Page 328 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 305

we only discuss the case of a two-dimensional parameter. Suppose that X is
an observable real valued random variable with the pmf or pdf f(x;θ) where
the parameter θ = (θ , θ ) ∈ Θ, an open rectangle ⊆ ℜ , and the χ space does
2
1 2
not depend upon θ. We assume throughout that f(x; θ) exists, i = 1, 2, for
all x ∈ χ, θ ∈ Θ, and that we can also interchange the partial derivative (with
respect to θ , θ ) and the integral (with respect to x).
1 2
Definition 6.4.2 Let us extend the earlier notation as follows. Denote
I (θ) = E , for i, j = 1, 2. The Fisher
ij θ
information matrix or simply the information matrix about θ is given by

Remark 6.4.2 In situations where f(x; θ) exists for all x ∈ χ, for all
i, j = 1, 2, and for all θ ∈ Θ, we can alternatively write

and express the information matrix I (θ) accordingly. We have left this as an
X
exercise.

Having a statistic T = T(X , ..., X ), however, the associated information
1
n
matrix about θ will simply be calculated as I (θ) where one would replace the
T
original pmf or pdf f(x; θ) by that of T, namely g(t;θ), t ∈ . When we
compare two statistics T and T in terms of their information content about a
2
1
single unknown parameter θ, we simply look at the two one-dimensional quan-
tities I (θ) and I (θ), and compare these two numbers. The statistic associ-
T1
T2
ated with the larger information content would be more appealing. But, when
θ is two-dimensional, in order to compare the two statistics T and T , we
2
1
have to consider their individual two-dimensional information matrices I (θ)
T1
and I (θ). It would be tempting to say that T is more informative about θ
T2
1
than T provided that
2