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5.8 Suggestions for Further Reading 169

(c) T (x) = (x 2 − x 1 , x 3 − x 1 ,..., x n − x 1 , x n+2 − x n+1 ,..., x n+m − x n+1 )
n+m
(d) T (x) = (x 1 − ¯ x, x 2 − ¯ x,..., x n+m − ¯ x) where ¯ x = j=1 x j /(n + m).
5.30 Let F 1 ,..., F n denote absolutely continuous distribution functions on the real line, where n is
afixed integer. Let denote the set of all permutations of (1,..., n) and consider the model P
for an n-dimensional random vector X = (X 1 ,..., X n ) consisting of distribution functions of
the form
(x n ),
F(x 1 ,..., x n ; θ) = F θ 1 (x 1 ) ··· F θ n
n
θ = (θ 1 ,...,θ n ) ∈ . The sample space of X, X, may be taken to be the subset of R in which,
for x = (x 1 ,..., x n ) ∈ X, x 1 ,..., x n are unique.
Let G denote the set of all permutations of (1,..., n) and, for g ∈ G and x ∈ X, define
).
gx = (x g 1 ,..., x g n
(a) Show that P is invariant with respect to G.
(b) For g ∈ G, describe the action of g on θ ∈ . That is, by part (a), if X has parameter θ ∈ ,
then gX has parameter θ 1 ∈ ; describe θ 1 in terms of g and θ.
(c) Let x ∈ X. Describe the orbit of x.In particular, for the case n = 3, give the orbit of
(x 1 , x 2 , x 3 ) ∈ X.
(d) Let T (x) = (x (1) ,..., x (n) ) denote the vector of order statistics corresponding to a point
x ∈ X. Show that T is an invariant statistic with respect to G.
(e) Is the statistic T defined in part (d) a maximal invariant statistic?
(f) For x ∈ X, let R(x) denote the vector of ranks of x = (x 1 ,..., x n ). Note that R takes values
in .Is R an equivariant statistic with respect to the action of g on ?

5.8 Suggestions for Further Reading

Statisticalmodelsandidentifiabilityarediscussedinmanybooksonstatisticaltheory;see,forexample,
Bickel and Doksum (2001, Chapter 1) and Gourieroux and Monfort (1989, Chapters 1 and 3). In the
approach used here, the parameter is either identified or it is not; Manski (2003) considers the concept
of partial identification.
Exponential families are discussed in Bickel and Doksum (2001, Section 1.6), Casella and Berger
(2002, Section 3.4), and Pace and Salvan (1997, Chapter 5). Comprehensive treatments of exponential
family models are given by Barndorff-Nielsen (1978) and Brown (1988). Exponential dispersion
models, considered briefly in Exercise 5.14 are considered in Pace and Salvan (1997, Chapter 6).
Schervish (1995, Chapter 8) contains a detailed treatment of hierarchical models and the statistical
inference in these models; see also Casella and Berger (2002, Section 4.4).
Regression models play a central role in applied statistics. See, for example, Casella and Berger
(2002, Chapters 11 and 12). Rao and Toutenburg (1999) is a comprehensive reference on statistical
inference in a wide range of linear regression models. McCullagh and Nelder (1989) considers a
general class of regression models that are very useful in applications. Transformation models and
equivariance and invariance are discussed in Pace and Salvan (1997, Chapter 7) and Schervish (1995,
Chapter 6); Eaton (1988) contains a detailed treatment of these topics.