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Statistical Method In Economics

2. with ci = 1 for all i.
n
Under what condition(s) is c1X1 +...+cnXn an unbiased estimator
a.
of µ?
Problem 1
Let X1, ..., Xn be a random sample (i.i.d.) of size n from a population with distribution ƒ(x) with
mean µ and variance σ2 (both finite). Consider a statistic formed by taking a linear combination of
the Xis, c1X1 +... +cnXn, where ci ≥ 0. For example, the sample mean, X, is the linear
combination
b. What is the variance of c1X1 + ...+ cnXn?
c. Find the linear combination (derive the c's) that minimizes the variance without violating
the condition(s) you derived in part a. (i.e. find the most efficient linear unbiased estimator of µ).
i (
Problem 2
Suppose that you have a sample of size n, X1, ..., Xn, from a population with mean µ and
variance σ2. You know that your draws from the popula- tion, the Xi, are identically distributed,
however, they are not independent
2
/
and have Cov(X ,X ) = gσ > 0, for all i = (.
a. What is E X ? Is the sample mean an unbiased estimator of µ?
Σ
i
i
b. What is the MSE of X ? You can use the fact that Var X =
Σ
i
Σ Σ
i (/=i
Var (Xi) + Cov(Xi,X().
c. Is the sample mean a consistent estimator of µ?consistent if g = 0? Would it be
1
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PROBLEMS

3. 2
Problem 3
Suppose that Z1,Z2,...,Zn is a random sample from an exponential dis-
tribution with parameter Z, so ƒ(x) =
Ze — λx x ≥ 0
0 elserhere
a. Find the Method of Moments estimator for Z.
b.
d.
Find the Maximum Likelihood estimator for Z.
c. Find the Maximum Likelihood estimator for
√
Z.
Is the estimator in part b unbiased? Is it consistent?
Problem 4
Assume a sample of continuous random variables: X1, X2, ...,Xn, where
i
E [X ] = µ, Var [X ] = σ2 > 0. Consider the following estimators: µ
^ =
n 2,n
X ,µ
^ = 1
n+1
i 1,n
n
Σ
i
X .
i=1
a. Are µ^1
,
n and µ^2
,
n unbiased?
b. Are µ
^ and µ
1,n 2,n
^ consistent?
c. What do you conclude about the relation between unbiased and consistent estimators?
d. For what values of n does µ^2
,
nhave a lower mean square error than
µ
^1,n ?
Problem 5
You have a sample of size n, X1, ...,Xn from a U [8l,8u] distribution.
a. Assume that it is known that 8l = 0. Find the MM and MLE estimators of 8u.
b. Now assumethat both 8l and 8u are unknown. estimators of 8l and
8u.
Find the MM
c. Determine whether the estimators in parts aand b are unbiased and consistent.
Problem 6
Let X1, ...,Xn be a random sample of size n from a U [0,8]population.
Consider the following estimators of 8:
^
8
1
^
8
2
= k1X2
= k2X
= k3X(n)
^
8
3
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4. 3
where the ki are constants and
X(n) = max(X1,...,Xn)
X(n) is known as the nth order statistic.
a. Calculate Pr X (n) ≤ x , given our knowledge of the population
distribution.
b. Calculate the pdf of X(n).
c. Choose k1 ,k2, and k3 such that all three estimators are unbiased;
call these values k
^ ^
1 2 3
^ ^ 1 1
h i
,k , and k (In other words, solve E k X = 8, and so
forth).
d. Calculate the following variances and compare them:
Var ^
k1X2
Var ^
k2X
Var ^
k3X(n)
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5. ANSWERS
Problem 1
a. For the statistic to be unbiased we must have
n
Σ
µ = E ciXi
i=1
!
n
Σ
i=1
= E (X ) c
i i
n
Σ
= µ c
i=1
i
n
Σ
i=1
So the statistic will be unbiased if and only if ci = 1.
b.
tistic is
Given that all of the Xi are independent, the variance of the sta-
Var
Σ
!
i=1
ciXi =
n n Σ
i=1
i
c2Var (Xi)
n
Σ
= σ c
2 2
i=1
i
c. We want to minimize
n
Σ
σ c
2 2
i=1
i
subject to the constraint
n
Σ
i
c = 1
i=1
The Lagrangian of this optimization problem is
n
Σ
i=1
2
i
G = σ2 c —Z
n
Σ
i=1
ci —1
!
1
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6. 2
And we have the first-order conditions
2
i
= σ 2c —Z = 0
6G
6ci
ci =
Z
2σ2
This implies that all of the ci are the same, that is c1 = c2 = ... = cn = c. Wecan then solve for
this common value c by substituting back into the
n
Σ 1
constraint c = 1, which implies that c = n.
i=1
Problem 2
a.
E X = E 1
n
n
Σ
i=1
Xi
!
= 1
n
n
Σ
i=1
i
E (X )
=
n
Σ
i=1
µ
1
n
= µ
So X is an unbiased estimator of µ.
b.
M SE X = E X —µ 2
+ Var X
= 0 + Var 1
n
n
Σ
i=1
Xi
!
1 n
Σ
i=1
= Var Xi n2
!
=
1
n2
@
i i (
=
/ i
i
Var (X ) +
Σ Σ Σ
i (
Cov(X ,X )
, 1
,
=
1
n2
,
2
Σ Σ Σ
σ + gσ 2
1
,
=
σ2
n
@
+
i i (/=i
(n —1)gσ2
n
economicshomeworkhelper.com

