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if X1,X2,...,Xn is a random sample from a distribution with pdf
f(x;θ)= 3θ^3/(x+θ)^4; 0<x<∞, 0<θ<∞; 0 elsewhere.
show that Y= 2X(bar) is an unbiased estimator of θand determine its efficiency.
Steps
Step 1 of 2
solution.) from the given data,
in order to show that is an unbiased estimator of
we have to prove that ,
please see the next step
Step 2 of 2
from the above data ,
C.R lower bound =
Efficiency
Hence solved
please refer to the above step
Final Answer
C.R lower bound =
Efficiency
y
= 2
x θ
,
E
(
Y
) = 2.
E
(
X
1) =
θ
EY
= 2
E
¯¯¯
X
= 2
EX
1= 2
∫
+ ∞
0
dx
=
θ
3
θ
3
x
(
x
+
θ
)4
var
(
Y
) = 4
var
(
¯¯¯
X
)
=
var
(
x
1)
4
n
EX
2
1=
∫
∞
0
dx
=
θ
2
3
θ
3
x
2
(
x
+
θ
)4
var
(
Y
) =
(
θ
2−
)
=
4
n
θ
2
4
3
θ
2
n
logf
=
log
3 + 3
logθ
− 4
log
(
x
+
θ
)
= − ,
∂log
f
∂
θ
3
θ
4
x
+
θ
= − +
∂2log
f
∂
θ
2
3
θ
2
4
(
x
+
θ
)2
E
= − + 4
∫
∞
0
dx
= − + =
∂2log
f
∂
θ
2
3
θ
2
1
(
x
+
θ
)2
(3
θ
)3
(
x
+
θ
)4
3
θ
2
12
5
θ
2
3
5
θ
2
5
θ
2
3
n
= =
5
θ
2
3
n
3
θ
2
n
5
9
5
θ
2
3
n
= =
5
θ
2
3
n
3
θ
2
n
5
9
