Page 313 - Probability and Statistical Inference

290 6. Sufficiency, Completeness, and Ancillarity

If part: Suppose that the factorization in (6.2.5) holds. Let us denote the
pmf of T by p(t; θ). Observe that the pmf of T is given by p(t; θ) = P {T(X)
θ
= t} = . It is easy to see that

For all x ∈ χ such that T(x) = t and p(t; θ) ≠ 0, we can express P {X = x
θ
|T(X) = t} as

because of factorization in (6.2.5). Hence, one gets

where q(x) does not depend upon θ. Combining (6.2.7)-(6.2.8), the proof of
the “if part” is now complete. !

In the statement of the Theorem 6.2.1, notice that we do not
demand that g(T(x , ..., x );θ) must be the pmf or the pdf of
1
n
T(X , ..., X ). It is essential, however, that the function
1 n
h(x , ..., x ) must be entirely free from θ.
1 n
It should be noted that the splitting of L(θ) may not be unique, that is
there may be more than one way to determine the function h(.) so that
(6.2.5) holds. Also, there can be different versions of the sufficient statis-
tics.
Remark 6.2.1 We mentioned earlier that in the Theorem 6.2.1, it was
not essential that the random variables X , ..., X and the unknown param-
1
n
eter θ be all real valued. Suppose that X , ..., X are iid p-dimensional ran-
n
1
dom variables with the common pmf or pdf f(x; θθ θθ θ) where the unknown
q
parameter θθ θθ θ is vector valued, θθ θθ θ ∈ Θ ⊆ ℜ . The Neyman Factorization Theo-
rem can be stated under this generality. Let us consider the likelihood
function,