# Sufficient statistic and complete sufficient statistic [duplicate]

I'm trying self-study some inference and now I'm trying to understand how to solve some problems on this topic but I found this basic problem that I'm not being able to solve.

Problem:

Let $$X_{1},...,X_{n}$$ i.i.d $$f(x | \theta)$$, where $$f(x | \theta) = \theta x ^{\theta -1 }$$.

Is $$\sum_{i}^{n} X_{i}$$ a sufficent statistic for $$\theta$$? How can I find a complete sufficient statistic?

Im really confused actually. I know these problems can be solved with the factorization theorem, so we have that $$f(x_{1},...,x_{n} | \theta) = \prod \theta x_{i} ^{\theta -1} = \theta ^{n} \prod x_{i}^{\theta} \prod x_{i}^{-1}$$, then $$h(x) = \prod x_{i} ^{-1}$$ and $$g[\theta,T(x_{i})] = \theta^{n}\prod x_{i}^{\theta}$$. So in this case $$T(x_{1},...,x_{n}) = \prod x_{i}$$ is a sufficient statistic right?

But I don't know hot to check if $$\sum X_{i}$$ is in fact a suff. statistic and I don't know how to prove that the statistic I found is complete.

• Are you asking about $\sum x_i$ or $\prod x_i$ ? Is $X_i$ restricted to $[0,1]$? Commented Jul 21, 2021 at 9:12
• Im refering to $\sum x_{i}$ and yes, X_{i} is restricted to $[0,1]$. Thanks for your comment :) Commented Jul 21, 2021 at 9:36
• Then you seem to have shown $\sum x_i$ is not a sufficient statistic but that $\prod x_i$ is - or some 1-1 function of it like $\sum \log(x_i)$ Commented Jul 21, 2021 at 9:39
• To check whether or not $T(X)$ is a sufficient statistic you may consider the conditional distribution of $X$ given $T(X)$. Commented Jul 21, 2021 at 11:34
• Or you may consider whether or not the likelihood ratio$$\ell(x)/\ell(x^\prime)$$is constant in $\theta$ when$$T(x)=T(x^\prime)$$ Commented Jul 21, 2021 at 11:36