Confdence Intervals: Lecture XXI
Charles B. Moss
October 19, 2010
I. Interval Estimation
A. As we discussed when we talked about continuous distribution
functions, the probability of a speciFc number under a continuous
distribution is zero.
B. Thus, if we conceptualize any estimator, either a nonparametric
estimate of the mean or a parametric estimate of a function, the
probability of the true value equal to the estimated value is obvi
ously zero.
C. Thus, usually talk about estimated values in terms of conFdence
intervals. SpeciFcally, as in the case when we discussed the proba
bility of a continuous variable, we deFne some range of outcomes.
However, this time we usually work the other way around deFning
a certain conFdence level and then stating the values that contain
this conFdence interval.
II. ConFdence Intervals
A. Amemiya notes a di±erence between conFdence and probability.
Most troubling is our classic deFnition of probability as ”a proba
bilistic statement involving parameters.” This is troublesome due
to our inability without some additional Bayesian structure to
state anything concrete about probabilities.
B.
Example 8.2.1.
Let
X
i
be distributed as a Bernoulli distribution,
i
=1
,
2
,
···
n
. Then,
T
=
¯
X
A
∼
N
p,
p
(1
−
p
)
n
!
(1)
1
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View Full DocumentAEB 6571 Econometric Methods I
Professor Charles B. Moss
Lecture XXI
Fall 2010
Table 1: Con±dence Levels
kγ
/
2
γ
1.0000
0.1587
0.3173
1.5000
0.0668
0.1336
1.6449
0.0500
0.1000
1.7500
0.0401
0.0801
1.9600
0.0250
0.0500
2.0000
0.0228
0.0455
2.3263
0.0100
0.0200
Therefore, we have
Z
=
T
−
p
s
p
(1
−
p
)
n
A
∼
N
(0
,
1)
(2)
1. Why? By the Central Limit Theory.
2. Given this distribution, we can ask questions about the prob
ability. Speci±cally, we know that if
Z
is distributed
N
(0
,
1)
, then we can de±ne
γ
k
=
P
(

Z

<k
)(
3
)
Building on the normal probability, the one tailed probabili
ties for the normal distribution are presented in Table 1.
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 Spring '10
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 Normal Distribution, Professor Charles, econometric methods, Professor Charles B. Moss, Lecture XXI

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