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Let Y1,Y2,...,Yn denote a random sample from the uniform distribution over the interval (0,theta). Show that Y(n)=max(Y1,Y2,...,Yn) is a sufficient statistic for theta by the factorization theorem.

Solution:

http://www.geocities.com/asdfasdf23135/stat10.JPG

1) While I understand that I_{A}(x)I_{B}(x)=I_{A intersect B}(x), I don't understand the equality circled in red above.

In the solutions, they say that I_{0,theta}(y1)...I_{0,theta}(yn)=I_{0,theta}(y_{(n)}). Is this really correct?

Shouldn't the right hand side be I_{0,theta}(y_{(n)})I_{0,infinity}(y_{(1)}) ? I believe that the second factor is necessary because the largest observation is greater than zero does not guarantee that the smallest observation is greater than zero.

Which one is correct?

2) Also, is I_{0,theta}(y_{(n)}) a function of y_{(n)}, a function of theta, or a function of both y_{(n)}and theta?

If it is a function of both y_{(n)}and theta, then there is something that I don't understand. Following the definition of indicator function that I_{A}(x) is a function of x alone (it is a function of only the stuff in the parenthesis), shouldn't I_{0,theta}(y_{(n)}) be a function of only y_{(n)}alone?

Thank you for explaining! I've been confused with these ideas for at least a week.

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# Factorization Theorem for Sufficient Statistics & Indicator Function

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