Let Y1, Y2, . . . , Yn be independent and identically distributed random variables having the same population distribution with density: f(y; θ) = ( θ(3^θ)/y^(θ+1) , y ⩾ 3; 0, elsewhere.) where θ is a positive unknown parameter.
So my question is, how can I use the Rao–Blackwell theorem to find the MVUE of θ? I was given the hint, "Show that the distribution of 2θ*Sum(i=1 to n) [ln(Yi/3)] is χ^2 with df = 2n.
I first showed that the density is an exponential family distribution as required, and I found that the sufficient statistic is K(y) = ln(y/3) which implies Y = Sum(i=1 to n) [ln(Yi/3)] I think. Im confident on that because it closely matches with the hint that was given.
Honestly, I have NO clue where to go from here since this is completely different from homework, class lectures, the additional examples etc. I've spent probably three hours online trying to find anything at all to help me solve this...I feel like I'm missing something so simple. Any ideas? Btw, I use the Book by Wackerly and Mendenhall (Probably the worst book I've ever had the displeasure of trying to learn from) BUT I also have the book by Hogg.