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Unformatted text preview: la to calculate q from sample data is called an
estimator VER. 9/11/2012. © P. KOLM 34 An Example Estimating the population mean by the sample average
n Let us start with the formula for E (X ) = å x i f (x i )
i =1 Since each point has an equal chance of being included in the sample, we
1
for f (x i ) (the probability weight). Then, the sample average for
n
our sample takes the form substitute 1n
ˆ
mX º X º å Xi
n i =1 We will see many more examples throughout the lecture sequence VER. 9/11/2012. © P. KOLM 35 What Makes a Good Estimator? Some estimators are better than others. The search for good estimators is a big
part of econometrics
Estimators are compared based on a number of attributes: Unbiasedness Efficiency Mean Square Error (MSE) Asymptotic properties (for large samples) Consistency We discuss each one of these attributes next VER. 9/11/2012. © P. KOLM 36 Unbiasedness Intuition: We want our estimator “to be right on average”
ˆ Formally, we say that an estimator, q = h(Y ) , of the parameter q is unbiased
if the mean of its sampling distribution is q . Mathematically, this means
that ˆ
E (q) = q VER. 9/11/2012. © P. KOLM 37 Example: The Sample Mean and Variance Let Y = (Y1,Y2,...,Yn ) be a random sample from a population with mean mY and
2
variance sY 1n The sample mean Y = åYi is unbiased:
n i =1
We take the expectation of the sample mean
æ1 n ö 1 n
11
÷=
ç
E (Y ) = E ç åYi ÷
E (Yi ) = å n mY =mY
÷
÷
ç n i =1 ø n å
n i =1
è
i =1 Note: The variance of Y (the sample variance) is given by
2
n
æ1 n ö
sY
2
÷= 1
ç
÷
Var (Y ) = Var ç åYi ÷
s=
ç n i =1 ÷ n 2 å Y
n
è
ø
i =1 Therefore, as n increases Y gets more precise VER. 9/11/2012. © P. KOLM 38 1n The estimator for the variance s =
å (Yi Y )2 is unbiased:
n  1 i =1
First, observe that
2
Y n n å (Yi Y ) = å ((Yi  mY )  (Y  mY ))
2 i =1 2 i =1
n n = å (Yi  mY )  2(Y  mY )å (Yi  mY ) + n(Y  mY )2
2 i =1
n i =1 = å (Yi  mY )2  2n(Y  mY )2 + n(Y  mY )2
i =1
n = å (Yi  mY )2  n(Y  mY )2
i =1 Therefore,
æ1 n
ö
2÷
ç
E (s ) = E ç
(Y Y ) ÷
÷
÷
çn  1 å i
è
ø
i =1
æ1 n
ö
n
2÷
ç
÷
E (Y  mY )2
=Eç
(Yi  mY ) ÷ å
÷ n 1
ç n  1 i =1
è
ø
2
Y ( ) n
n
E (Yi  mY )2 E (Y  mY )2
n 1
n 1
2
ææ n
öö
çç 1
n
n
÷÷
çç
÷÷
÷
=
Var (Yi ) E ç å (Yi  mY )÷ ÷
çç
÷÷
÷
n 1
n  1 çè n i =1
øø
è = VER. 9/11/2012. © P. KOLM ( ) ( ) 39 Since Y = (Y1,Y2,...,Yn ) are iid, we obtain 2
ææ n
öö
çç 1
n
n
2
÷÷
çç
÷÷
E (sY ) =
Var (Yi ) E ç å (Yi  mY )÷ ÷
çç n i =1
÷÷
n 1
n  1 çè
ø÷
÷
è
ø
æ1 n
ö
n
n
2÷
ç
=
Var (Yi ) E ç 2 å (Yi  mY ) ÷
÷
÷
n 1
n  1 ç n i =1
è
ø
n
n
n
Var (Yi ) =
⋅ 2 E (Yi  mY )2
n 1
n 1 n
n
n
n
Var (Yi ) =
⋅ 2 E (Yi  mY )2
n 1
n 1 n
= Var (Yi ) ( ) ( ) which shows the unbiasedness of the estimator. 1n
ˆ
Note: The estimator s = å (Yi Y )2 is biased downward. In particular,
n i =1
n 1 2
n 1
1
2
ˆ2
ˆ2
s . However, Var (sY ) = 2 2 s 4 < Var (sY ) = 2
E (sY ) =
s 4 (For you:
n
n 1
n
Show this.) This means that the biased estimator has a smaller variance. The
2
Y difference is negligible in large samples, but, for example is about 10% in a
sample of 16. VER. 9/11/2012...
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This document was uploaded on 02/17/2014 for the course COURANT G63.2751.0 at NYU.
 Fall '14

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