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MIT14_30s09_lec20

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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 20 Konrad Menzel April 30, 2009 Confidence Intervals (continued) The following example illustrates one way of constructing a confidence interval when the distribution of the estimator is not normal. Example 1 Suppose X 1 , . . . , X n are i.i.d. with X i U [0 , θ ] , and we want to construct a 90% confidence interval for θ 0 . Let θ ˆ = max { X 1 , . . . , X n } = X ( n ) the n th order statistic (as we showed last time, this is also the maximum-likelihood estimator). Even though, as we saw, θ ˆ is not unbiased for θ , we can use it to construct a confidence interval for θ . From results for order statistics, we saw that the c.d.f. of θ ˆ is given by the c.d.f. of θ ˆ is given by 0 θ 0 ⎨ � θ n F θ ˆ ( θ ) = θ 0 if 0 < θ θ 0 1 if θ > θ 0 where we plugged in the c.d.f. of a U [0 , θ 0 ] random variable, F ( x ) = θ x 0 . In order to obtain the functions for A and B , let us first find constants a and b such that P θ 0 ( a θ ˆ b ) = F θ ˆ ( b ) F θ ˆ ( b ) = 0 . 95 0 . 05 = 0 . 9 We can find a and b by solving F θ ˆ ( a ) = 0 . 05 and F θ ˆ ( b ) = 0 . 95 n n so that we obtain a = 0 . 05 θ 0 and b = 0 . 95 θ 0 . This doesn’t give us a confidence interval yet, since looking at the definition of a CI, we want the true parameter θ 0 in the middle of the inequalities, and the functions on either side depend only on the data and other known quantities. However, we can rewrite n θ ˆ θ ˆ 0 . 9 = P θ 0 ( a θ ˆ b ) = 0 . 05 θ 0 ˆ 0 . 95 θ 0 = P θ 0 n P θ 0 n θ n 0 . 95 θ 0 ≤ √ 0 . 05 Therefore max { X 1 , . . . , X n } max { X 1 , . . . , X n } [ A, B ] = [ A ( X 1 , . . . , X n ) , B ( X 1 , . . . , X n )] = n , n 0 . 95 0 . 05 is a 90% confidence interval for θ 0 . Notice that in this case, the bounds of the confidence intervals depend on the data only through the estimator θ ˆ ( X 1 , . . . , X n ) . This need not be true in general. 1
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Let’s recap how we arrived at the confidence interval: 1. first get estimator/statistic θ ˆ ( X 1 , . . . , X n ) and the distribution of θ ˆ . 2. find a ( θ ) , b ( θ ) such that P ( a ( θ ) θ ˆ b ( θ )) = 1 α 3. rewrite the event by solving for θ P ( A ( X ) θ B ( X )) = P ( A ( θ ˆ ) θ B ( θ ˆ )) = 1 α 4. evaluate A ( X ) , B ( X ) for the observed sample X 1 , . . . , X n 5. the 1 α confidence interval is then given by CI = [ A ( X 1 , . . . , X n ) , B ( X 1 , . . . , X n )] 1.1 Important Cases 1. θ ˆ is
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