MIT14_30s09_lec20

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Unformatted text preview: 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 . 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 , ] , and we want to construct a 90% confidence interval for . Let = 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, is not unbiased for , we can use it to construct a confidence interval for . From results for order statistics, we saw that the c.d.f. of is given by the c.d.f. of is given by 0 n F ( ) = if 0 < 1 if > where we plugged in the c.d.f. of a U [0 , ] random variable, F ( x ) = x . In order to obtain the functions for A and B , let us first find constants a and b such that P ( a b ) = F ( b ) F ( b ) = 0 . 95 . 05 = 0 . 9 We can find a and b by solving F ( a ) = 0 . 05 and F ( b ) = 0 . 95 n n so that we obtain a = . 05 and b = . 95 . This doesnt give us a confidence interval yet, since looking at the definition of a CI, we want the true parameter 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 . 9 = P ( a b ) = . 05 . 95 = P n P n n . 95 . 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 . 95 . 05 is a 90% confidence interval for . Notice that in this case, the bounds of the confidence intervals depend on the data only through the estimator ( X 1 , . . . , X n ) . This need not be true in general. 1 Lets recap how we arrived at the confidence interval: 1. first get estimator/statistic ( X 1 , . . . , X n ) and the distribution of ....
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