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In statistics , the estimation error describes the deviation of an estimation function from the unknown parameter of the population . It is a measure of the quality of the estimator (or interpolation ).
ϑ
^
{\ displaystyle {\ hat {\ vartheta}}}
ϑ
{\ displaystyle \ vartheta}
definition
It is defined as:
e
: =
ϑ
^
-
ϑ
{\ displaystyle e: = {\ hat {\ vartheta}} - \ vartheta}
If the true parameter is unknown, the estimation error is also unknown. Nevertheless, it is possible to make a statement about the precision of the estimation error.
Parameter of the estimation error
The expected value of the estimation error is called the bias .
The standard deviation of the estimation error is equal to the standard error .
Examples
If the mean is in the population, the variance and the proportional value are in a dichotomous population, then the following table shows estimators, estimation errors and biases. Here referred to the normal distribution with expected value and variance .
μ
{\ displaystyle \ mu}
σ
2
{\ displaystyle \ sigma ^ {2}}
π
{\ displaystyle \ pi}
N
(
μ
,
σ
2
)
{\ displaystyle N (\ mu, \ sigma ^ {2})}
μ
{\ displaystyle \ mu}
σ
2
{\ displaystyle \ sigma ^ {2}}
Parameters of population
Sample variables
Estimator
θ
^
{\ displaystyle {\ hat {\ theta}}}
Estimation error
e
{\ displaystyle e}
distortion
E.
(
e
)
{\ displaystyle \ operatorname {E} (e)}
μ
{\ displaystyle \ mu}
X
i
∼
N
(
μ
,
σ
2
)
{\ displaystyle X_ {i} \ sim N (\ mu, \ sigma ^ {2})}
X
¯
=
X
1
+
...
+
X
n
n
{\ displaystyle {\ bar {X}} = {\ frac {X_ {1} + \ ldots + X_ {n}} {n}}}
X
¯
-
μ
{\ displaystyle {\ bar {X}} - \ mu}
0
{\ displaystyle 0}
μ
{\ displaystyle \ mu}
X
i
∼
(
μ
,
σ
2
)
{\ displaystyle X_ {i} \ sim (\ mu, \ sigma ^ {2})}
and ZGS fulfilled
X
¯
=
X
1
+
...
+
X
n
n
{\ displaystyle {\ bar {X}} = {\ frac {X_ {1} + \ ldots + X_ {n}} {n}}}
X
¯
-
μ
{\ displaystyle {\ bar {X}} - \ mu}
0
{\ displaystyle 0}
π
{\ displaystyle \ pi}
X
i
{\ displaystyle X_ {i}}
dichotomous
Π
=
X
1
+
...
+
X
n
n
{\ displaystyle \ Pi = {\ frac {X_ {1} + \ ldots + X_ {n}} {n}}}
Π
-
π
{\ displaystyle \ Pi - \ pi}
0
{\ displaystyle 0}
σ
2
{\ displaystyle \ sigma ^ {2}}
X
i
∼
N
(
μ
,
σ
2
)
{\ displaystyle X_ {i} \ sim N (\ mu, \ sigma ^ {2})}
and known
μ
{\ displaystyle \ mu}
S.
∗
2
=
1
n
∑
i
=
1
n
(
X
i
-
μ
)
2
{\ displaystyle S ^ {* 2} = {\ frac {1} {n}} \ sum _ {i = 1} ^ {n} (X_ {i} - \ mu) ^ {2}}
S.
∗
2
-
σ
2
{\ displaystyle S ^ {* 2} - \ sigma ^ {2}}
0
{\ displaystyle 0}
σ
2
{\ displaystyle \ sigma ^ {2}}
X
i
∼
N
(
μ
,
σ
2
)
{\ displaystyle X_ {i} \ sim N (\ mu, \ sigma ^ {2})}
and unknown
μ
{\ displaystyle \ mu}
S.
2
=
1
n
-
1
∑
i
=
1
n
(
X
i
-
X
¯
)
2
{\ displaystyle S ^ {2} = {\ frac {1} {n-1}} \ sum _ {i = 1} ^ {n} (X_ {i} - {\ bar {X}}) ^ {2 }}
S.
2
-
σ
2
{\ displaystyle S ^ {2} - \ sigma ^ {2}}
0
{\ displaystyle 0}
σ
2
{\ displaystyle \ sigma ^ {2}}
X
i
∼
N
(
μ
,
σ
2
)
{\ displaystyle X_ {i} \ sim N (\ mu, \ sigma ^ {2})}
and unknown
μ
{\ displaystyle \ mu}
S.
′
2
=
1
n
∑
i
=
1
n
(
X
i
-
X
¯
)
2
{\ displaystyle S ^ {'2} = {\ frac {1} {n}} \ sum _ {i = 1} ^ {n} (X_ {i} - {\ bar {X}}) ^ {2} }
S.
′
2
-
σ
2
{\ displaystyle S ^ {'2} - \ sigma ^ {2}}
-
σ
2
n
{\ displaystyle - {\ frac {\ sigma ^ {2}} {n}}}
See also
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