lec4 - Lecture 4 Let us go back to the example of...

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Lecture 4 Let us go back to the example of exponential distribution E ( α ) from the last lecture and recall that we obtained two estimates of unknown parameter α 0 using the Frst and second moment in the method of moments. We had: 1. Estimate of α 0 using Frst moment: g ( X ) = X, m ( α ) = α g ( X ) = 1 α , ˆ α 1 = m - 1 g ) = 1 ¯ X . 2. Estimate of α using second moment: g ( X ) = X 2 , m ( α ) = α g ( X 2 ) = 2 α 2 , ˆ α 2 = m - 1 g ) = r 2 ¯ X 2 . How to decide which method is better? The asymptotic normality result states: n ( m - 1 g ) - θ 0 ) N ± 0 , Var θ 0 ( g ( X )) ( m 0 ( θ 0 )) 2 ² . It makes sense to compare two estimates by comparing their asymptotic variance. Let us compute it in both cases: 1. In the Frst case: Var α 0 ( g ( X )) ( m 0 ( α 0 )) 2 = Var α 0 ( X ) ( - 1 α 2 0 ) 2 = 1 α 2 0 ( - 1 α 2 0 ) 2 = α 2 0 . In the second case we will need to compute the fourth moment of the exponential distribution. This can be easily done by integration by parts but we will show a di±erent way to do this. The moment generating function of the distribution E ( α ) is: ϕ ( t ) = α e tX = Z 0 e tx αe - αx dx = α α - t = X k =0 t k α k , 13
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LECTURE 4. 14 where in the last step we wrote the usual Taylor series. On the other hand, writing
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.

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lec4 - Lecture 4 Let us go back to the example of...

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