Transcript Supplement #1B

ESTIMATORS
Estimators and estimates:
An estimator is a mathematical formula.
An estimate is a number obtained by applying
this formula to a set of sample data.
It is important to distinguish between estimators and estimates. Definitions are given
above.
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ESTIMATORS
Population characteristic
Estimator
Mean: mX
1 n
X   xi
n i 1
A common example of an estimator is the sample mean, which is the usual estimator of the
population mean.
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ESTIMATORS
Population characteristic
Estimator
Mean: mX
1 n
X   xi
n i 1
Here it is defined for a random variable X and a sample of n observations.
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ESTIMATORS
Population characteristic
Estimator
Mean: mX
1 n
X   xi
n i 1
Population variance: s X2
n
1
2
s2 
(
x

X
)
 i
n  1 i 1
Another common estimator is s2, defined above. It is used to estimate the population
variance, sX2.
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ESTIMATORS
Estimators are random variables
1 n
1
X   x i  ( x1  ...  x n )
n i 1
n
An estimator is a special kind of random variable. We will demonstrate this in the case of
the sample mean.
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ESTIMATORS
Estimators are random variables
1 n
1
X   x i  ( x1  ...  x n )
n i 1
n
xi  m X  ui
We saw in the previous sequence that each observation on X can be decomposed into a
fixed component and a random component.
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ESTIMATORS
Estimators are random variables
1 n
1
X   x i  ( x1  ...  x n )
n i 1
n
xi  m X  ui
1
1
X  ( m X  ...  m X )  ( u1  ...  un )
n
n
1
 ( nm X )  u  m X  u
n
So the sample mean is the average of n fixed components and n random components.
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ESTIMATORS
Estimators are random variables
1 n
1
X   x i  ( x1  ...  x n )
n i 1
n
xi  m X  ui
1
1
X  ( m X  ...  m X )  ( u1  ...  un )
n
n
1
 ( nm X )  u  m X  u
n
It thus has a fixed component mX and a random component u, the average of the random
components in the observations in the sample.
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ESTIMATORS
probability density
probability density
function of X
function of X
mX
X
mX
X
The graph compares the probability density functions of X and X. As we have seen, they
have the same fixed component. However the distribution of the sample mean is more
concentrated.
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ESTIMATORS
probability density
probability density
function of X
function of X
mX
X
mX
X
Its random component tends to be smaller than that of X because it is the average of the
random components in all the observations, and these tend to cancel each other out.
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UNBIASEDNESS AND EFFICIENCY
Unbiasedness of X:
1
 1
E ( X )  E  ( x1  ... xn )  E ( x1  ...  xn )
n
 n
1
1
 E ( x1 )  ...  E ( xn )  nm X  m X
n
n
Suppose that you wish to estimate the population mean mX of a random variable X given a
sample of observations. We will demonstrate that the sample mean is an unbiased
estimator, but not the only one.
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UNBIASEDNESS AND EFFICIENCY
Unbiasedness of X:
1
 1
E ( X )  E  ( x1  ... xn )  E ( x1  ...  xn )
n
 n
1
1
 E ( x1 )  ...  E ( xn )  nm X  m X
n
n
We use the second expected value rule to take the (1/n) factor out of the expectation
expression.
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UNBIASEDNESS AND EFFICIENCY
Unbiasedness of X:
1
 1
E ( X )  E  ( x1  ... xn )  E ( x1  ...  xn )
n
 n
1
1
 E ( x1 )  ...  E ( xn )  nm X  m X
n
n
Next we use the first expected value rule to break up the expression into the sum of the
expectations of the observations.
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UNBIASEDNESS AND EFFICIENCY
Unbiasedness of X:
1
 1
E ( X )  E  ( x1  ... xn )  E ( x1  ...  xn )
n
 n
1
1
 E ( x1 )  ...  E ( xn )  nm X  m X
n
n
Each expectation is equal to mX, and hence the expected value of the sample mean is mX.
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UNBIASEDNESS AND EFFICIENCY
probability
density
function
estimator B
estimator A
mX
How do we choose among them? The answer is to use the most efficient estimator, the one
with the smallest population variance, because it will tend to be the most accurate.
