Page 181 - Elements of Distribution Theory

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5.7 Exercises 167

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and η takes values in the natural parameter space H. Let S = T (Y j ).
j=1
Let A ≡ A(Y 1 ,..., Y n ) denote a statistic such that for each η ∈ H the moment-generating
function of A, M A (t; η), exists for t in a neighborhood of 0 and let M S (t; η) denote the moment-
generating function of S.
(a) Find an expression for the joint moment-generating function of (A, S),
M(t 1 , t 2 ; η) = E η (exp{t 1 A + t 2 S})
in terms of M A and M S .
(b) Suppose that for a given value of η, η 0 ,
∂
= 0.

∂η E(A; η) η=η 0
Find an expression for Cov(S, A; η 0 ).
5.18 Let Y 1 ,..., Y n denote independent binomial random variables each with index m and success
probability θ.Asan alternative to this model, suppose that Y j is a binomial random variable
with index m and success probability φ where φ has a beta distribution. The beta distribution is
an absolutely continuous distribution with density function
(α + β) α−1 β−1
φ (1 − φ) , 0 <φ < 1
(α) (β)
where α> 0 and β> 0. The distribution of Y 1 ,..., Y n is sometimes called the beta-binomial
distribution.
(a) Find the density function of Y j .
(b) Find the mean and variance of Y j .
(c) Find the values of the parameters of the distribution for which the distribution reduces to
the binomial distribution.
5.19 Let (Y j1 , Y j2 ), j = 1, 2,..., n, denote independent pairs of independent random variables such
that, for given values of λ 1 ,...,λ n , Y j1 has an exponential distribution with mean ψ/λ j and
that Y j2 has an exponential distribution with mean 1/λ j . Suppose further that λ 1 ,...,λ n are
independent random variables, each distributed according to an exponential distribution with
mean 1/φ, φ> 0. Show that the pairs (Y j1 , Y j2 ), j = 1,..., n, are identically distributed and
find their common density function.
5.20 Let λ and X 1 ,..., X n denote random variables such that, given λ, X 1 ,..., X n are independent
and identically distributed. Show that X 1 ,..., X n are exchangeable.
Suppose that, instead of being independent and identically distributed, X 1 ,..., X n are only
exchangeable given λ. Are X 1 ,..., X n exchangeable unconditionally?
5.21 Consider a linear exponential family regression model, as discussed in Example 5.22. Find an
expression for the mean and variance of T (Y j ).
5.22 Show that G s , G l , and G ls each satisfy (G1)–(G4) and (T1) and (T2).
5.23 Let P denote the family of normal distributions with mean θ and standard deviation θ, where
θ> 0. Is this model invariant with respect to the group of location transformations? Is it invariant
with respect to the group of scale transformations?
5.24 Let Y 1 ,..., Y n denote independent, identically distributed random variables, each uniformly
distributed on the interval (θ 1 ,θ 2 ), θ 1 <θ 2 .
(a) Show that this is a transformation model and identify the group of transformations. Show
the correspondence between the parameter space and the transformations.
(b) Find a maximal invariant statistic.
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5.25 Let X denote an n-dimensional random vector and, for g ∈ R , define the transformation
gX = X + g.