Consider a random sample of size n n that follows a density probability function given...

Lorena Becker

Lorena Becker

Answered

2022-11-22

Consider a random sample of size n n that follows a density probability function given by:
f ( x , θ ) = 1 θ x θ + 1 θ 1 ( 1 , + ) , θ > 0
where θ is unknown.

Answer & Explanation

Averi Shaffer

Averi Shaffer

Expert

2022-11-23Added 10 answers

Indeed you can use logarithms if you want to, and you don’t need to consider log f instead of f in order to do so. Since
f ( x θ ) = i = 1 n f ( x i , θ ) = 1 θ n i = 1 n x i θ + 1 θ = 1 θ n exp ( θ + 1 θ i = 1 n ln x i ) ,
we have
f ( x θ ) = g ( T ( x ) θ ) h ( x ) ,
with
T ( x ) = i = 1 n ln x .
Thus T ( x ) is a sufficient statistic.
But you don’t have to use logarithms. You could also just write
f ( x θ ) = i = 1 n f ( x i , θ ) = 1 θ n i = 1 n x i θ + 1 θ = 1 θ n ( i = 1 n x i ) θ + 1 θ ,
which factorizes with S ( x ) = i = 1 n x i . That S ( x ) and T ( x ) are likewise sufficient statistics is not surprising, since one is an injective function of the other S ( x ) = exp ( T ( x ) ) and T ( x ) = log ( S ( x ) ), and an injective function of a sufficient statistic is again a sufficient statistic.

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