Minimum Kolmogorov distance estimates for multivariate parametrized families

In a previous paper the authors introduced minimum Kolmogorov distance estimates of distribution parameters and distribution densites for arbitrarily parametrized univariate statistical models. This paper is an extension to the multivariate case. Some results extend straightforwardly, the extension of other ones is quite difficult. It is shown that the Kolmogorov distance parameter estimates are strongly consistent if the parameter space metric is topologically weaker than the metric induced by the Kolmogorov distance of distributions from the statistical model. If the parameter space metric is locally uniformly bounded above by the induced metric then the estimates are consistent of order . Similar results are proved for the Kolmogorov distance estimates of densities from parametrized families. In this case the consistency is considered in the -norm.