Page 372 - Probability and Statistical Inference

7. Point Estimation 349

wish to find the MLE for θ. The likelihood function is

which is maximized at an end point. By drawing a simple picture of L(θ) it
should be apparent that L(θ) is maximized when θ = x . That is the MLE of
n:n
θ is the largest order statistic. !
Example 7.2.10 Suppose that X , ..., X are iid Poisson(λ) where 0 < λ <
1
n
χ
∞ is the unknown parameter. Here = {0, 1, 2, ...} and Θ = ℜ . We wish to
+
find the MLE for θ. The likelihood function is
so that one has

Now L(λ) is to be maximized with respect to λ and that is equivalent to
maximizing logL(λ) with respect to λ. We have logL(µ) = −n +
which when equated to zero provides the solution If > 0, then d/dµ
logL(µ) = 0 when . In this situation it is easy to verify that is
negative. Hence the MLE is whenever But there is a fine point
here. If which is equivalent to saying that x = ... = x = 0, the
1
n
likelihood function in (7.2.5) does not have a global maximum. In this case
L(λ) = e which becomes larger as λ(> 0) is allowed to become smaller. In
-nλ
other words if an MLE for λ can not be found. Observe, however,
that for all i = 1, ..., n} = exp( -nλ) which will be
negligible for “large” values of nλ.

Table 7.2.1. Values of

nλ: 3 4 5 6 7
: .0498 .01∞3 .0067 .0025 .0009

By looking at these entries, one may form some subjective opinion about
what values of nλ should perhaps be considered “large” in a situation like
this.!
In the Binomial(n,p) situation where 0 < p < 1 is the unknown parameter,
the problem of deriving the MLE of p would hit a snag similar to what we
found in the Example 7.2.10 when is 0 or 1. Otherwise the MLE of p
would be If the parameter space is replaced by 0 ≤ p ≤ 1, then of
course the MLE of p would be We leave this as the Exercise 7.2.8.