TY - JOUR

T1 - On estimation of parameters in the case of discontinuous densities

AU - Borovkov, A. A.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - This paper is concerned with the problem of construction of estimators of parameters in the case when the density fθ (x) of the distribution Pθ 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 θg
∗ that depends on the parameter g such that (1) for sufficiently large n, the probabilities P(θg
∗ − 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 θg
∗ 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 fθ (x). With an available “auxiliary” consistent estimator (formula presented), simple rules are suggested for finding estimators θg
∗ which are asymptotically equivalent to(formula presented). The limiting distribution of n(θg
∗ −θ) as n → ∞ is studied.

AB - This paper is concerned with the problem of construction of estimators of parameters in the case when the density fθ (x) of the distribution Pθ 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 θg
∗ that depends on the parameter g such that (1) for sufficiently large n, the probabilities P(θg
∗ − 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 θg
∗ 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 fθ (x). With an available “auxiliary” consistent estimator (formula presented), simple rules are suggested for finding estimators θg
∗ which are asymptotically equivalent to(formula presented). The limiting distribution of n(θg
∗ −θ) as n → ∞ is studied.

KW - Change-point problem

KW - Distribution with discontinuous density

KW - Estimators of parameters

KW - Infinitely divisible factorization

KW - Maximum likelihood estimator

KW - ASYMPTOTIC REPRESENTATION

KW - MAXIMUM

KW - change-point problem

KW - DISTRIBUTIONS

KW - maximum likelihood estimator

KW - infinitely divisible factorization

KW - CONVERGENCE

KW - distribution with discontinuous density

KW - LIKELIHOOD RATIO

KW - estimators of parameters

UR - http://www.scopus.com/inward/record.url?scp=85056951809&partnerID=8YFLogxK

U2 - 10.1137/S0040585X97T98899X

DO - 10.1137/S0040585X97T98899X

M3 - Article

AN - SCOPUS:85056951809

VL - 63

SP - 169

EP - 192

JO - Theory of Probability and its Applications

JF - Theory of Probability and its Applications

SN - 0040-585X

IS - 2

ER -