Research output: Contribution to journal › Article › peer-review
On estimation of parameters in the case of discontinuous densities. / Borovkov, A. A.
In: Theory of Probability and its Applications, Vol. 63, No. 2, 01.01.2018, p. 169-192.Research output: Contribution to journal › Article › peer-review
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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 -
ID: 17562413