## Abstract

This paper is concerned with the problem of construction of estimators of parameters in the case when the density f_{θ} (x) of the distribution P_{θ} of a sample X of size n has at least one point of discontinuity x(θ), x^{′}(θ) ≠ 0. It is assumed that either (a) from a priori considerations one can specify a localization of the parameter θ (or points of discontinuity) satisfying easily verifiable conditions, or (b) there exists a consistent estimator (formula presented) of the parameter θ (possibly constructed from the same sample X), which also provides some localization. Then a simple rule is used to construct, from the segment of the empirical distribution function defined by the localization, a family of estimators θ_{g}
^{∗} that depends on the parameter g such that (1) for sufficiently large n, the probabilities P(θ_{g}
^{∗} − θ > v/n)and P(θ^{∗}
_{g} − θ < −v/n) can be explicitly estimated by a v-exponential bound; (2) in case (b) under suitable conditions (see conditions I–IV in Chap. 5 of [I. A. Ibragimov and R. Z. Has’minskiĭ, Statistical Estimation. Asymptotic Theory, Springer, New York, 1981], where maximum likelihood estimators were studied), a value of g can be given such that the estimator θ_{g}
^{∗} is asymptotically equivalent to the maximum likelihood estimator (formula presented) for any v and n → ∞; (3) the value of g can be chosen so that the inequality (formula presented) is possible for sufficiently large n. Effectively no smoothness conditions are imposed on f_{θ} (x). With an available “auxiliary” consistent estimator (formula presented), simple rules are suggested for finding estimators θ_{g}
^{∗} which are asymptotically equivalent to(formula presented). The limiting distribution of n(θ_{g}
^{∗} −θ) as n → ∞ is studied.

Original language | English |
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Pages (from-to) | 169-192 |

Number of pages | 24 |

Journal | Theory of Probability and its Applications |

Volume | 63 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

## Keywords

- Change-point problem
- Distribution with discontinuous density
- Estimators of parameters
- Infinitely divisible factorization
- Maximum likelihood estimator
- ASYMPTOTIC REPRESENTATION
- MAXIMUM
- change-point problem
- DISTRIBUTIONS
- maximum likelihood estimator
- infinitely divisible factorization
- CONVERGENCE
- distribution with discontinuous density
- LIKELIHOOD RATIO
- estimators of parameters