Renewal of shift families by the distribution of the maximal invariant distribution (Q1600647)

The aim of this paper is to give some general conditions for the characterization, based on certain statistics, of random population distributions which take values on an invariant space of a Lie group \(G\) and are generated by \(G\) transformations. The novelty of the proposed approach consists in not using the classical method of characteristic functions, but imposing some more restrictive assumptions (dimensionality preserving by a certain mapping with the \(n\)-product measure probability having the value one). Several examples of the characterization by distribution of invariants are discussed for the gamma function with known form parameter, for the generalized Wishart distribution, and for \(p\)-variate families with (un)known mean and (un)known covariance matrix.