- #1
BlueKazoo
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Homework Statement
Let Y1,Y2,...,Yn denote independent and identically distributed random variables from a power family distribution with parameters α and θ. Then, if α, θ > 0,
f(y|α, θ)={αy(α-1)/θα, 0≤y≤θ; 0, otherwise.
If θ is known, show that ∏i=1n Yi is sufficient for α.
Homework Equations
Well, I know that the likelihood function is L(Y1,Y2,...,Yn|θ)=∏i=1nf(yi|θ).
I also know that a statistic is sufficient if L(Y1,Y2,...,Yn|θ)=g(u,θ) x h(y1,y2,...,yn) Where g(u, θ) is a function of only u and θ, and h(y1,y2,...,yn) doesn't have θ. I am not 100% clear on what exactly this entails in practice since there are very few examples in my book which deal with proving sufficiency this way.
The Attempt at a Solution
I simply plugged the first formula into the the second, so I got:
∏i=1nαyiα-1/θα
(α/θα)(∏i=1nyi)α-1
I have no idea how to handle the remaining product function. I have some more problems in my homework where I reach an impasse at this point. Can I stop here, or am I missing some steps here?