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On weak structural sufficiency Mehdi Shams1 Received: 18 February 2019 / Accepted: 4 September 2019 © Sociedad Matemática Mexicana 2019

Abstract In this paper, we consider sufficiency of the pair (t, π ) where π is a canonical projection which is maximal invariant and t is a maximum likelihood estimator which is weakly equivariant. We define weak structural sufficiency for (t, π ) and then we will consider the conditions under which (t, π ) is weakly structurally sufficient and investigate its properties. For the case in which the group on the parameter space is transitive, and the stabilizer group is characteristic and also for trivial-transitive spaces obtain new results on weak structural sufficiency. Keywords Topological group · Characteristic subgroup · Invariance · Weak equivariance · Weak structural sufficiency · Maximum likelihood estimator · Trivial-transitive space Mathematics Subject Classification 62F10 · 54H11

1 Introduction Statistical decisions should not be affected by transformations on the data, so we study invariance. In this paper, however, we are interested in the weakly equivariant estimators, so we will generalize the concept of equivariance in statistics to weak equivariance which will be considered from a topological group point of view. The concept of weak equivariance has already found applications in harmonic analysis and manifold geometry, but we shall introduce its application to statistical inference and estimation theory in particular. A sufficient statistic provides a reduction of the data without loss of information. Jespersen [2] introduced a generalization of sufficiency, called structural sufficiency. In his paper, he considered the pair (t, π ) in which t is the maximum likelihood estimator, which is equivariant, and π is the canonical projection, which is maximal invariant and obtained some results about conditions that (t, π ) is structural sufficient.

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Mehdi Shams [email protected] Department of Statistics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

M. Shams

In this paper, we will generalize the above concept to the case where t is a weakly G-equivariant maximum likelihood estimator and shall introduce a new concept of weak structural sufficiency. Then we will give some conditions under which (t, π ) is weakly structurally sufficient and improve Jespersen’s works, so that our results contain Jespersen’s results as special cases, also we correct one of Jespersen’s results (see Remark 4). When the group acts transitively on the parameter space and the isotropy subgroup is characteristic, also when the sample space is trivial-transitive space, the concept of weak structural sufficiency turns out to be rather trivial. Finally, we give some results for weak structural sufficiency. In this paper, we focus on transformation models. Statisticians commonly examine the invariance theory from the point of view of transformation models, but we study invariance from another perspective, namely from the perspective of topological groups. Thus, it i