7. 3
c. To determine whether the sample mean is a consistent estimator, we need to see
whether the MSE goes to zero as n approaches infinity.
σ2
n → ∞ n → ∞ n
lim MSE X = lim +
(n —1)gσ 2
n
σ2
= lim + lim n→ ∞
2
(n —1)gσ
n → ∞ n n
= 0 + gσ2 = gσ2
Thus (assuming a positive variance) the sample mean is not a consistent estimator of the
population mean unless g = 0.
Problem 3
a. We know that the first moment of Z is
∫ ∞
O
E (Z) = zƒ(z) dz
∫ ∞
O
— λx
= zZe dz
=
1
Z
1
Z
= 1
n
Then, we calculate our method of moments estimator by setting E (Z) =
Z and solving for Z.
n
Σ
i=1
Z i
Z^MM =
n
Σ n
i=1 zi
1
=
Z
b. The exponential pdf is ƒ(z) = Ze—λx. Thus, the likelihood function for Z is
G(Z;z) =
Y n
i=1
Ze— λxi
.
Then, the log likelihood function is
ln G(Z;z) = ln
Y n
i=1
Ze—λxi
=
Σ n
i=1 i
(—
Zz ) + nln Z
Differentiating the function with respect to Z and set it to 0,
6lnG(Z;z)
6Z
= —
Σ n
i=1
i
Z
n
(z ) + = 0
=⇒ Σ n
i=1
zi =
n
Z
economicshomeworkhelper.com

8. 4
So we find that the MLE estimator is the same as the MM estimator:
^
Z = Σ
M LE n
i=1 zi
n 1
=
Z
c. Using the invariance property, the MLE for
√
Z is simply
^
√
ZMLE = M L E
Z
q
^
r
1 n
Z = √ = Σ n
i=1 i
.
Z
d. ^
We know that Z M LE will be consistent because all MLE estimators
are consistent. ^
But is it unbiased? A direct calculation of E(Z M LE ) =
E 1 is untractable (note that E
7 7
/
=
1 1
E (7 ) , in general). But we can
use Jensens's inequality to help us out, which tells us that for any random variable X, if g(x) is
a convex function, then
E (g(X )) ≥ g(E (X ))
with a strict inequality if g(x) is strictly convex. Thus, using Z as our
1
7
random variable and the strictly convex function g Z = , we have
E
Z E Z
> =
1 1 1
E (Z)
= Z
and we know that our estimator is biased.
Note that if we had instead used the version of the exponetial pdf that
β
replaces Z with 1 , and estimated the parameter β by either method, we
^ 1
λ
would have found β = Z, which is both unbiased (since E (Z) = = β) and
consistent (by the law of large numbers).
Problem 4
a. Bias µ 1,n
^ = E µ
^ 1,n n
—µ = E [X ] —µ = µ —µ = 0 so µ 1,n
^ is
unbiased for every n.
Bias µ^2
,
n = E µ2,n ^ —µ = E 1
n+1
n
Σ
" #
i=1
Xi —µ = 1
n+1
n
Σ
i=1
E [Xi]—µ =
n
n+1
— µ
n+1 /
µ —µ = = 0 so µ 2,n
^ is biased for for every n.
b. lim . .
n → ∞ 1,n n → ∞ n
P µ
^ —µ < o = lim P (|X —µ| < o) = P (µ —o< X n < µ + o) =
P (Xn ≤ µ + o)—P (Xn ≤ µ —o)(because the R.V. are continuous)= F (µ + o)—
F (µ —o) /= 0 for some o> 0. Thus µ 1,n
^ is not consistent.
As for µ 2,n 2,n
^ : Var µ
^ = Var
"
1
n+1
n
Σ
i=1
i
X =
#
1
n+1
2 n
Σ
i=1
i
Var [X ] =
1
n+1
2
nσ2 so:
economicshomeworkhelper.com

