Measure Compact Sets of Functions and Consistency of Statistical Models

Sufficient conditions for consistency of statistical models are deduced by using Fréchet-Šmulian’s necessary and sufficient conditions for conditionally compactness relative to the topology of convergence in measure being imposed on the family of associated densities by so-called a control measure . The method relies upon the facts established in [4] where sufficient conditions for consistency are deduced by using necessary and sufficient conditions for conditionally compactness of the family of associated densities relative to the topology of pointwise convergence. In the case where a statistical model under consideration admits a finite control measure the present conditions for conditionally compactness become weaker and easily verified. However the present approach relies upon the fact that any control measure that yields consistency must be nice enough in such a way that the upper oscillation of densities on infinitesimally small balls behaves smoothly relative to the distribution of the random phenomenon under consideration.