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On estimation of parameters in the case of discontinuous densities
A. A. Borovkovab a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University
Abstract:
This paper is concerned with the problem of construction of estimators of parameters
in the case when
the density
$f_\theta(x)$ of the distribution
$\mathbf{P}_\theta$ of a sample
$\mathrm X$ of size
$n$
has at least one point of discontinuity
$x(\theta)$,
$x'(\theta)\neq 0$. It is assumed that either
(a) from a priori considerations one can specify a localization of the parameter
$\theta$
(or points of discontinuity) satisfying easily verifiable conditions,
or (b) there exists a consistent estimator
$\widetilde{\theta}$
of the parameter
$\theta$ (possibly constructed from the same sample
$\mathrm{X}$),
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
$\theta^*_{g}$ that depends on the parameter
$g$ such that
(1) for sufficiently large
$n$, the probabilities
$\mathbf{P}(\theta^*_{g}-\theta>v/n)$ and
$\mathbf{P}(\theta^*_{g}-\theta<-v/n)$ 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
$\theta^*_{g}$ is asymptotically equivalent
to the maximum likelihood estimator
$\widehat{\theta}$; i.e.,
$\mathbf{P}_\theta(n(\theta^*_{g}-\theta)>v)\sim
\mathbf{P}_\theta(n(\widehat{\theta}-\theta)>v)$ for any
$v$ and
$n\to\infty$;
(3) the value of
$g$ can be chosen so that the inequality
$\mathbf{E}_\theta(\theta^*_{g}-\theta)^2<
\mathbf{E}_\theta(\widehat{\theta}-\theta)^2$ is possible for sufficiently large
$n$.
Effectively no smoothness conditions are imposed on
$f_\theta(x)$.
With an available “auxiliary” consistent estimator
$\widetilde{\theta}$,
simple rules are suggested for finding estimators
$\theta^*_g$ which are asymptotically equivalent to
$\widehat{\theta}$.
The limiting distribution of
$n(\theta^*_g-\theta)$ as
$n\to\infty$ is studied.
Keywords:
estimators of parameters, maximum likelihood estimator, distribution with discontinuous density,
change-point problem, infinitely divisible factorization. Received: 23.03.2017
Revised: 03.04.2017
Accepted: 29.08.2017
DOI:
10.4213/tvp5141