Page 359 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 336

the lines of the Example 6.5.8. Next, along the lines of the Example 6.5.11,
recover the lost information in X by means of conditioning on the ancillary
statistic Y.
6.5.15 (Example 6.5.8 Continued) Suppose that (X, Y) is distributed as
N (0, 0, σ , σ , ρ) where θ = (σ , ρ) with the unknown parameters σ , ρ
2
2
2
2
2
where 0 < σ < ∞, –1 < ρ < 1. Evaluate the expression for the information
matrix I (θ). {Hint: Does working with U = X + Y, V = X - Y help in the
X,Y
derivation?}
6.5.16 Suppose that X′ = (X , ..., X ) where X is distributed as multivariate
p
1
2
normal N (0, σ [(1 – ρ)I + ρ11′]) with 1′ = (1, 1, ....1), σ ∈ ℜ and – (p –
+
p
p×p
1) < ρ < 1. We assume that θ = (σ , ρ) where σ , ρ are the unknown
–1
2
2
parameters, 0 < σ < ∞, –1 < ρ < 1. Evaluate the expression for the informa-
tion matrix I (θ). {Hint: Try the Helmert transformation from the Example
X
2.4.9 on X to generate p independent normal variables each with zero mean,
and variances depending on both σ and ρ, while (p - 1) of these variances are
2
all equal but different from the p one. Is it then possible to use the Theorem
th
6.4.3?}
6.5.17 (Example 6.5.8 Continued) Suppose that (X, Y) is distributed as
N (0, 0, 1, 1, ρ) where ρ is the unknown parameter, –1 < ρ < 1. Start with the
2
pdf of (X, Y) and then directly apply the equivalent formula from the equation
(6.4.9) for the evaluation of the expression of I (ρ).
X,Y
6.6.1 Suppose that X , X are iid Poisson(λ) where λ(> 0) is the unknown
1
2
parameter. Consider the family of distributions induced by the statistic T =
(X , X ). Is this family, indexed by λ, complete?
1 2
6.6.2 (Exercise 6.3.5 Continued) Let X , ..., X be iid Geometric(p), that is
1
n
the common pmf is given by f(x; p) = p(1 - p) , x = 0, 1, 2, ... where 0 < p <
x
1 is the unknown parameter. Is the statistic complete sufficient for p?
{Hint: Is it possible to use the Theorem 6.6.2 here?}
2
6.6.3 Let X , ..., X be iid N(θ, θ ) where 0 < θ < ∞ is the unknown
n
1
parameter. Is the minimal sufficient statistic complete? Show that and S 2
are independent. {Hint: Can the Example 4.4.9 be used here to solve the
second part?}
6.6.4 Let X , ..., X be iid N(θ, θ) where 0 < θ < ∞ is the unknown param-
n
1
2
eter. Is the minimal sufficient statistic complete? Show that and S are
independent. {Hint: Can the Example 4.4.9 be used here to solve the second
part?}
6.6.5 Let X , ..., X be iid having a negative exponential distribution
1
n
with the common pdf θ exp{–(x – θ)/θ}I(x > θ) where 0 < θ < ∞ is the
–1
unknown parameter. Is the minimal sufficient statistic complete? Show