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UNBIASEDNESS AND EFFICIENCY
probability
density
function
estimator B
estimator A
mX
In the diagram, A and B are both unbiased estimators but B is superior because it is more
efficient.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
probability
density
function
estimator B
estimator A
q
Suppose that you have alternative estimators of a population characteristic q, one unbiased,
the other biased but with a smaller population variance. How do you choose between
them?
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
loss
error (negative)
error (positive)
One way is to define a loss function which reflects the cost to you of making errors, positive
or negative, of different sizes.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
probability
density
function
MSE( Z )  E ( Z  q ) 2   s Z2  ( m Z  q ) 2
estimator B
q
A widely-used loss function is the mean square error of the estimator, defined as the
expected value of the square of the deviation of the estimator about the true value of the
population characteristic.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
probability
density
function
MSE( Z )  E ( Z  q ) 2   s Z2  ( m Z  q ) 2
estimator B
bias
q
mZ
The mean square error involves a trade-off between the population variance of the estimator
and its bias. Suppose you have a biased estimator like estimator B above, with expected
value mZ.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
probability
density
function
MSE( Z )  E ( Z  q ) 2   s Z2  ( m Z  q ) 2
estimator B
bias
q
mZ
The mean square error can be shown to be equal to the sum of the population variance of
the estimator and the square of the bias.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
MSE( Z )  E ( Z  q ) 2 
 E ( Z  m Z  m Z  q ) 2 
 E ( Z  m Z ) 2  ( m Z  q ) 2  2( Z  m Z )( m Z  q )
 E ( Z  m Z ) 2   E ( m Z  q ) 2   E 2( Z  m Z )( m Z  q )
 s Z2  ( m Z  q ) 2  2( m Z  q ) E ( Z  m Z )
 s Z2  ( m Z  q ) 2  2( m Z  q )( m Z  m Z )
 s Z2  ( m Z  q ) 2
To demonstrate this, we start by subtracting and adding mZ .
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
MSE( Z )  E ( Z  q ) 2 
 E ( Z  m Z  m Z  q ) 2 
 E ( Z  m Z ) 2  ( m Z  q ) 2  2( Z  m Z )( m Z  q )
 E ( Z  m Z ) 2   E ( m Z  q ) 2   E 2( Z  m Z )( m Z  q )
 s Z2  ( m Z  q ) 2  2( m Z  q ) E ( Z  m Z )
 s Z2  ( m Z  q ) 2  2( m Z  q )( m Z  m Z )
 s Z2  ( m Z  q ) 2
We expand the quadratic using the rule (a + b)2 = a2 + b2 + 2ab, where a = Z - mZ and b = mZ q.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
MSE( Z )  E ( Z  q ) 2 
 E ( Z  m Z  m Z  q ) 2 
 E ( Z  m Z ) 2  ( m Z  q ) 2  2( Z  m Z )( m Z  q )
 E ( Z  m Z ) 2   E ( m Z  q ) 2   E 2( Z  m Z )( m Z  q )
 s Z2  ( m Z  q ) 2  2( m Z  q ) E ( Z  m Z )
 s Z2  ( m Z  q ) 2  2( m Z  q )( m Z  m Z )
 s Z2  ( m Z  q ) 2
We use the first expected value rule to break up the expectation into its three components.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
MSE( Z )  E ( Z  q ) 2 
 E ( Z  m Z  m Z  q ) 2 
 E ( Z  m Z ) 2  ( m Z  q ) 2  2( Z  m Z )( m Z  q )
 E ( Z  m Z ) 2   E ( m Z  q ) 2   E 2( Z  m Z )( m Z  q )
 s Z2  ( m Z  q ) 2  2( m Z  q ) E ( Z  m Z )
 s Z2  ( m Z  q ) 2  2( m Z  q )( m Z  m Z )
 s Z2  ( m Z  q ) 2
The first term in the expression is by definition the population variance of Z.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
MSE( Z )  E ( Z  q ) 2 
 E ( Z  m Z  m Z  q ) 2 
 E ( Z  m Z ) 2  ( m Z  q ) 2  2( Z  m Z )( m Z  q )
 E ( Z  m Z ) 2   E ( m Z  q ) 2   E 2( Z  m Z )( m Z  q )
 s Z2  ( m Z  q ) 2  2( m Z  q ) E ( Z  m Z )
 s Z2  ( m Z  q ) 2  2( m Z  q )( m Z  m Z )
 s Z2  ( m Z  q ) 2
(mZ - q) is a constant, so the second term is a constant.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
MSE( Z )  E ( Z  q ) 2 
 E ( Z  m Z  m Z  q ) 2 
 E ( Z  m Z ) 2  ( m Z  q ) 2  2( Z  m Z )( m Z  q )
