Dr. Hackney STA Solutions pg 112

Dr. Hackney STA Solutions pg 112 - Second Edition 7-15 b....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Second Edition 7-15 b. ∂θ log L ( θ | x ) = ∂θ log Y i log θ θ - 1 θ x i = ∂θ X i [loglog θ - log( θ - 1) + x i log θ ] = X i ± 1 θ log θ - 1 θ - 1 ² + 1 θ X i x i = n θ log θ - n θ - 1 + n ¯ x θ = n θ ³ ¯ x - ± θ θ - 1 - 1 log θ ²´ . Thus, ¯ X is the UMVUE of θ θ - 1 - 1 log θ and attains the Cram´ er-Rao lower bound. Note: We claim that if ∂θ log L ( θ | X ) = a ( θ )[ W ( X ) - τ ( θ )], then E W ( X ) = τ ( θ ), because under the condition of the Cram´ er-Rao Theorem, E ∂θ log L ( θ | x ) = 0. To be rigorous, we need to check the “interchange differentiation and integration“ condition. Both (a) and (b) are exponential families, and this condition is satisfied for all exponential families. 7.39 E θ ³ 2 ∂θ 2 log f ( X | θ ) ´ = E θ ³ ∂θ ± ∂θ log f ( X | θ ) ²´ = E θ " ∂θ µ ∂θ f ( X | θ ) f ( X | θ ) !# = E θ 2 ∂θ 2 f ( X | θ ) f ( X | θ ) - µ ∂θ f ( X | θ ) f ( X | θ ) ! 2 . Now consider the first term: E θ " 2 ∂θ 2 f ( X | θ ) f ( X | θ ) # = Z ³ 2 ∂θ 2 f ( x | θ ) ´ d x = d Z ∂θ f ( x | θ ) d x (assumption) = d E θ ³ ∂θ log f ( X | θ ) ´ = 0
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online