Glasnik Matematicki, Vol. 36, No.1 (2001), 139-153.
A GENERAL THEOREM ON APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATION
Miljenko Huzak
Department of Mathematics, University of Zagreb, Bijenicka 30,
10000 Zagreb, Croatia
e-mail: huzak@math.hr
Abstract. In this paper a version of the general theorem
on approximate maximum likelihood estimation is proved. We assume that
there exists a log-likelihood function
L(θ) and
a sequence (Ln(θ))
of its estimates defined on some statistical structure parametrized by
θ from an open set
Θ ⊆ Rd, and dominated by a probability
P.
It is proved that if
L(θ)
and Ln(θ)
are random functions of class
C2(Θ)
such that there exists a unique point
θ ∈ Θ
of the global maximum of
L(θ)
and the first and second derivatives of
Ln(θ)
with the respect to θ
converge to the corresponding derivatives of
L(θ)
uniformly on compacts in Θ
with the order
OP(γn),
limn γn = 0,
then there exists a sequence of
Θ-valued random
variables θn
which converges to θ
with the order
OP(γn),
and such that
θn
is a stationary point of
Ln(θ)
in asymptotic sense. Moreover, we prove that under two more
assumptions on L and Ln, such
estimators could be chosen to be measurable with respect to the
σ-algebra
generated by
Ln(θ).
1991 Mathematics Subject Classification.
62F10, 62F12.
Key words and phrases. Parameter estimation, consistent
estimators, approximate likelihood function.
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