NTHU MATH 2820, 2008, Lecture Notes
Ch8, p.53
•
Data reduction 
the concepts of sufficiency, minimal sufficiency, and completeness
Question 6.4
(information and data reduction)
raw (original) data
order statistics
histogram
sample mean
sample variance
To present concrete information, the data reduction through useful transfor
mations is required. However, noninvertible transformations can cause the
loss of information. ( Example? ) The lost information can be important or
worthless to the objective of studying the data. How to examine whether the
important information lose in transformation? Furthermore, what is
impor
tant information
?
1
n
n
i
=1
(
X
i
−
X
n
)
2
X
1
, X
2
, . . . , X
n
X
n
X
(1)
, X
(2)
, . . . , X
(
n
)
(numerical or graphical) transformations of data appear all the time in statis
tics for o
ff
ering a summary of information contained in data. For example,
made by ShaoWei Cheng (NTHU, Taiwan)
Ch8, p.54
Summary
(formulation of information and data reduction problem, TBp. 305)
•
Let
X
1
, X
2
, . . . , X
n
be a sample with joint pdf/pmf
f
(
x

Θ
), wherer
Θ
is unknown parameter.
—
X
1
, X
2
, . . . , X
n
contains two types of information:
∗
information related to
Θ
∗
information irrelevant to
Θ
—
For example, toss a coin
n
times, i.e.,
X
1
, X
2
, . . ., X
n
are i.i.d.
from Bernoulli
B
(
θ
),
∗
X
n
or
T
=
n
i
=1
X
i
contains information about
θ
∗
When
T
is known, the information that at which trials the
head’s occur is irrelevant to
θ
∗
n
=5, consider the following possible results:
B
(0, 1, 1, 1, 1),
T
= 4; (1, 0, 1, 1, 1),
T
= 4;
(1, 1, 0, 1, 1),
T
= 4; (1, 1, 1, 0, 1),
T
= 4;
(1, 1, 1, 1, 0),
T
= 4
B
(1, 0, 0, 0, 0),
T
= 1; (0, 1, 0, 0, 0),
T
= 1;
(0, 0, 1, 0, 0),
T
= 1; (0, 0, 0, 1, 0),
T
= 1;
(0, 0, 0, 0, 1),
T
= 1