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 }.

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http://dx.doi.org/10.1090/gsm/119/01