Dr. Hackney STA Solutions pg 112

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

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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
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## This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.

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