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EXST7034  Regression Techniques
Page 1
Joint Estimation of
and
""
!"
Recall that we have assumed that
is independent of the model, and that the
's are independent of each other
%%
33
We have NOT assumed that the regression coefficients (
and
) are
independent of each other.
However, we place confidence intervals on each as if they were independent
Consider,
Each confidence interval is done with a 5% chance of error.
If we do 2 CI, EACH
has a 5% chance of error.
If we do 20 CI each and every one has a 5%
chance of error.
As we do more CI's, the OVERALL chance of making
an error increases.
Therefore, we want to obtain a confidence interval such that we are, say, 95%
confident that BOTH
and
are contained in the interval.
We start
with the simple, individual confidence intervals.
P
(
bt
s
b
t
s
)
1

!
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8
#
!
!
#
Ÿ
Ÿ
œ
!!
##
bb
"!
P
(
s
t
s
)
1

"
#
"
"
#
Ÿ
œ
Individually, each is correct with probability 1
!
b  tS
0
b
1
1
b
1
b +tS
1
b
1
b
0
b
1
0
b
1
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Page 2
Now we wish to calculate the probability that BOTH are correct.
There is a page of probability theorems in Chapter 1, Section 1.2.
The probability statements are
P(A ) =
P(A ) =
"#
!!
where A and A are some events (eg. probability of error)
P(A
A )
P(A )
P(A )
P(A
A )
"
#
"#"
#
œ
= the union of the events; it is the probability of EITHER event occurring
= the intersection of the events; it is the probability of BOTH events occurring
From this we can derive (see text) the
, which is the
Bonferroni inequality
probability that BOTH CI ARE CORRECT, as
P(A
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This note was uploaded on 12/29/2011 for the course EXST 7034 taught by Professor Geaghan,j during the Fall '08 term at LSU.
 Fall '08
 Geaghan,J

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