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I have a problem for finding a sufficient and a minimal sufficient statistics for the next density: let $X_1,...,X_n$ a sample with $$f_X(x;\alpha,\theta)=\frac{\alpha x^{\alpha-1}}{\theta^{\alpha}}I_{(0,\theta)}(x)$$

First of all, I tried to use the Factorization theorem (single sufficient statistics) ($f_{X_1,...,X_n}(x_1,...,x_n;\boldsymbol{\theta})=g(s(x_1,...,x_n);\boldsymbol{\theta})h(x_1,...,x_n)$) but I only can factorized it in terms of jointly sufficient statistics.

On the other hand, assuming that $\alpha$ is known (that is not the case), it is clear that $\max\{X_1,...,X_n\}$ (or some function of that) is a good candidate for a single sufficient statistics for $\theta$.

How can I find a (single) minimal sufficient statistics for $\boldsymbol{\theta}=(\alpha,\theta)$?

Thanks in advance.

sinbadh
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  • Why should there be a one-dimensional (assuming this is what you mean by 'single') sufficient statistic? – StubbornAtom Nov 19 '21 at 05:50
  • @StubbornAtom "single" means that the conditional distribution of $X_1,...,X_n$ given $T=t$ does not depend on $\boldsymbol{\theta}=(\alpha,\theta)$ for any value $t$ of $T$. It is not about one-dimensional – sinbadh Nov 19 '21 at 17:23
  • You say "I only can factorized it in terms of jointly sufficient statistics". So have you or have you not found your sufficient statistic? – StubbornAtom Nov 19 '21 at 19:13
  • @StubbornAtom a jointly sufficient statistics is a set $T_1,T_2,...,T_r$ of statistics such that the conditional distribution of $X_1,X_2,...,X_n$ given $T_1=t_1,T_2=t_2,...,T_r=t_r$ does not depend on $\boldsymbol{\theta}$. I want to find a single sufficient statistic, but I find two jointly sufficient statistics – sinbadh Nov 19 '21 at 19:45
  • That's what I was referring to in my first comment. When you say $(T_1,T_2)$ is jointly sufficient, it means your sufficient statistic is two-dimensional (or vector-valued). You cannot find a scalar-valued sufficient statistic here. – StubbornAtom Nov 19 '21 at 20:04

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