Suppose

Inside Collection (Course): Statistical Signal Processing

Determining a sufficient statistic directly from the definition can be a tedious process. The following result can simplify this process by allowing one to spot a sufficient statistic directly from the functional form of the density or mass function.

Let

Suppose

The next example illustrates the appliction of the theorem to a continuous random variable.

Consider a normally distributed random sample

First, suppose

Suppose the probability mass function for

From the proof, the Fisher-Neyman
factorization gives us a formula for the conditional
probability of x x
given t t . In the
discrete case we have
f x |
t
= a x ∑ T x = t a x
f
t
x
a
x
x
T
x
t
a
x
An analogous formula holds for continuous random variables
(Scharf, pp.82).

The following exercises provide additional examples where the Fisher-Neyman factorization may be used to identify sufficient statistics.

Suppose

Express the likelihood
f
θ
x
f
θ
x
in terms of indicator functions.

Suppose

Find a sufficient statistic

What is the conditional probability
mass function of

Consider

Use the same trick as in Example 2.

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