 E ( Z  m Z ) 2   E ( m Z  q ) 2   E 2( Z  m Z )( m Z  q )
 s Z2  ( m Z  q ) 2  2( m Z  q ) E ( Z  m Z )
 s Z2  ( m Z  q ) 2  2( m Z  q )( m Z  m Z )
 s Z2  ( m Z  q ) 2
In the third term, (mZ - q) may be brought out of the expectation, again because it is a
constant, using the second expected value rule.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
MSE( Z )  E ( Z  q ) 2 
 E ( Z  m Z  m Z  q ) 2 
 E ( Z  m Z ) 2  ( m Z  q ) 2  2( Z  m Z )( m Z  q )
 E ( Z  m Z ) 2   E ( m Z  q ) 2   E 2( Z  m Z )( m Z  q )
 s Z2  ( m Z  q ) 2  2( m Z  q ) E ( Z  m Z )
 s Z2  ( m Z  q ) 2  2( m Z  q )( m Z  m Z )
 s Z2  ( m Z  q ) 2
Now E(Z) is mZ, and E(-mZ) is -mZ.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
MSE( Z )  E ( Z  q ) 2 
 E ( Z  m Z  m Z  q ) 2 
 E ( Z  m Z ) 2  ( m Z  q ) 2  2( Z  m Z )( m Z  q )
 E ( Z  m Z ) 2   E ( m Z  q ) 2   E 2( Z  m Z )( m Z  q )
 s Z2  ( m Z  q ) 2  2( m Z  q ) E ( Z  m Z )
 s Z2  ( m Z  q ) 2  2( m Z  q )( m Z  m Z )
 s Z2  ( m Z  q ) 2
Hence the third term is zero and the mean square error of Z is shown be the sum of the
population variance of Z and the bias squared.
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CONFLICTS BETWEEN UNBIASEDNESS AND MINIMUM VARIANCE
probability
density
function
estimator B
estimator A
q
In the case of the estimators shown, estimator B is probably a little better than estimator A
according to the MSE criterion.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
The sample mean is the usual estimator of a population mean, for reasons discussed in the
previous sequence. In this sequence we will see how its properties are affected by the
sample size.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
Suppose that a random variable X has population mean 100 and standard deviation 50, as in
the diagram. Suppose that we do not know the population mean and we are using the
sample mean to estimate it.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
The sample mean will have the same population mean as X, but its standard deviation will
be 50/ n , where n is the number of observations in the sample.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
The larger is the sample, the smaller will be the standard deviation of the sample mean.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
0.08
1
50
0.06
0.04
0.02
n=1
50
100
150
200
If n is equal to 1, the sample consists of a single observation. X is the same as X and its
standard deviation is 50.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
0.08
1
4
50
25
0.06
0.04
n=4
0.02
50
100
150
200
We will see how the shape of the distribution changes as the sample size is increased.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
0.08
1
4
25
50
25
10
0.06
n = 25
0.04
0.02
50
100
150
200
The distribution becomes more concentrated about the population mean.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
n = 100
0.08
1
4
25
100
0.06
50
25
10
5
0.04
0.02
50
100
150
200
To see what happens for n greater than 100, we will have to change the vertical scale.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
0.8
1
4
25
100
0.6
50
25
10
5
0.4
n = 100
0.2
50
100
150
200
We have increased the vertical scale by a factor of 10.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
0.8
1
4
25
100
1000
0.6
n = 1000
50
25
10
5
1.6
0.4
0.2
50
100
150
200
The distribution continues to contract about the population mean.
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EFFECT OF INCREASING THE SAMPLE SIZE ON THE DISTRIBUTION OF x
probability density
function of X
n sX
n = 5000
0.8
1
4
25
100
1000
5000
0.6
50
25
10
5
1.6
0.7
0.4
0.2
50
100
150
200
In the limit, the variance of the distribution tends to zero. The distribution collapses to a
spike at the true value. The sample mean is therefore a consistent estimator of the
population mean.
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