Proving sufficiency via likelihood functions

[BlueKazoo]

Homework Statement

Let Y₁,Y₂,...,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=1ⁿ Y_i is sufficient for α.

Homework Equations

Well, I know that the likelihood function is L(Y₁,Y₂,...,Y_n|θ)=∏_i=1ⁿf(y_i|θ).

I also know that a statistic is sufficient if L(Y₁,Y₂,...,Y_n|θ)=g(u,θ) x h(y₁,y₂,...,y_n) Where g(u, θ) is a function of only u and θ, and h(y₁,y₂,...,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=1ⁿαy_i^α-1/θ^α
(α/θ^α)(∏_i=1ⁿy_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?

 

[micromass]

I also know that a statistic is sufficient if L(Y₁,Y₂,...,Y_n|θ)=g(u,θ) x h(y₁,y₂,...,y_n) Where g(u, θ) is a function of only u and θ, and h(y₁,y₂,...,y_n) doesn't have θ.

What is u in the first place? Is it your statistic?? What is your statistic in this case??

Secondly, θ is the unknown parameter. What is the unknown parameter in this case?? From this sentence

If θ is known,

I already know that θ won't be the unkown parameter.

 

[BlueKazoo]

Ah, whoops, I should have used different variables in the problem than in the set of given equations.

I believe "u" in g(u|θ) would be the ∏_i=1ⁿY_i i this case. In g(u|θ) and L(...|θ), θ is just whatever variable I need to prove the first one is sufficient for.

So I guess I'm trying to find g(∏_i=1ⁿY_i|α) and h(Y₁...Y_n), the latter of which would not contain α). I apologize for the confusion, though I believe the presentation of the book is a bit wonky to begin with and I'm not entirely sure how to use these equations.

 

[micromass]

OK, next. What is the joint distribution of Y₁,...,Y_n??

I know you said

\prod_{i=1} \frac{\alpha y_i^{\alpha-1}}{\theta^\alpha}

but that isn't correct. The reason that it's not correct is because this form only holds for 0\leq y_i\leq \theta. But we want the formula to hold for all y_i.

To make it work, you'll need to use characteristic functions. These are defined for a set A as

I_A:\Omega\rightarrow \mathbb{R}:x\rightarrow \left\{\begin{array}{l}0 ~\text{if} ~x\notin A\\ 1~\text{if}~x\in A\end{array}\right.

Try to use this with a suitable A and \Omega.

 

[BlueKazoo]

Oh, so you basically mean that it's incorrect because I didn't take into account where the function is 0 otherwise.

If any y is not between 0 and θ, the value of the function is only dependent on that specific y and therefore dependent on the product of all the y's. So the function is sufficient for α, since all other values besides that product y are inconsequential to the final value, right?

Buuut, let's assume all y's fall between 0 and θ. Then would I solve it like this?:
\prod_{i=1}^{n}\frac{\alpha y_{i}^{\alpha - 1}}{\theta^{\alpha}}
\frac{\alpha^{n}}{\theta^{n\alpha}}(\prod_{i=1}^{n}y_{i})^{\alpha - 1}
Therefore
g(\prod_{i=1}^{n}y_{i}|\alpha)=\frac{\alpha^{n}}{ \theta ^{n\alpha}}(\prod_{i=1}^{n}y_{i})^{\alpha - 1}
h(y_{1},y_{2},...,y_{n})=1
Since n and θ are constants.

I should have probably set up the indicator function earlier, and I'm not too familiar with them, but I'll try anyways. I guess A could be the range of numbers between 0 and θ? And Ω would be some form of my function? Honestly, I don't understand how to set them up in this case, and none of my resources have examples of this (in this context anyways).

 

[micromass]

Oh, so you basically mean that it's incorrect because I didn't take into account where the function is 0 otherwise.

If any y is not between 0 and θ, the value of the function is only dependent on that specific y and therefore dependent on the product of all the y's. So the function is sufficient for α, since all other values besides that product y are inconsequential to the final value, right?

Buuut, let's assume all y's fall between 0 and θ. Then would I solve it like this?:
\prod_{i=1}^{n}\frac{\alpha y_{i}^{\alpha - 1}}{\theta^{\alpha}}
\frac{\alpha^{n}}{\theta^{n\alpha}}(\prod_{i=1}^{n}y_{i})^{\alpha - 1}
Therefore
g(\prod_{i=1}^{n}y_{i}|\alpha)=\frac{\alpha^{n}}{ \theta ^{n\alpha}}(\prod_{i=1}^{n}y_{i})^{\alpha - 1}
h(y_{1},y_{2},...,y_{n})=1
Since n and θ are constants.

OK, that seems good if all y are between 0 and θ. But now you need to work with indicator functions.

I should have probably set up the indicator function earlier, and I'm not too familiar with them, but I'll try anyways. I guess A could be the range of numbers between 0 and θ? And Ω would be some form of my function? Honestly, I don't understand how to set them up in this case, and none of my resources have examples of this (in this context anyways).

OK. Let's try an example. Let's say that I have f(x)=x^3 for 0≤x≤1 and f(x)=0 otherwise. I would write this as

f(x)=x^3 I_{[0,1]}(x)

So in this case, A=[0,1] and \Omega=\mathbb{R}. Do you see why this works??
What is it going to be in your case?