Consider a random sample of size n n that follows a density probability function given...
Consider a random sample of size n that follows a density probability function given by:
where is unknown.
Answer & Explanation
Indeed you can use logarithms if you want to, and you don’t need to consider instead of in order to do so. Since
Thus is a sufficient statistic.
But you don’t have to use logarithms. You could also just write
which factorizes with . That and are likewise sufficient statistics is not surprising, since one is an injective function of the other and , and an injective function of a sufficient statistic is again a sufficient statistic.