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**tttcomrader** Let $\displaystyle X_1,...,X_n$ be a random variable from an absolutely continuous distribution with density:

$\displaystyle f_ \theta (x) = \left \{ \begin {array}{rcl} \frac {2x}{ \theta ^2} & x \in (0, \theta ) \\ 0 & \mbox {otherwise} \end {array} \right. $

Find the minimum sufficient statistics T and its density.

The solution is as of follows:

Let $\displaystyle T(x) = \mbox {max} \{ X_i \} , M(x) = \mbox {min} \{ X_i \} $

Then $\displaystyle f_ \theta (x_1, \ldots ,x_n) = f _ \theta (x_1) \cdots f_ \theta (x_n) $

$\displaystyle = \frac {2x_1}{ \theta ^2} I(0<x_1<\theta)\cdots \frac {2x_n}{ \theta ^2} I(0<x_n<\theta)$

$\displaystyle = \prod ^n_{i=1} \frac {2x_i}{ \theta ^2}I(0<x_i<\theta) $

$\displaystyle = \frac {2^n}{ \theta ^{2n}} \cdot \prod ^n_{i=1}x_i I(0<x_i<\theta)$

$\displaystyle = \frac {2^n}{ \theta ^{2n}} \cdot \prod ^n_{i=1} x_i \cdot 1 _{ \{ M(x)>0 \} } \cdot 1_{ \{ T(x)< \theta \} } $, with $\displaystyle T = T(X) $

I understand that this density gives non zero values only when $\displaystyle M(X)>0 $ and $\displaystyle T(X) < \theta $ but I don't understand how that last line with the two indicator functions come into being. Thank you!