Estimation of transformations that preserve a quadratic metric in Euclidean space (Q1112483)

Let \((R^ m,{\mathcal A},P)\) be a probabilistic model, where \(R^ m\) is the m-dimensional Euclidean space and \({\mathcal A}\) the \(\sigma\)-algebra of Borel sets on \(R^ m\). Let \(\epsilon =\{\epsilon_ i\}\), \(i=1,...m\), be an orthonormal basis in \(R^ m\). We let \(G^ m\) denote the group of all transformations of \(R^ m\) which preserve a given quadratic metric. We start with the measure P and construct the family of measures \[ \{P_ g:\quad P_ g(A)=P(g^{-1}A),\quad g\in G^ m,\quad A\in {\mathcal A}\}. \] Our goal is to estimate the parameter \(g\in G^ m\) from a given iterated sample \(X=(x_ 1,...,x_ n)\) taken from the family \(\{P_ g\}\).