Discrete examples of sufficient or non-sufficient statistic $$\left| X\right| $$ X for unknown $$\mathbf {\theta }$$ θ via conditioning

Mukhopadhyay (Calc Stat Assoc Bul 66:55–62, 2014a), illustrated non-sufficiency of a continuously distributed statistic $$\left| X\right| $$ X via conditioning arguments. In a continuous case, however, finer arguments behind the success of conditioning often remains out of reach for some beginners. Since publication of Mukhopadhyay (2014a), queries poured in over the years seeking analogous clarifications with the help of discrete examples. In discrete situations, calculations of conditional probabilities appear more tangible and transparent. In this paper, we begin with a general construction and then focus on explicit discrete probability models leading to a decisive conclusion of sufficiency or non-sufficiency of $$\left| X\right| $$ X for an unknown parameter $$\theta $$ θ when X has a probability mass function. The discrete constructions from this paper and the continuous analogs from Mukhopadhyay (2014a) form a solid set of basic supplements to Mukhopadhyay and Banerjee (Metron 71:33–38, 2013).