ABSTRACT
In this chapter we consider a wide class of estimators which can be used when
the data are discrete, either the underlying distribution is discrete or is contin-
uous but the observations are classified into groups. The latter situation can
occur either by experimental reasons or because the estimation problem with-
out grouped data is not easy to solve; see Fryer and Robertson (1972). Some
examples in which it is not possible to find the maximum likelihood estimator
based on the original data can be seen in Le Cam (1990). For example, when
we consider distributions resulting from the mixture of two normal populations,
whose probability density function is given by
fθ(x) = w 1√ 2πσ1
exp
à −1 2
µ x− µ1 σ1
¶2! +(1−w) 1√
2πσ2 exp
à −1 2
µ x− µ2 σ2
¶2! ,
the likelihood function is not a bounded function. In this situation
θ = (µ1, µ2,σ1,σ2, w), µ1, µ2 ∈ R,σ1,σ2 > 0 and w ∈ (0, 1),
and the likelihood function for a random sample of size n, x1, ..., xn is given by
L(θ;x1, . . . , xn) = nY j=1 fθ(xj).
