11-Joint CI

11-Joint CI - EXST7034 Regression Techniques Page 1 Joint Estimation of and Recall that we have assumed that%3 is independent of the model and that

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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 - ! "ß 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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EXST7034 - Regression Techniques 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.

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11-Joint CI - EXST7034 Regression Techniques Page 1 Joint Estimation of and Recall that we have assumed that%3 is independent of the model and that

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