Theory of Statistics - George Mason University

458 6 Statistical Inference Based on Likelihood
n
i=1
2(xi − θ)
= 0.
1 + (xi − θ) 2
This may have multiple roots (depending on the sample), and so the one
yielding the maximum would be the MLE. Depending on the sample, however,
multiple roots can yield the same value of the likelihood function.
Another example in which the MLE is not unique is U(θ − 1/2, θ + 1/2).
Example 6.12 likelihood in a uniform family with fixed range
Given the sample x1, . . ., xn, the likelihood function for U(θ − 1/2, θ + 1/2)
is
I[x (n)−1/2, x (1)+1/2](θ).
It is maximized at any value between x(n) − 1/2 and x(1) + 1/2.
Nonexistence and Other Properties
We have already mentioned situations in which the likelihood approach does
not seem to be the logical way, and have seen that sometimes in nonparametric
problems, the MLE does not exist. This often happens when there are more
“things to estimate” than there are observations. This can also happen in
parametric problems. It may happen that the maximum does not exist because
the likelihood is unbounded from above. In this case the argmax does not exist,
and the maximum likelihood estimate does not exist.
Example 6.13 nonexistence of MLE
Consider the normal family of distributions with parameters µ and σ 2 . Suppose
we have one observation x. The log-likelihood is
lL(µ, σ 2 ; x) = − 1
2 log(2πσ2 ) −
(x − µ)2
2σ2 ,
which is unbounded when µ = x and σ 2 approaches zero. It is therefore clear
that no MLE of σ 2 exists. Strictly speaking, we could also say that no MLE
of µ exists either; however, for any fixed value of σ 2 in the (open) parameter
space, µ = x maximizes the likelihood, so it is reasonable to call x the MLE
of µ.
Recall from Example 5.14 that the degree of the variance functional is 2.
In this case, some people prefer to say that the likelihood function does not
exist; that is, they suggest that the definition of a likelihood function include
boundedness.
Theory of Statistics c○2000–2013 James E. Gentle