Page 337 - Probability and Statistical Inference

6. Sufficiency, Completeness, and Ancillarity 314

A direct verification of (6.5.6) using the expression of f(x, y; ρ) is left as the
Exercise 6.5.17. !
Example 6.5.9 Let X , X , ... be a sequence of iid Bernoulli(p), 0 < p < 1.
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One may think of X = 1 or 0 according as the i toss of a coin results in a
th
i
head (H) or tail (T) where P(H) = 1 – P(T) = p in each independent toss, i =
1, ..., n. Once the coin tossing experiment is over, suppose that we are only
told how many times the particular coin was tossed and nothing else. That is,
we are told what n is, but nothing else about the random samples X , ..., X . In
n
1
this situation, knowing n alone, can we except to gain any knowledge about
the unknown parameter p? Of course, the answer should be “no,” which
amounts to saying that the sample size n is indeed ancillary for p. !
6.5.1 The Location, Scale, and Location-Scale Families
In Chapter 3, we discussed a very special class of distributions, namely the
exponential family. Now, we briefly introduce a few other special ones which
are frequently encountered in statistics. Let us start with a pdf g(x), x ∈ ℜ
and construct the following families of distributions defined through the
g(.) function:

The understanding is that θ, δ may belong to some appropriate subsets of ℜ,
ℜ respectively. The reader should check that the corresponding members
+
f(.) from the families F , F , F are indeed pdf’s themselves.
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The distributions defined via parts (i), (ii), and (iii) in (6.5.7)
are respectively said to belong to the
location, scale, and location-scale family.

We often say that the families F , F , F are respectively indexed by (i) the
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parameter θ, (ii) the parameter δ, and (iii) the parameters θ and δ. In part (i) θ
is called a location parameter, (ii) δ is called a scale parameter, and (iii) θ, δ
are respectively called the location and scale parameters.
For example, the collection of N(µ, 1) distributions, with µ ∈ ℜ, forms
a location family. To see this, let g(x) = φ (x) = exp{–x /2}, x ∈ ℜ
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