Page 373 - Probability and Statistical Inference

350 7. Point Estimation

Remark 7.2.1 In the Examples 7.2.6-7.2.10, the MLE of the unknown
parameter θ turned out to be either a (minimal) sufficient statistic or its func-
tion. This observation should not surprise anyone. With the help of Neyman
factorization one can rewrite L(θ) as g(T; θ)h(x) where T is a (minimal) suf-
ficient statistic for θ. Now, any maximizing technique for the likelihood func-
tion with respect to θ would reduce to the problem of maximizing g(T; θ) with
respect to θ. Thus the MLE of θ in general will be a function of the associated
(minimal) sufficient statistic T. The method of moment estimator of θ may
lack having this attractive feature. Refer to the Examples 7.2.4-7.2.5.

The MLE has a very attractive property referred to as its
invariance property: That is, if is the MLE of θ,
then is the MLE of g(θ).

The MLE can be quite versatile. It has a remarkable feature which is known
as its invariance property. This result will be useful to derive the MLE of
parametric functions of θ without a whole lot of effort as long as one already

has the MLE . We state this interesting result as a theorem without giving its
proof. It was proved by Zehna (1966).
Theorem 7.2.1 (Invariance Property of MLE) Consider the likelihood
function L(θ) defined in (7.2.2). Suppose that the MLE of θ (∈ Θ ⊆ ℜ ) exists
k
and it is denoted by . Let g(.) be a function, not necessarily one-to-one,
from ℜ to a subset of ℜ . Then, the MLE of the parametric function g(θ) is
m
k
given by .
Example 7.2.11 (Example 7.2.7 Continued) Suppose that X , ..., X are iid
1
n
N(µ, σ ) with θ = (µ σ ) where µ, σ are both unknown, −∞ < µ < ∞, 0 < σ <
2
2
2
χ
+
∞, n ≥ 2. Here = ℜ and Θ = ℜ × ℜ . We recall that the MLE of µ and σ are
2
respectively and Then, in view of
the MLE’s invariance property, the MLE of (i) σ would be (ii) µ + σ
would be (iii) µ /σ would be !
2
2
Note that the function g(.) in the Theorem 7.2.1 is not necessarily
one-to-one for the invariance property of the MLE to hold.
Example 7.2.12 (Example 7.2.9 Continued) Suppose that X , ..., X are iid
n
1
χ
Uniform(0, θ) where 0 < θ < ∞ is the unknown parameter. Here = (0, θ) and
Θ = ℜ . We recall that the MLE for θ is X . In view of the MLE’s invariance
+
n:n
property, one can verify that the MLE of (i) θ would be (ii) 1/θ would
2
be 1/X , (iii) would be !
n:n