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hw6 - Homework Due iid Assume X1 Xn Exponential Use this...

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Homework Due Nov 29, 2011 Assume X 1 . . . X n iid Exponential ( λ ) . Use this from of the exponential distribution: f ( x ) = 1 λ e - x λ 1. Find the MLE for λ 2. Find the asymptotic variance of the MLE, V ( ˆ λ ) 3. Is ˆ λ biased or unbiased? 4. What is the MSE of ˆ λ 5. What is the MLE for λ 2 , the variance of the exponential distribution? 6. Is ˆ λ 2 biased or unbiased? 7. If ˆ λ 2 is biased, propose a modification to make it unbiased and call this estimator ˆ λ 2 2 8. The exponential distribution is often used for survival analysis with X the time until death. Consider the following data, the observed time until cancer mortality following therapeutic intervention in years: 13 . 84 , 4 . 52 , 18 . 43 , 5 . 51 , 8 . 58 , 8 . 43 , 0 . 13 , 0 . 01 , 0 . 01 , 3 . 44 Calculate ˆ λ and its 95 % . Interpret both of these. 9. Plot the likelihood and log-likelihood of these data. Point out where the MLE occurs on the plot 10. Make an excel (or whatever program you like) spreadsheet with λ ranging from 0 to 20 (with increments of 0.1; that is, 0, 0.1, 0.2, 0.3, etc) as one column and

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hw6 - Homework Due iid Assume X1 Xn Exponential Use this...

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