9. 5
→ ∞
limn MSE µ 2,n n → ∞ 2,n
^ = lim Var µ
^ +lim n→ ∞ Bias µ
^ 2,n
2
=
0, proving that µ 2,n
^ is consistent.
c. An unbiased estimator is not necessarily consistent; a consistent estimator is not
necessarily unbiased.
Problem 5
a. MM: Since we know that 8l = 0, we only need to use the first
moment equation:
0 + 8h
E(X) = , 2
Then, the MM estimator is obtained by solving
8h
2
=
^ Σ
1
n
2
n
i
X = X
^h
8 =
Σ
n
i=1
n
i=1
i
X = 2X
MLE: The uniform pdf is
ƒ(x) = =
1 1
8h —8l 8h
and the likelihood function and log likelihood functions are given (respec- tively) by
G(0,8h;x1,x2,...,xn) =
1
8h
n
ln G(0,8h;x) = —
nln 8h
In order to maximize the log likelihood function above, weneed to
minimize 8h subject to the constraint xi ≤ 8h 6 xi.
Then, it must be that
^
8h = max(xi)
b. The first two moments are
E(X) =
2
8l + 8h
,
E(X2) = l h
82 + 82 + 8l8h
3
.
Then, the MM estimators are obtained by solving
l
^ ^
8 + 8 h
2
=
n
i=1
Σ
Σ
i
X = X,
^2
l
^2
h
^^
8 + 8 + 88 l h
3
=
1
n
1
n
n
i=1 i
X2 = X2.
economicshomeworkhelper.com

10. 6
They are given by
^l
2
8 = X —
q
3(X 2 —X ),
^
8h
2
= X +
q
3(X 2 —X ).
c. We begin with the MM estimator for part a:
^h
^
8h
2
^
E 8 = E 2X = 2 = 8 h
So this estimator is unbiased. Then we consider the variance.
^h
Var 8 = Var 2X =
4
n2
n
Σ
i=1
Var (Xi) = 1
2 2
2
4 8 + 8 —28 8 1 2
n 12
Because the bias is zero and the variance approaches zero as n gets large, the MSE also
approaches zero, and the estimator is consistent.
For the MLE estimator, we use the fact (shown in problem 6) that, for
8n
this estimator, ƒ(x) = nxn—1
. Then we can see that the MLE estimator is
biased:
^
E 8h =
∫ h
8 n
O h
nx n
8n dx =
n + 1
8h
n
n+1
However, as n gets large, —
→ 1, so the bias approaches zero, and we
know that in general, MLE estimators are consistent.
For the MM estimators in part b, finding the expected value is not par- ticularly tractable nor
particularly interesting, so I will retract this portion of the question.
8 8 2
Problem 6
Note that for a U [0,8]distribution, ƒ(x) = 1, F (x) = x , E (X) = 8,
and Var (X ) = 82
.
a.
12
It will be important that the sample draws are independent (and
identical). Then,
Pr X(n) ≤ x = Pr (max(Xi) ≤ x)
n
i=1
i
= П Pr (X ≤ x)
= F (x)n
=
x
8
n
economicshomeworkhelper.com

11. 7
b. In part a, we found the cdf of X(n), so the pdf is just the derivaive of this:
ƒ(n) (x) = dF(n) (x)
dx
=
d x
dx 8
n
=
nxn— 1
8n
c. We want to choose constants k1,k2, and k3 such that our estimators are unbiased. For the
first estimator,
E k^1X2
= 8
= 8
= 8
= 2
^
1 2
k E (X )
8
k
^
1
2
k
^
1
Then, for the second estimator
E k^2X
= 8
= 8
=
=
8
2
^
2
k E X
8
k
^
2
2
k
^
2
And for the third estimator
E k^3X(n) = 8
^
3
k E X (n) = 8
k
^
3
∫
O
x
8 nxn—1
8n
^
k3
8n
n 8 n+1
n + 1
dx = 8
= 8
k
^
3 =
n + 1
n
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12. 8
d. Now we calculate the variances of our estimators, using the con- stants that we found
in part c.
Var (2X2)
Var 2X = 4Var X =
82
= 4Var (X2) =
3
2
8
3n
Var
n + 1
X(n)
n + 1
n n
2
= Var X(n)
To find Var X(n) , we will need E(n) X 2 :
2
X =
∫
O
8 n+1
E(n)
8n
nx n
dx = 8
n + 2
2
so
Var X(n) =
n
n + 2
82 —
n n
+ 1
2
82 =
n82
(n + 2)(n + 1)2
and
Var
n + 1
n
X(n) =
n + 1
n
2
n82
82
(n + 2)(n + 1)2 =
n(n + 2)
Thus, whenever n > 1,
Var k
^
3X(n) < Var k
^
2X < Var k
^
1X2
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