On sufficient statistics for families of distribution with variable support

Let $X_1,\dots,X_n$ be independent random vectors with density of distribution $f(x-\theta)$, where
$$ f(x-\theta)=\exp\{\sum_{i=1}^lc_i(\theta)f_i(x)+r(x-\theta)\}h(x)c_0(\theta), $$
if $x\in H+\theta$, and $f(x-\theta)=0$ if $x\bar\in H+\theta$. It is supposed, that function $r$ is constant on some open sets $H_1,\dots,H_k$ and $H=\bigcup_{i=1}^kH_i$. This condition gives possibility function $f$ to have discontinuities into support. Sufficient statistics are considered in that situation.