I have came across a question regarding sufficient statistics, but I cannot understand it under this context.
Let $P$ be a finite family with densities $p_i, i = 0, \dots , k$ and for any x, let $S(x)$ be the set of pairs of subscripts $(i, j)$ for which $p_i(x) + p_j(x) > 0$. Then, the statistic \begin{equation} T(X)=\frac{p_j(X)}{p_i(X)},\quad i<j \text{ and } (i,j)\in S(X) \end{equation} is minimal sufficient. Here, $p_j(x)/p_i(x) = \infty$ if $p_i(x) = 0$ and $p_j(x) > 0$
I am confused about the settings of this question. Based on my understanding of sufficient statistics, usually, it's discussed in a way saying "The sufficient statistics of parameter $\theta$", however, there seem to be multiple distributions presenting and it is unclear about what the parameters of interest is. Can someone help explain what the question is trying to ask here and correct my understanding of sufficient statistics? Thanks a million!