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statistics, Let and let be independent

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Let
0
λ
>
and let
1 2
, ,...,
n
X X X
be independent R.V each with pdf
( )
1
;f x
x
λ
λ
λ
+
=
Suppose
1
λ
>
(a) Obtain the method of moments estimator of
,
λ λ
:
(b) Is
λ
:
a consistent estimator of
λ
? Justify your answer
(c) Is
λ
:
an unbiased estimator of
?
λ
Justify your answer
(d) Suppose
2
λ
>
, show that
λ
:
is asymptotically normally distributed ( as
n → ∞
). Find the
parameters
(d)
Consider
( )
1
;f x
x
λ
λ
λ
+
=
By invariance of ML estimators to reparameterization, or first principles, the ML
estimator of
( )
1
;f x
x
λ
λ
λ
+
=
is
For Cramer’s Theorem ( Delta Method )
Let
( )
1
;g x
x
λ
λ
λ
+
=
, so that
( )
( )
1
1
;g x
xx
λ
λ λ
λ
+
+
= −
÷
g
Thus
1 1
2
,
n
T AN
x nx
λ λ
λ λ
λ
+ +
=
÷ ÷
(a)
We have first moments of the random variable:
( )
__
1
1
1 1
1
1 1
1
E X
1
1
i
x
x dx
x
dx
x x
dx
x
λ
λ
λ
λ
λ
λ
∞
+
∞
+ −
∞
+ −
=
=
=
=
∫
∫
∫
So that
__
1
x
λ
λ
=
−
( )
__
__
1
1
x
x
λ
λ
λ λ
=
−
− =
__ __
__ __
__ __
__
__
1
1
x x
x x
x x
x
x
λ λ
λ λ
λ
λ
− =
− =
− =
÷
=
−
÷
Therefore, the moment esmate is
__
__
1
x
x
λ
=
−
÷
(b)
We have
Var 0
1
x
x
÷
≠
÷
−
:
:
as
n → ∞
,Since
( )
2
Var
2 1
i
X
λ λ
λ λ
= −
÷
− −
does not tend to
zero as
n
→ ∞
(C)
We have
E
1
x
x
λ
÷
≠
÷
−
:
:
Therefore,
λ
:
is not unbiased estimator of
λ

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Let and let be independent R.V each with pdf Suppose (a) Obtain the method of moments estimator of (b) Is a consistent estimator of ? Justify your answer (c) Is an unbiased estimator of Justify your answer (d) Suppose , show that is asymptotically normally distributed ( as ). Find the parameter ...
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