Asymptotic confidence interval for a discontinuity point of a probability density function

We consider the problem of interval estimation of an unknown parameter $\theta\in\Theta\subset R$ of a distribution density $f(x,\theta)$ (with respect to the Lebesgue measure) for a sample $X_1,\dots,X_n$ of large size. It is assumed that the density has a discontinuity of the first kind at the point $x=\theta$. We construct a confidence interval based on a known maximum likelihood estimator $\theta_n^*$ and the distribution function $G(x,\theta)$ found by the authors earlier, which is the limit of the sequence of distribution functions of normalized maximum likelihood estimators\linebreak $n(\theta_n^*-\theta)$. It is proved that the resulting confidence interval is asymptotically exact. We also describe a method for the “fast” calculation of maximum likelihood estimators for a discontinuity point of a density.