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## Homework Statement

Let Y

_{1},Y

_{2},...,Y

_{n}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)}/θ

^{α}, 0≤y≤θ; 0, otherwise.

If θ is known, show that ∏

_{i=1}

^{n}Y

_{i}is sufficient for α.

## Homework Equations

Well, I know that the likelihood function is L(Y

_{1},Y

_{2},...,Y

_{n}|θ)=∏

_{i=1}

^{n}f(y

_{i}|θ).

I also know that a statistic is sufficient if L(Y

_{1},Y

_{2},...,Y

_{n}|θ)=g(u,θ) x h(y

_{1},y

_{2},...,y

_{n}) Where g(u, θ) is a function of only u and θ, and h(y

_{1},y

_{2},...,y

_{n}) 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=1}

^{n}αy

_{i}

^{α-1}/θ

^{α}

(α/θ

^{α})(∏

_{i=1}

^{n}y

_{i})

^{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?