Page 180 - Elements of Distribution Theory

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166 Parametric Families of Distributions

5.12 Let Y denote a random variable with an absolutely continuous distribution with density p(y; θ),
where p is given in Exercise 5.11. Find the cumulant-generating function of Y + 1/Y − 2.
5.13 Let Y denote a random variable with density or frequency function of the form
exp{ηT (y) − k(η)}h(y)
where η ∈ H ⊂ R. Fix η 0 ∈ H and let s be such that E{exp[sT (Y)]; η 0 } < ∞. Find an expres-
sion for Cov(T (Y), exp[sT (Y)]; η 0 )in terms of the function k.
5.14 Afamily of distributions that is closely related to the exponential family is the family of expo-
nential dispersion models. Suppose that a scalar random variable X has a density of the form
2
2
2
p(x; η, σ ) = exp{[ηx − k(η)]/σ }h(x; σ ),η ∈ H
2
where for each fixed value of σ > 0 the density p satisfies the conditions of a one-parameter
2
exponential family distribution and H is an open set. The set of density functions {p(·; η, σ ): η ∈
2
H,σ > 0} is said to be an exponential dispersion model.
(a) Find the cumulant-generating function of X.
(b) Suppose that a random variable Y has the density function p(·; η, 1), that is, it has the
2
distribution as X except that σ is known to be 1. Find the cumulants of X in terms of the
cumulants of Y.
5.15 Suppose Y is a real-valued random variable with an absolutely continuous distribution with
density function
p Y (y; η) = exp{ηy − k(η)}h(y), y ∈ Y,
where η ∈ H ⊂ R, and X is a real-valued random variable with an absolutely continuous dis-
tribution with density function
p X (x; η) = exp{ηx − ˜ k(η)} ˜ h(x), x ∈ X,
where η ∈ H. Show that:
(a) if k = ˜ k, then Y and X have the same distribution, that is, for each η ∈ H, F Y (·; η) = F X (·; η)
where F Y and F X denote the distribution functions of Y and X, respectively
(b) if E(Y; η) = E(X; η) for all η ∈ H then Y and X have the same distribution
(c) if Var(Y; η) = Var(X; η) for all η ∈ H then Y and X do not necessarily have the same
distribution.
5.16 Let X denote a real-valued random variable with an absolutely continuous distribution with
density function p. Suppose that the moment-generating function of X exists and is given by
M(t), |t| < t 0 .
Let Y denote a real-valued random variable with an absolutely continuous distribution with
density function of the form
p(y; θ) = c(θ)exp(θy)p(y),

where c(·)isa function of θ.
(a) Find requirements on θ so that p(·; θ) denotes a valid probability distribution. Call this
set .
(b) Find an expression for c(·)in terms of M.
(c) Show that {p(·; θ): θ ∈ } is a one-parameter exponential family of distributions.
(d) Find the moment-generating function corresponding to p(·; θ)in terms of M.
5.17 Let Y 1 , Y 2 ,..., Y n denote independent, identically distributed random variables such that Y 1 has
density p(y; θ) where p is of the form
p(y; θ) = exp{ηT (y) − k(η)}h(y), y ∈ Y