5 Must-Read On Minimal Sufficient Statistics
e.
(1) For all n and all strings x of length n there is a program x
∗
for x with
\(\textit {CT}(x^{*}|x)\leqslant 2\log n+O(1)\)
and
\(|x^{*}|\leqslant C(x)+O(1)\).
The joint density of the sample takes the form required by the Fisher–Neyman factorization theorem, by letting
Since
h
(
x
n
)
{\displaystyle h(x_{1}^{n})}
does not depend on the parameter
(
,
)
{\displaystyle (\alpha ,\beta )}
and
g
(
,
)
(
x
1
n
)
{\displaystyle g_{(\alpha \,,\,\beta )}(x_{1}^{n})}
depends only on
x
1
n
{\displaystyle x_{1}^{n}}
through the function
T
(
X
1
n
)
=
(
my website min
1
i
n
X
i
,
max
n
X
i
)
,
{\displaystyle T(X_{1}^{n})=\left(\min _{1\leq i\leq n}X_{i},\max _{1\leq i\leq n}X_{i}\right),}
the Fisher–Neyman factorization theorem implies
T
(
X
1
n
)
=
(
min
1
i
n
X
i
,
max
1
i
n
X
i
)
{\displaystyle T(X_{1}^{n})=\left(\min _{1\leq i\leq n}X_{i},\max _{1\leq i\leq n}X_{i}\right)}
is a sufficient statistic for
(
,
)
{\displaystyle (\alpha \,,\,\beta )}
. .