#### 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 (*L*_{n}(θ))
of its estimates defined on some statistical structure parametrized by
θ from an open set
Θ ⊆ **R**^{d}, and dominated by a probability
*P*.
It is proved that if
*L*(θ)
and *L*_{n}(θ)
are random functions of class
*C*^{2}(Θ)
such that there exists a unique point
θ ∈ Θ
of the global maximum of
*L*(θ)
and the first and second derivatives of
*L*_{n}(θ)
with the respect to θ
converge to the corresponding derivatives of
*L*(θ)
uniformly on compacts in Θ
with the order
*O*_{P}(γ_{n}),
lim_{n} γ_{n} = 0,
then there exists a sequence of
Θ-valued random
variables θ_{n}
which converges to θ
with the order
*O*_{P}(γ_{n}),
and such that
θ_{n}
is a stationary point of
*L*_{n}(θ)
in asymptotic sense. Moreover, we prove that under two more
assumptions on *L* and *L*_{n}, such
estimators could be chosen to be measurable with respect to the
σ-algebra
generated by
*L*_{n}(θ).

**1991 Mathematics Subject Classification.**
62F10, 62F12.

**Key words and phrases.** Parameter estimation, consistent
estimators, approximate likelihood function.

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