A location class , also known as a location family , translation class or translation family , is a special distribution class in mathematical statistics . Location classes are clearly created by shifting a given probability distribution by a certain value. The set of all these shifted probability distributions then forms the location class. A stochastic model whose distribution class is a location class is called a location model . Location classes are used, for example, in the investigation of equivariant estimators and translation invariant estimators and belong to the Q-invariant distribution classes .
definition
On the real numbers
A probability distribution is given . Define
P
{\ displaystyle P}
(
R.
,
B.
(
R.
)
)
{\ displaystyle (\ mathbb {R}, {\ mathcal {B}} (\ mathbb {R}))}
P
ϑ
(
B.
)
: =
P
(
B.
-
ϑ
)
For
ϑ
∈
R.
{\ displaystyle P _ {\ vartheta} (B): = P (B- \ vartheta) {\ text {for}} \ vartheta \ in \ mathbb {R}}
or equivalent to the Dirac distribution in
δ
ϑ
{\ displaystyle \ delta _ {\ vartheta}}
ϑ
{\ displaystyle \ vartheta}
P
ϑ
: =
(
P
∗
δ
ϑ
)
{\ displaystyle P _ {\ vartheta}: = (P * \ delta _ {\ vartheta})}
.
Here denotes the convolution of the probability measures. Then is called
∗
{\ displaystyle *}
P
L.
: =
{
P
ϑ
|
ϑ
∈
R.
}
{\ displaystyle {\ mathcal {P}} _ {L}: = \ {P _ {\ vartheta} \, | \, \ vartheta \ in \ mathbb {R} \}}
the location class generated by.
P
{\ displaystyle P}
In higher dimensions
For a probability measure on one defines
P
{\ displaystyle P}
(
R.
n
,
B.
(
R.
n
)
)
{\ displaystyle (\ mathbb {R} ^ {n}, {\ mathcal {B}} (\ mathbb {R} ^ {n}))}
P
ϑ
(
B.
)
: =
P
(
B.
-
1
⋅
ϑ
)
{\ displaystyle P _ {\ vartheta} (B): = P (B- \ mathbf {1} \ cdot \ vartheta)}
,
where the one vector called, so a vector in all ones as entries. Analogous to above then means
1
{\ displaystyle \ mathbf {1}}
R.
n
{\ displaystyle \ mathbb {R} ^ {n}}
P
L.
: =
{
P
ϑ
|
ϑ
∈
R.
}
{\ displaystyle {\ mathcal {P}} _ {L}: = \ {P _ {\ vartheta} \, | \, \ vartheta \ in \ mathbb {R} \}}
the location class generated by.
P
{\ displaystyle P}
example
Let be a standard normal distribution , i.e. in distribution. Then
P
{\ displaystyle P}
P
=
N
(
0
,
1
)
{\ displaystyle P = {\ mathcal {N}} (0,1)}
P
ϑ
(
B.
)
=
P
(
B.
-
ϑ
)
=
∫
B.
-
ϑ
1
2
π
exp
(
-
1
2
x
2
)
d
λ
(
x
)
=
∫
B.
1
2
π
exp
(
-
1
2
(
x
-
ϑ
)
2
)
d
λ
(
x
)
{\ displaystyle P _ {\ vartheta} (B) = P (B- \ vartheta) = \ int _ {B- \ vartheta} {\ tfrac {1} {\ sqrt {2 \ pi}}} \ exp \ left ( - {\ tfrac {1} {2}} x ^ {2} \ right) \ mathrm {d} \ lambda (x) = \ int _ {B} {\ tfrac {1} {\ sqrt {2 \ pi} }} \ exp \ left (- {\ tfrac {1} {2}} (x- \ vartheta) ^ {2} \ right) \ mathrm {d} \ lambda (x)}
.
So , the location class consists exactly of the normal distributions with variance one and expected value :
P
ϑ
=
N
(
ϑ
,
1
)
{\ displaystyle P _ {\ vartheta} = {\ mathcal {N}} (\ vartheta, 1)}
ϑ
{\ displaystyle \ vartheta}
P
L.
=
{
N
(
ϑ
,
1
)
|
ϑ
∈
R.
}
{\ displaystyle {\ mathcal {P}} _ {L} = \ {{\ mathcal {N}} (\ vartheta, 1) \, | \, \ vartheta \ in \ mathbb {R} \}}
.
It should be noted, however, that for all distributions, as in the example above, a shift by on the x-axis does not correspond to a change in the position parameter of the distribution by . An example of this would be the geometric distribution with the expected value as the location parameter.
ϑ
{\ displaystyle \ vartheta}
ϑ
{\ displaystyle \ vartheta}
properties
A probability measure and the generated location class are given . Then:
P
{\ displaystyle P}
P
L.
{\ displaystyle {\ mathcal {P}} _ {L}}
P
L.
{\ displaystyle {\ mathcal {P}} _ {L}}
is a dominated distribution class if and only if is dominated, that is, absolutely continuous with respect to a σ-finite measure .
P
{\ displaystyle P}
The following applies more strongly: is absolutely continuous with respect to a σ-finite measure if and only if all are absolutely continuous with respect to.
P
{\ displaystyle P}
λ
{\ displaystyle \ lambda}
P
ϑ
∈
P
L.
{\ displaystyle P _ {\ vartheta} \ in {\ mathcal {P}} _ {L}}
λ
{\ displaystyle \ lambda}
Denoting by generated Lokationsklasse so true .
P
L.
(
P
)
{\ displaystyle {\ mathcal {P}} _ {L} (P)}
P
{\ displaystyle P}
P
L.
(
P
⊗
n
)
=
P
L.
(
P
)
⊗
n
{\ displaystyle {\ mathcal {P}} _ {L} (P ^ {\ otimes n}) = {\ mathcal {P}} _ {L} (P) ^ {\ otimes n}}
Each location class is a Q-invariant distribution class with respect to
Q
: =
{
T
ϑ
|
ϑ
∈
R.
}
With
T
ϑ
(
x
)
=
x
-
ϑ
⋅
1
{\ displaystyle {\ mathcal {Q}}: = \ {T _ {\ vartheta} \, | \, \ vartheta \ in \ mathbb {R} \} {\ text {with}} T _ {\ vartheta} (x) = x- \ vartheta \ cdot \ mathbf {1}}
.
literature
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">