Local properties of the limiting distribution of the statistical estimator for jump point of a density

Consider a random sample from the probability density function with exactly one jump point, which depends on an unknown parameter to be evaluated. The local properties of the limiting distribution of the normalized maximum likelihood estimators for the parameter are investigated. The so-called integro-local estimates for this limiting distribution are obtained. As an application it is shown that these estimates allow simplifying the problem of obtaining the rates of convergence for the distributions of the normalized maximum likelihood estimators.