Suﬃcient Statistics
Charles B. Moss
July 23, 2010
I. Data Reduction
A. The typical mode of operation in statistics is to use information
from a sample
X
1
,
· · ·
X
2
to make inferences about an unknown
parameter
θ
.
1. Put slightly differently, the researcher summarizes the infor
mation in the sample (or the sample values) with a statistic.
2. Thus, any statistic
T
(
X
) summarizes the data, or reduces the
information in the sample to a single number.
We use only
the information in the statistic instead of the entire sample.
3. Put in a slightly more mathematical formulation, the statistic
partitions the sample space into two sets
a) Defining the sample space for the statistic
T
=
{
t
:
t
=
T
(
x
)
, x
∈
X
}
(1)
b) Thus, a given value of a sample statistic
T
(
x
) implies that
the sample comes from a space of sets
A
t
such that
t
∈
T
,
A
t
=
{
x
:
T
(
x
) =
t
}
.
c) The second possibility (that is ruled out by observing a
sample statistic of
T
(
x
) ) is
A
C
t
=
{
x
:
T
(
x
) =
t
}
.
4. Thus, instead of presenting the entire sample, we could report
the value of the sample statistic.
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 Spring '10
 Staff
 Professor Charles B. Moss, Cassela

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