{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Dr. Hackney STA Solutions pg 94

Dr. Hackney STA Solutions pg 94 - 6-8 Solutions Manual for...

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

6-8 Solutions Manual for Statistical Inference (ii) If α is fixed, T = i X i is a complete sufficient statistic for β by Theorem 6.2.25. Because β is a scale parameter, if Z 1 , . . . , Z n is a random sample from a gamma( α, 1) distribution, then X ( i ) /T has the same distribution as ( βZ ( i ) ) / ( β i Z i ) = Z ( i ) / ( i Z i ), and this distribution does not depend on β . Thus, X ( i ) /T is ancillary, and by Basu’s Theorem, it is independent of T . We have E( X ( i ) | T ) = E X ( i ) T T T = T E X ( i ) T T indep . = T E X ( i ) T part (i) = T E( X ( i ) ) E T . Note, this expression is correct for each fixed value of ( α, β ), regardless whether α is “known” or not. 6.32 In the Formal Likelihood Principle, take E 1 = E 2 = E . Then the conclusion is Ev( E, x 1 ) = Ev( E, x 2 ) if L ( θ | x 1 ) /L ( θ | x 2 ) = c . Thus evidence is equal whenever the likelihood functions are equal, and this follows from Formal Sufficiency and Conditionality. 6.33 a. For all sample points except (2 , x * 2 ) (but including (1 , x * 1 )), T ( j, x j ) = ( j, x j ). Hence, g ( T ( j, x j ) | θ ) h ( j, x j ) = g (( j, x j ) | θ )1 = f * (( j, x j ) | θ ) .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online