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Unformatted text preview: ML estimate for θ is ˆ θ mle = n − 1 P n i =1 x i . 2. Determine the asymptotic normal distribution for ˆ θ mle . 1 3. Derive the Wald, LR and LM statistics for testing the above hypothesis. These statistics have the form Wald = ( ˆ θ mle − θ ) 2 I ( ˆ θ mle | x ) LM = S ( θ | x ) 2 I ( θ | x ) − 1 LR = − 2[ln L ( θ | x ) − ln L ( ˆ θ mle | x )] Are these statistics numerically equivalent? For a 5% test, what is the decision rule to reject H for each statistic ? 4. Prove that the Wald and LM statistics have asymptotic χ 2 (1) distributions. 5. How do your results compare to those that you derived for GMM in the previous lab? 2...
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- Fall '09
- Maximum likelihood, maximum likelihood estimation, Wald, Bernoulli random variables, LM statistics, Eric Zivot Fall