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Mean square error
The mean square error of an estimator
θ
^
of the parameter
θ
is given by
M
S
E
(
θ
^
)
=
E
[
(
θ
^
−
θ
)
2
]
=
E
[
(
θ
^
−
E
(
θ
^
)
+
E
(
θ
^
)
−
θ
^
)
2
]
=
E
[
(
θ
^
−
E
(
θ
^
)
2
+
2
(
θ
^
−
E
(
θ
^
)
)
(
E
(
θ
^
−
θ
)
)
+
(
E
(
θ
^
)
−
θ
)
2
)
]
=
E
(
(
θ
^
−
E
(
θ
^
)
)
2
)
+
2
E
(
(
θ
^
−
E
(
θ
^
)
)
(
E
(
θ
^
)
−
θ
)
)
+
E
(
(
E
(
θ
^
−
θ
)
)
2
)
=
v
a
r
(
θ
^
)
+
2
(
E
(
θ
^
)
−
θ
)
(
E
(
θ
^
)
−
E
(
θ
^
)
)
+
b
i
a
s
2
(
θ
^
)
=
v
a
r
(
θ
^
)
+
0
+
b
i
a
s
2
(
θ
^
)
=
v
a
r
(
θ
^
)
+
[
E
(
θ
^
)
−
θ
]
2
So basically
E
[
(
X
¯
−
μ
)
2
]
=
V
a
r
(
X
¯
)
=
σ
2
n