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6. Sufficiency, Completeness, and Ancillarity 295

statistics which were substantially “reduced” compared with X. As a prin-
ciple, we should use the “shortest sufficient” summary in lieu of handling the
original data. Two pertinent questions may arise: What is a natural way to
define the “shortest sufficient” summary statistic? The next question is how
to get hold of such a “shortest sufficient” summary, if indeed there is one?
Lehmann and Scheffé (1950) developed a precise mathematical formula-
tion of the concept known as minimal sufficiency and they gave a technique
that helps to locate minimal sufficient statistics. Lehmann and Scheffé (1955,
1956) included important followups.
Definition 6.3.1 A statistic T is called minimal sufficient for the unknown
parameter θθ θθ θ or simply minimal sufficient if and only if
(i) T is sufficient for θθ θθ θ, and
(ii) T is minimal or “shortest” in the sense that T is a function
of any other sufficient statistic.
Let us think about this concept for a moment. We do want to summarize
the whole data X by reducing it to some appropriate statistic such as , the
median (M), or the histogram, and so on. Suppose that in a particular situa-
tion, the summary statistic T = ( , M) turns out to be minimal sufficient for
θ θ θ θ θ. Can we reduce this summary any further? Of course, we can. We may
simply look at, for example, T = or T = M or T = 1/2 ( + M). Can T ,
1
2
3
1
T , or T individually be sufficient for θθ θθ θ? The answer is no, none of these
2
3
could be sufficient for θθ θθ θ. Because if, for example T by itself was sufficient
1
for θθ θθ θ, then T = ( , M) would have to be a function of T in view of the
1
requirement in part (ii) of the Definition 6.3.1. But, T = ( , M) can not be a
function of T because we can not uniquely specify the value of T from our
1
knowledge of the value of T alone. A minimal sufficient summary T can not
1
be reduced any further to another sufficient summary statistic. In this sense,
a minimal sufficient statistic T may be looked upon as the best sufficient
statistic.
In the Definition 6.3.1, the part (i) is often verified via Neyman factoriza-
tion, but the verification of part (ii) gets more involved. In the next subsec-
tion, we state a theorem due to Lehmann and Scheffé (1950) which provides
a direct approach to find minimal sufficient statistics for θθ θθ θ.
6.3.1 The Lehmann-Scheffé Approach

The following theorem was proved in Lehmann and Scheffé (1950). This
result is an essential tool to locate a minimal sufficient statistic when it exists.
Its proof, however, requires some understanding of the correspondence