Page 182 - Elements of Distribution Theory

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168 Parametric Families of Distributions

n
Let G denote the set of such transformations with g restricted to a set A ⊂ R and define g 1 g 2
to be vector addition, g 1 + g 2 . Find conditions on A such that G is a group and that (T1) and
(T2) are satisfied.
5.26 Consider the set of binomial distributions with frequency function of the form

n x n−x
θ (1 − θ) , x = 0, 1,..., n
x
where n is fixed and 0 <θ < 1. For x ∈ X ≡{0, 1,..., n}, define transformations g 0 and g 1
by g 0 x = x and g 1 x = n − x.
(a) Define g 0 g 1 , g 1 g 0 , g 0 g 0 , and g 1 g 1 so that {g 0 , g 1 }, together with the binary operation defined
by these values, is a group satisfying (T1) and (T2). Call this group G.
(b) Show that the set of binomial distributions described above is invariant with respect to G.
Find g 0 θ and g 1 θ for θ ∈ (0, 1).
(c) Is the set of binomial distributions a transformation model with respect to G?
(d) Let x ∈ X. Find the orbit of x.
(e) Find a maximal invariant statistic.
5.27 Suppose that the random vector (X 1 ,..., X n ) has an absolutely continuous distribution with
density function of the form
n
p 0 (x j − θ), −∞ < x j < ∞, j = 1,..., n,
j=1
where θ ∈ R and p 0 is a density function on the real line. Recall that this family of distributions
is invariant with respect to G l ; see Example 5.35.
Let T denote an equivariant statistic. Show that the mean and variance of T do not depend on
θ. Let S denote an invariant statistic. What can be said about the dependence of the mean and
variance of S on θ?
5.28 Let X = R n+m , let v denote the vector in X with the first n elements equal to 1 and the remaining
elements equal to 0, and let u denote the vector in X with the first n elements equal to 0 and
2
the remaining elements equal to 1. Let G = R and for an element g = (a, b) ∈ G, define the
transformation
gx = x + av + bu, x ∈ X.
For each of the models given below, either show that the model is invariant with respect to G or
show that it is not invariant with respect to G.Ifa model is invariant with respect to G, describe
the action of G on the parameter space ; that is, for g ∈ G and θ ∈ ,give gθ.
(a) X 1 , X 2 ,..., X n+m are independent, identically distributed random variables, each with a
normal distribution with mean θ, −∞ <θ < ∞.
(b) X 1 , X 2 ,..., X n+m are independent random variables, such that X 1 ,..., X n each have a
normal distribution with mean µ 1 and X n+1 ,..., X n+m each have a normal distribution
2
with mean µ 2 ; here θ = (µ 1 ,µ 2 ) ∈ R .
(c) X 1 , X 2 ,..., X n+m are independent random variables such that, for each j = 1,..., n + m,
X j has a normal distribution with mean µ j ; here θ = (µ 1 ,...,µ n+m ) ∈ R n+m .
(d) X 1 , X 2 ,..., X n+m are independent random variables such that X 1 ,..., X n each have an
exponential distribution with mean µ 1 and X n+1 ,..., X n+m each have an exponential dis-
+ 2
tribution with mean µ 2 ; here θ = (µ 1 ,µ 2 ) ∈ (R ) .
5.29 Consider the model considered in Exercise 5.28. For each of the statistics given below, either
show that the statistic is invariant with respect to G or show that it is not invariant with respect
to G.Ifa statistic is invariant, determine if it is a maximal invariant.
n n+m

(a) T (x) = x j /n − x j /m
j=1 j=n+1
n
(b) T (x) = x 1 − j=1 x j /n