We prove three theorems for iid discrete randomvariables taking two values, three values, and k (3\leq k<\infty) valuesby using the technique of indicator function. Under some specifications of the probabilities, we prove that the sum is a minimal sufficient statistics for the unknown parameter of interest of the discrete random variable taking two values, three values, and k (3\leq k<\infty) values. For the dice example, a figure shows that the specifications of the six probabilities are between 0 and 1 and sum to 1, and a fair dice is possible.
The research was supported by the Fundamental Research Funds for the Central Universities (Grant Nos. 2019CDXYST0016; 2018CDXYST0024), the China Scholarship Council (Grant No. 201606055028), the National Natural Science Foundation of China (Grant No. 11671060) and the MOE Project of Humanities and Social Sciences on the West and the Border Area (Grant No. 14XJC910001).
ZHANG Yingying,RONG Tengzhong,LI Manman. When the Sum is a Minimal Sufficient Statistics for the Parameter of a Discrete Distribution with Finite Values?[J]. CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST, 2019, 35(6): 611-620.