Page 329 - Probability and Statistical Inference

306 6. Sufficiency, Completeness, and Ancillarity

We add that a version of the Theorem 6.4.2 holds in the multiparameter
case as well. One may refer to Section 5a.3 of Rao (1973).
Example 6.4.7 Let X , ..., X be iid N(µ, σ ) where µ ∈ (–∞, ∞) and σ ∈
2
2
1 n
2
(0, ∞) are both unknown parameters. Denote θ = (µ, σ ), X = (X , ..., X ). Let
1 n
us evaluate the information matrix for X. First, a single observation X has its
1
pdf
so that one has

Hence we obtain

–2
2
2
–2
–2
since σ (X – µ) is so that E[σ (X – µ) ] = 1, V[σ (X – µ) ] = 2. Next,
2
1
1
1
we have
so that combining (6.4.14)-(6.4.15) with (6.4.10), we obtain the following
information matrix for one single observation X :
1

Utilizing (6.4.12), we obtain the information matrix,

for the whole data X. !