V.E. Mosyagin, N.A. Shvemler. Asymptotic confidence interval for a discontinuity point of a probability density function ... P. 194-199

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.


1.   Borovkov A.A. Large sample change-point estimation when distributions are unknown. Theory Probability Appl., 2009, vol. 53, no. 3, pp. 402–418. doi: 10.1137/S0040585X97983729 .

2.   Ibragimov I.A., Khas’minski R.Z. Statistical estimation. Asymptotic theory. N Y: Springer-Verlag, 1981, 403 p. doi: 10.1007/978-1-4899-0027-2 . Original Russian text published in Ibragimov I.A., Khas’minskii R.Z. Asimptoticheskaya teoriya otsenivaniya. Moscow, Nauka Publ., 1979, 528 p.

3.   Mosyagin V.E. Asymptotic representation of the process of the likelihood ratio in the case of a discontinuous density. Sib. Math. J., 1994, vol. 35, no. 2, pp. 375–382. doi: 10.1007/BF02104785 .

4.   Mosyagin V.E. Estimation of the convergence rate for the distributions of normalized maximum likelihood estimators in the case of a discontinuous density. Sib. Math. J., 1996, vol. 37, no. 4, pp. 788–796. doi: 10.1007/BF02104670 .

5.   Mosyagin V.E., Shvemler N.A. Distribution of the time of attaining the maximum for the difference of the two Poisson’s processes with negative linear drift. Sib. Electron. Math. Reports, 2016, vol. 13, pp. 1229–1248 (in Russian). doi: 10.17377/semi.2016.13.096 .

6.   Mosyagin V.E., Shvemler N.A. Local properties of the limiting distribution of the statistical estimator for jump point of a density. Sib. Electron. Math. Reports, 2017, vol. 14, pp. 1307–1316 (in Russian). doi: 10.17377/semi.2017.14.111 .

7.   Mosyagin V.E., Shvemler N.A. The average minimum method for construction of consistent estimates for jump point of a probability density. Abstr. Internet. Conf. “Lomonosovskie chteniya na Altae: fundamental’nye problemy nauki i obrazovaniya”. [Lomonosov’s Readings in the Altai: Fundamental Problems of Science and Education], Barnaul, 2017, pp. 453–455 (in Russian).

8.   Skorokhod A.V. Stochastic processes with independent increments. Dordrecht: Kluwer Acad. Publ. Group, 1991, 279 p. ISBN: 0-7923-0340-7 . Original Russian text (1st ed.) published in Skorokhod A.V., Sluchainye protsessy s nezavisimymi prirashcheniyami, Moscow, Nauka Publ., 1964, 280 p.

9.   Shvemler N.A., Mosyagin V.E. Program for estimating the unknown parameter of the point of discontinuity in the probability density. Registered Rospatent Certificate No. 2017660490 dated 22.09.2017 (in Russian).

10.   Borovkov A.A., Linke Yu. Yu. Change–point problem for large samples and incomplete information on distributions. Math. Methods Statist., 2005, vol. 14, no. 4, pp. 404–430.

11.   Hawkins D.L. A simple least squares method for estimating a change in mean Comm. Statist. Simulation, 1986, vol. 15, pp. 655–679.