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6. Sufficiency, Completeness, and Ancillarity 334

communication from C. R. Rao. Note that the likelihood function f(x; θ) can
be written as g(t; θ)h(X | T = t; θ) where g(t; θ) is the pdf or pmf of T and h(x
| T = t; θ) is the conditional pdf or pmf of X given that T = t. From this
equality, first derive the identity: I (θ) = I (θ) + I (θ) for all θ ∈ Θ. This
X|T
X
T
implies that I (θ) ≥ I (θ) for all θ ∈ Θ. One will have I (θ) = I (θ) if and only
X
T
X
T
if I (θ) = 0, that is h(x | T = t;θ) must be free from θ. It then follows from
X|T
the definition of sufficiency that T is sufficient for θ.}
6.4.16 Suppose that X , X are iid N(θ, 1) where the unknown parameter θ
2
1
∈ ℜ. ‘We know that T = X + X is sufficient for θ and also that I (θ) = 2.
2
1
T
Next, consider a statistic U = X + pX where p is a known positive number.
p 1 2
2
(i) Show that I (θ) = (1 + p) /(1 + p );
2
Up
(ii) Show that 1 < I (θ) ≤ 2 for all p(> 0);
Up
(iii) Show that I (θ) = 2 if and only if p = 1;
Up
(iv) An expression such as I (θ)/I (θ) may be used to quantify the ex-
Up
T
tent of non-sufficiency or the fraction of the lost information due to
using U instead of utilizing the sufficient statistic T. Analytically,
p
study the behavior of I (θ)/I (θ) as a function of p(> 0);
Up T
(v) Evaluate I (θ)/I (θ) for p = .90, .95, .99, 1.01, 1.05, 1.1. Is U too
T
p
Up
non-sufficient for θ in practice when p = .90, .95, .99, 1.01?
(vi) In practice, if an experimenter is willing to accept at the most 1%
lost information compared with the full information I (θ), find the
T
range of values of p for which the non-sufficient statistic U will
p
attain the goal.
{Note: This exercise exploits much of what has been learned conceptually
from the Theorem 6.4.2.}
6.5.1 (Example 6.5.2 Continued) Find the pdf of U defined in (6.5.1).
3
Hence, show that T is ancillary for θ.
3
6.5.2 (Example 6.5.3 Continued) Show that T is ancillary for λ.
3
6.5.3 (Example 6.5.4 Continued) Show that T = (T , T ) is distributed as
1
2
N (0, 0, 2, 6, 0). Are T , T independent?
2 1 2
6.5.4 (Example 6.5.5 Continued) Show that T is ancillary for θ.
6.5.5 (Exercise 6.3.7 Continued) Let X , ..., X be iid having the common
n
1
Uniform distribution on the interval (θ – ½, θ + ½) where –∞ < θ < ∞ is the
unknown parameter. Show that X – X is an ancillary statistic for θ.
n:n n:1