e^{-\lambda(S-n\theta)} \, d\lambda \, d\theta = \frac{1}{\pi} \int_{-\infty}^{x_{(1)}}\frac{\Gamma(n-1,S-n\theta)}{(S-n\theta)^{n-1}(1+\theta^2)} \, d\theta $
In this case $\Gamma(n-1,s-n\theta)$ is the Incomplete Gamma Function. Changing the order of integration doesn't help: $ c_n = \int_{1}^\infty \lambda^{n-2} e^{-\lambda S} \int_{-\infty}^{x_{(1)}} \frac{e^{n\theta\lambda}}{\pi(1+\theta^2)} \, d\theta \, d\lambda $
Of course, I am also unable to find the (approximate) marginal posterior distributions.
Thanks in advance for your help!