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.
Keywords: estimation of a discontinuity point of a probability density, maximum likelihood estimators, asymptotic confidence interval, limiting distributions of statistical estimators.
The paper was received by the Editorial Office on March 31, 2018.
Vyacheslav Evgen’evich Mosyagin, Cand. Sci. (Phys.-Math.), Tyumen State University, Tyumen, 625003 Russia, e-mail: vmosyagin@mail.ru.
Natal’ya Aleksandrovna Shvemler, Tyumen State University, Tyumen, 625003 Russia,
e-mail: shvemler.natalya@mail.ru.
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