Partitions without small blocks and $r$-associated Bell polynomials in a parametric model: probabilistic-statistical analysis

On the set of all partitions of an $n$-element set $X_n = \{1, 2,\dots, n\}$ into blocks with sizes exceeding the number $r\geqslant 0$ a probability measure is defined such that for each partition with $k$ blocks its probability is proportional to $\theta^k$, where $\theta>0$ is the parameter of the measure. The asymptotic normality of the number of blocks in a random partition of $X_n$ in this model is proved, a statistical test for the uniformity hypothesis $H_0 :\, \theta = 1$ against the alternatives $H_1 :\, \theta \ne 1$ is constructed.