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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