Page 179 - Elements of Distribution Theory

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052184472Xc05 CUNY148/Severini May 24, 2005 17:53

5.7 Exercises 165

5.5 Consider a parametric model {P θ : θ ∈ } for a random variable X. Such a model is said to be
complete if, for a real-valued function g,
E[g(X); θ] = 0,θ ∈ ,
implies
Pr[g(X) = 0; θ] = 1,θ ∈ .

For each of the models given below, determine if the model is complete.
(a) P θ is the uniform distribution on (0,θ) and = (0, ∞)
(b) P θ is the absolutely continuous distribution with density
2θ θ
2
x 2θ−1 exp(−θx ), x > 0,
(θ)
and = (0, ∞)
(c) P θ is the binomial distribution with frequency function of the form
3 x 3−x

p(x; θ) = θ (1 − θ) , x = 0, 1, 2
x
and = (0, 1).
5.6 Consider the family of absolutely continuous distributions with density functions
(α + β)
p(y; θ) = y α−1 (1 − y) β−1 , 0 < y < 1,
(α) (β)
where θ = (α, β) and = (0, ∞) × (0, ∞). This is known as the family of beta distributions.
Show that this is a two-parameter exponential family by putting p(y; θ)in the form (5.1). Find
the functions c, T, A, h and the set Y.
5.7 Consider the family of gamma distributions described in Example 3.4 and Exercise 4.1. Show
that this is a two-parameter exponential family. Find the functions c, T, A, h and the set Y in
the representation (5.1).
5.8 Consider the family of absolutely continuous distributions with density function
θ
p(y; θ) = , y > 1
y θ+1
where θ> 0. Show that this is a one-parameter exponential family of distributions and write p
in the form (5.1), giving explicit forms for c, T , A, and h.
5.9 Consider the family of discrete distributions with density function
θ 1 + y − 1 y

θ 1
p(y; θ) = θ (1 − θ 2 ) , y = 0, 1,...
2
y
where θ = (θ 1 ,θ 2 ) ∈ (0, ∞) × (0, 1). Is this an exponential family of distributions? If so, write
p in the form (5.1), giving explicit forms for c, T , A, and h.
5.10 Let X denote a random variable with range X such that the set of possible distributions of
X is a one-parameter exponential family. Let A denote a subset of X and suppose that X is
only observed if X ∈ A; let Y denote the value of X given that it is observed. Is the family of
distributions of Y a one-parameter exponential family?
5.11 Consider the family of absolutely continuous distributions with density functions
√ 2
λ φ λ
p(y; θ) = 1 exp − y + φ − , y > 0
3
(2πy ) 2 2λ 2y
+ 2
where θ = (φ, λ) ∈ (R ) .Is this an exponential family of distributions? If so, write p in the
form (5.1), giving explicit forms for c, T , A, and h.