An approximate pivot statistic is a sequence of functions in mathematical statistics that is used to construct approximate confidence ranges . It thus forms the asymptotic counterpart to pivot statistics , which are used to construct (non-approximate) confidence ranges .
definition
Framework
For were measuring rooms and families of probability measures on . Be another measuring room as well
n
∈
N
{\ displaystyle n \ in \ mathbb {N}}
(
X
n
,
A.
n
)
{\ displaystyle (X_ {n}, {\ mathcal {A}} _ {n})}
(
P
ϑ
,
n
)
ϑ
∈
Θ
{\ displaystyle (P _ {\ vartheta, n}) _ {\ vartheta \ in \ Theta}}
(
X
n
,
A.
n
)
{\ displaystyle (X_ {n}, {\ mathcal {A}} _ {n})}
(
Γ
,
A.
Γ
)
{\ displaystyle (\ Gamma, {\ mathcal {A}} _ {\ Gamma})}
G
:
Θ
→
Γ
{\ displaystyle g \ colon \ Theta \ to \ Gamma}
the function to be estimated.
In most cases, the measurement rooms and families of probability measures are -fold product models . Would be typical example of this , and as probability a corresponding product measure of a probability measure on .
n
{\ displaystyle n}
X
n
=
R.
n
{\ displaystyle X_ {n} = \ mathbb {R} ^ {n}}
P
ϑ
n
{\ displaystyle P _ {\ vartheta} ^ {n}}
P
ϑ
{\ displaystyle P _ {\ vartheta}}
R.
{\ displaystyle \ mathbb {R}}
Formalization
A series of statistics with
(
T
n
)
n
∈
N
{\ displaystyle (T_ {n}) _ {n \ in \ mathbb {N}}}
T
n
:
X
n
×
Γ
→
Γ
{\ displaystyle T_ {n} \ colon X_ {n} \ times \ Gamma \ to \ Gamma}
is called an approximate pivot statistic for if:
G
{\ displaystyle g}
There is a probability distribution on , so that the distribution of all to converge . So it is
Q
{\ displaystyle Q}
(
Γ
,
A.
Γ
)
{\ displaystyle (\ Gamma, {\ mathcal {A}} _ {\ Gamma})}
T
n
(
⋅
,
G
(
ϑ
)
)
{\ displaystyle T_ {n} (\ cdot, g (\ vartheta))}
ϑ
∈
Θ
{\ displaystyle \ vartheta \ in \ Theta}
Q
{\ displaystyle Q}
P
ϑ
∘
T
n
(
⋅
,
G
(
ϑ
)
)
-
1
→
D.
Q
{\ displaystyle P _ {\ vartheta} \ circ T_ {n} (\ cdot, g (\ vartheta)) ^ {- 1} {\ stackrel {\ mathcal {D}} {\ rightarrow}} Q}
for and for everyone .
n
→
∞
{\ displaystyle n \ to \ infty}
ϑ
∈
Θ
{\ displaystyle \ vartheta \ in \ Theta}
For all levels is in included.
B.
∈
A.
Γ
{\ displaystyle B \ in A _ {\ Gamma}}
{
x
∈
X
n
∣
T
n
(
x
,
G
(
ϑ
)
)
∈
B.
}
{\ displaystyle \ {x \ in X_ {n} \ mid T_ {n} (x, g (\ vartheta)) \ in B \}}
A.
n
{\ displaystyle {\ mathcal {A}} ^ {n}}
The second condition guarantees that all sets can be assigned probabilities in meaningful ways by means of the probability measures , that is, the distribution of is well-defined for all .
A.
Γ
{\ displaystyle {\ mathcal {A}} _ {\ Gamma}}
P
ϑ
{\ displaystyle P _ {\ vartheta}}
T
n
(
⋅
,
G
(
ϑ
)
)
{\ displaystyle T_ {n} (\ cdot, g (\ vartheta))}
ϑ
{\ displaystyle \ vartheta}
example
Consider a Bernoulli product model, so
X
=
{
0
,
1
}
and
A.
=
P
{
0
,
1
}
{\ displaystyle X = \ {0.1 \} {\ text {and}} {\ mathcal {A}} = {\ mathcal {P}} \ {0.1 \}}
provided with the Bernoulli distribution for the parameter .
ϑ
∈
(
0
,
1
)
{\ displaystyle \ vartheta \ in (0,1)}
The -fold product model is then . The parameter of the Bernoulli distribution is to be estimated, so the function to be estimated is
n
{\ displaystyle n}
(
X
n
,
A.
n
,
(
Ber
ϑ
n
)
ϑ
∈
(
0
,
1
)
)
{\ displaystyle (X ^ {n}, {\ mathcal {A}} ^ {n}, (\ operatorname {Ber} _ {\ vartheta} ^ {n}) _ {\ vartheta \ in (0,1)} )}
G
(
ϑ
)
=
ϑ
{\ displaystyle g (\ vartheta) = \ vartheta}
.
Let be the sample variable . They are independently distributed identically and it is
X
=
(
X
1
,
X
2
,
...
,
X
n
)
{\ displaystyle X = (X_ {1}, X_ {2}, \ dots, X_ {n})}
X
i
{\ displaystyle X_ {i}}
T
n
(
X
,
ϑ
)
=
n
(
(
1
n
∑
i
=
1
n
X
i
)
-
ϑ
ϑ
(
1
-
ϑ
)
)
{\ displaystyle T_ {n} (X, \ vartheta) = {\ sqrt {n}} \ left ({\ frac {\ left ({\ tfrac {1} {n}} \ sum _ {i = 1} ^ {n} X_ {i} \ right) - \ vartheta} {\ sqrt {\ vartheta (1- \ vartheta)}}} \ right)}
an approximate pivot statistic, since it converges to the standard normal distribution according to the Moivre-Laplace theorem . So it is .
Q
=
N
(
0
,
1
)
{\ displaystyle Q = {\ mathcal {N}} (0,1)}
swell
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