Characterization properties based on the Fisher information for weighted distributions

The Fisher information on θθ of the rr-size weighted pdf fr(x;θ)fr(x;θ) and its parent pdf f(x;θ)f(x;θ) are compared leading to some characterization properties for f(x;θ)f(x;θ). Additionally, some bounds for the Fisher information in terms of rr are also presented.