Econ 140
Inference about a Mean
Lecture 5
Lecture 5
1
Today’s Plan
Econ 140
• Start to explore more concrete ideas of statistical inference
• Look at the process of generalizing from the sample mean
to the population value
• Consider properties of the sample mean Y as a point
estimator
• Properties of an estimator: BLUE (Best Linear Unbiased
Estimator)
Lecture 5
2
Sample and Population Differences
Econ 140
• So far we’ve seen how weights connect a sample and a
population
• But what if all we have is a sample without any weights?
• What can the estimation of the mean tell us?
• We need the sample to be composed of independently
random and identically drawn observations
Lecture 5
3
Estimating the Expected value
Econ 140
• We’ve dealt with the expected value µy=E(Y) and the
variance V(Y) = 2
• Previously, our estimation of the expected value of the
mean Y was
Y

Y 
n
– But this is only a good estimate of the true expected
value if the sample is an unbiased representation of the
population
– What does the actual estimator tell us?
Lecture 5
4
BLUE
Econ 140
• We need to consider the properties of Y as a point
estimator of µ
• Three properties of an estimator : BLUE
– Best (efficiency)
– Linearity
– Unbiasedness
– (also Consistency)
• We’ll look at linearity first, then unbiasedness and
efficiency
Lecture 5
5
BLUE: Linearity
Econ 140
• Y is a linear function of sample observations
n
 Yi
Y Y ...Yn
Y  i 1  1 2
n
n
• The values of Y are added up in a linear fashion such that
all Y values appear with weight equal
Lecture 5
6
BLUE: Unbiasedness
Econ 140
• Proving that Y is an unbiased estimator of the expected
value of µ
• We can rewrite the equation for Y
n
 Yi
Y  i 1   ciYi
n
where ci  1
n
– This expression says that each Y has an equal weight of
1/n
• Since ci is a constant, the expectation of Y is
1
E (Y )   ci E (Y )   ci (  ) n   
n
Lecture 5
7
Proving Unbiasedness
Econ 140
• Lets examine an estimator that is biased and inefficient
• We can define some other estimator m as
m   ciYi
where di  ci 'ci
ci '  ci  di
• We can then plug the equation for c’ into the equation for
m and take its expectation
• The expected value of this new estimator m is biased if
 di  0
Lecture 5
8
BLUE: Best (Efficiency)
Econ 140
• To look at efficiency, we want to consider the variance of Y
• We can redefine Y as Y   ciY
• Our variance can be written as
V (Y )   ci2V (Y )  h  j chc j C (YhY j )
– Where the last term is the covariance term
– Covariance cancels out because we are assuming that
the sample was constructed under independence. So
there should be no covariance between the Y values
C (YhY j )  0
– Note: we’ll see later in the semester that covariance will
not always be zero
Lecture 5
9
BLUE: Best (Efficiency) (2)
Econ 140
• So how did we get the equation for the variance of Y ?
V (Y )   2
V (Y )   2
2 
c
 i
2
n
  Yi 
 i  1
1
2
2
V Y   V 
  2  V (Yi )  2 n 
n
n
 n  n i


Lecture 5
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Variance
Econ 140
• Our expression for variance shows that the variance of Y
is dependent on the sample size n
• How is this different from the variance of Y?
2
n
2
y
Lecture 5
Y
Y
11
Variance (2)
Econ 140
• Before when we were considering the distribution around
µy we were considering the distribution of Y
• Now we are considering Y as a point estimator for µy
– The estimate for Y will have its own probability
distribution much like Y had its own
– The difference is that the distribution for Y has a
variance of 2/n whereas Y has a variance of 2
Lecture 5
12
Proving Efficiency
Econ 140
• The variance of m looks like this
V(m) = Sici’2V(Y) + ShSich’ci’C(YhYi)
• Why is this not the most efficient estimate?
• We have an inefficient estimator if we use anything other
than ci for weights
Lecture 5
13
Consistency
Econ 140
• This isn’t directly a part of BLUE
• The idea is that an optimal estimator is best, linear, and
unbiased
• But, an estimator can be biased or unbiased and still be
consistent
• Consistency means that with repeated sampling, the
estimator tends to the same value for Y
Lecture 5
14
Consistency (2)
Econ 140
• We write our estimator of µ as
Y

Y 
n
• We can write a second estimator of µ
Y

Y* 
n
• The expected value of Y* is
1
E (Y1)  ...  E (Yn )
n 1
1
n
1  ...   n  


n 1
n 1
E (Y ) 
Lecture 5
15
Consistency (3)
Econ 140
• If n is small, say 10,
n
10

n  1 11
– Y* will be a biased estimator of µ
– But, Y* will be a consistent estimator
– So as n approaches infinity Y* becomes an unbiased
estimator of µ
Lecture 5
16
Law of Large Numbers
Econ 140
• Think of this picture:
PDF
n=500
n=1000
n=200
n=50
As you draw samples of
larger and larger size, the
law of large numbers
says that your estimation
of the sample mean will
become a better
approximation of µy
Y
The law only hold if you are drawing random samples
Lecture 5
17
Central Limit Theorem
Econ 140
• Even if the underlying population is not normally
distributed, the sampling distribution of the mean tends to
normality as sample size increases
– This is an important result if n < 30
PDF
Population
Sample
Y
Lecture 5
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What have we done today?
Econ 140
• Examined the properties of an estimator.
• Estimator was for the estimation of a value for an unknown
population mean.
• Desirable properties are BLUE: Best Linear Unbiased.
• Also should include consistency.
Lecture 5
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