Consider independent observations ?1,…,?? y 1 , … , y n from the model ??∼Poisson(?) Y i ∼ Poisson ( μ ) . Using likelihood ?(?) L ( μ ) and log-likelihood ?(?) l ( μ ) as appropriate, compute the following items. a. Derive the maximum likelihood estimate ?̂ μ ^ . b. Write the second derivative of log-likelihood ?(?) l ( μ ) . c. Give an expression of the approximated asymptotic standard error of ?̂ μ ^ by plugging in the estimate ?̂ μ ^ . To this end, estimate the Fisher Information Matrix by ?̂=−∂2∂?2?(?)∣∣?=?̂ V ^ = − ∂ 2 ∂ μ 2 l ( μ ) | μ = μ ^ and then ?.?.(?̂)=?̂−1‾‾‾‾√ s . e . ( μ ^ ) = V ^ − 1 . d. Consider data 23, 14, 16, 22, 18, 22, 24, 30. Using your formulæ, compute and write numerical estimates ?̂ μ ^ , ?.?.(?̂) s . e . ( μ ^ ) and give a 95% confidence interval for ? μ using the normal approximation.
Consider independent observations ?1,…,?? y 1 , … , y n from the model ??∼Poisson(?) Y i ∼ Poisson ( μ ) . Using likelihood ?(?) L ( μ ) and log-likelihood ?(?) l ( μ ) as appropriate, compute the following items.
a. Derive the maximum likelihood estimate ?̂ μ ^ .
b. Write the second derivative of log-likelihood ?(?) l ( μ ) .
c. Give an expression of the approximated asymptotic standard error of ?̂ μ ^ by plugging in the estimate ?̂ μ ^ . To this end, estimate the Fisher Information Matrix by ?̂=−∂2∂?2?(?)∣∣?=?̂ V ^ = − ∂ 2 ∂ μ 2 l ( μ ) | μ = μ ^ and then ?.?.(?̂)=?̂−1‾‾‾‾√ s . e . ( μ ^ ) = V ^ − 1 .
d. Consider data 23, 14, 16, 22, 18, 22, 24, 30. Using your formulæ, compute and write numerical estimates ?̂ μ ^ , ?.?.(?̂) s . e . ( μ ^ ) and give a 95% confidence interval for ? μ using the normal approximation.
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