I'm trying to estimate the parameters of a Pareto distribution (actually the paretian tail of a generic distribution) via Metropolis-Hastings.
The problem is that the loglikelihood,
$$ l(\alpha, x_m) = n\log(\alpha) + n\alpha\log(x_m) - (1 + \alpha)\sum\log(x),$$
is monotonically increasing with $x_m$, so that the greater the value of $x_m$, the greater the value of the likelihood function.
Thus, the chain for the parameter $x_m$ never converges. Any idea about overcoming such a problem?
EDIT: the following picture illustrates the kind of distribution I'm dealing with. In this case, $x_m = 260$, and $\alpha = 2.5$.