Minimality of a complete sufficient statistic

Abstract Bahadur (Sankhyā 18:211–224, 1957) proved that a complete sufficient statistic is minimal sufficient in the case of a statistical model given by a family of densities on a Euclidean sample space. More recently, completeness was used by Hoff (Bernoulli 29:901–928, 2023) to prove that conformal prediction procedures used in modern machine learning are Bayes-optimal in a nonparametric model. It is known that Bahadur’s theorem and proof are valid more generally as required in modern data contexts. In this paper, we state and prove Bahadur’s theorem in a form that is more general than what is typically found in the literature. The proof relies on the Lehmann-Scheffé and Rao-Blackwell theorems and is simplified compared to proofs of less general results.