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288 6. Sufficiency, Completeness, and Ancillarity
is, merely knowing the value of T after the experiment, some information
about the unknown parameter θ would be lost. !
Example 6.2.7 Suppose that X has the exponential pdf given by f(x) = λe
λx I(x > 0) where λ(> 0) is the unknown parameter. Instead of the original data,
suppose that we are only told whether X ≤ 2 or X > 2, that is we merely
observe the value of the statistic T ≡ I(X > 2). Is the statistic T sufficient for
λ? In order to check, let us proceed as follows: Note that we can express
P{X > 3 | T = 1} as
which depends on λ and hence T is not a sufficient statistic for λ. !
The methods we pursued in the Examples 6.2.1-6.2.7 closely followed
the definition of a sufficient statistic. But, such direct approaches to verify
the sufficiency or non-sufficiency of a statistic may become quite cumber-
some. More importantly, in the cited examples we had started with specific
statistics which we could eventually prove to be either sufficient or non-
sufficient by evaluating appropriate conditional probabilities. But, what is
one supposed to do in situations where a suitable candidate for a sufficient
statistic can not be guessed readily? A more versatile technique follows.
6.2.2 The Neyman Factorization Theorem
Suppose that we have at our disposal, observable real valued iid random vari-
ables X , ..., X from a population with the common pmf or pdf f(x; θ). Here,
n
1
the unknown parameter is θ which belongs to the parameter space Θ.
Definition 6.2.4 Consider the (observable) real valued iid random vari-
ables X , ..., X from a population with the common pmf or pdf f(x; θ), where
n
1
the unknown parameter θ ∈ Θ. Once we have observed X = x , i = 1, ..., n, the
i
i
likelihood function is given by
In the discrete case, L(θ) stands for P {X = x ∩ ... ∩ X = x }, that is the
θ
1
n
n
1
probability of the data on hand when θ obtains. In the continuous case, L(θ)
stands for the joint pdf at the observed data point (x , ..., x ) when θ obtains.
1 n
It is not essential however for the Xs to be real valued or that they be
iid. But, in many examples, they will be so. If the Xs happen to be vector
valued or if they are not iid, then the corresponding joint pmf or pdf of
