  In this note we introduce the notion of a $C^k$-diffeological statistical
model, which allows us to apply the theory of diffeological spaces to (possibly
singular) statistical models. In particular, we introduce a class of almost
2-integrable $C^k$-diffeological statistical models that encompasses all known
statistical models for which the Fisher metric is defined. This class contains
a statistical model which does not appear in the Ay-Jost-L\^e-Schwachh\"ofer
theory of parametrized measure models. Then we show that for any positive
integer $k$ the class of almost 2-integrable $C^k$-diffeological statistical
models is preserved under probabilistic mappings. Furthermore, the monotonicity
theorem for the Fisher metric also holds for this class. As a consequence, the
Fisher metric on an almost 2-integrable $C^k$-diffeological statistical model
$P \subset {\cal P}({\cal X})$ is preserved under any probabilistic mapping $T:
{\cal X}\leadsto {\cal Y}$ that is sufficient w.r.t. $P$. Finally we extend the
Cram\'er-Rao inequality to the class of 2-integrable $C^k$-diffeological
statistical models.
