On optimal estimators of shift population density functions and conditions of their consistency

Abstract

Let ~or(x) = (~ol (z), ~o2(z) ..... ~o~(x)) be a linear independent (with constant) system of functions on a homogeneous space X of a coherent Lie group G satisfying the second countability axiom. We shall assume that the components of ~o(x) are functions of C(t)-class and the linear space generated by the functions 1, ~o(z) is invariant with respect to the left shift operators lb(x) ::r ~b(g-tz), g G G. If V(g) is a matrix of the group G representation in the above-mentioned base, then it has the block form , where e is the unit of the group G, will be called the generating density of the considered family and the population ~ = g~0, where the random element ~0 has f(x) as its density function, will be called the exponential shift family. In our previous paper [4], it was shown that the structure elements of an exponential shift family satisfy rather strong restrictions. Namely, cr(g) = c r (e)V(g) + at(g), where a(g) satisfies the following set of equations:r(g~g~) = ar(g~)+ar(g2)V(g~) ' a(e) = 0for arbitrary elements gl, g2 E G. If a(g) is given, then h(x) is uniquely determined from the set of equations