Chapter 1
The Fisher Efficiency
1.1. Statistical Experiment
A classical statistical experiment
(
X1,...,Xn; p(x, θ); θ Θ
)
is composed of
the following three elements: (i) a set of independent observations X1,...,Xn
where n is the sample size, (ii) a family of probability densities p(x, θ) de-
fined by a parameter θ, and (iii) a parameter set Θ of all possible values of
θ.
Unless otherwise stated, we always assume that θ is one-dimensional,
that is, Θ R. For discrete distributions, p (x, θ) is the probability mass
function. In this chapter we formulate results only for continuous distri-
butions. Analogous results hold for discrete distributions if integration is
replaced by summation. Some discrete distributions are used in examples.
Example 1.1. (a) If n independent observations X1,...,Xn have a normal
distribution with an unknown mean θ and a known variance σ2, that is,
Xi N (θ,
σ2),
then the density is
p (x, θ) = (2 π
σ2)−1/2
exp (x
θ)2/(2σ2)
, −∞ x, θ ∞,
and the parameter set is the whole real line Θ = R.
(b) If n independent observations have a normal distribution with a known
mean μ and an unknown variance θ, that is, Xi N (μ, θ), then the density
is
p(x, θ) = (2 π
θ)−1/2
exp (x
μ)2/(2θ)
, −∞ x , θ 0,
and the parameter set is the positive half-axis Θ = { θ R : θ 0 }.
3
http://dx.doi.org/10.1090/gsm/119/01
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