91Ó°ÊÓ

A Beta distribution is defined by two parameters, \(\alpha\) and \(\beta\). Given that the problem mentions \(\alpha=\beta=\theta>0\), we have a symmetric Beta distribution with parameter \(\theta\). The pdf of a Beta distribution is given by: \(f(x;\alpha,\beta)=\frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}\), where \(B(\alpha,\beta)\) is the Beta function. Here, since \(\alpha=\beta=\theta\), the pdf simplifies to \(f(x;\theta,\theta)=\frac{x^{\theta-1}(1-x)^{\theta-1}}{B(\theta,\theta)}\).

A statistic \(T(X)\) is said to be sufficient for \(\theta\) if the conditional probability distribution of the data \(X\), given the statistic \(T(X)\), does not depend on \(\theta\). In the case of a Beta distribution, the sufficient statistic is generally the order statistics. Now, given \(n\) i.i.d Beta random variables \(X_i\), \(i=1,2,...,n\), with parameters \(\theta\) (i.e., \(\alpha=\beta\)=\(\theta\)), the joint density function of \(X_i\) is as follows: \(f(X_1,...,X_n;\theta)=\prod_{i=1}^{n}\frac{x_i^{\theta-1}(1-x_i)^{\theta-1}}{B(\theta,\theta)}\). We can see that for the i.i.d sample \(X_i\), the joint density function can be factorized into a function of \(\theta\) and a function of data \(X_i\). This shows that the data does not provide any other information about \(\theta\) beyond the sample observations, thus all i.i.d observations \(X_i\) are a complete sufficient statistics for \(\theta\) on its own.

Hence, the sample independent observations arising from the Beta distribution in which \(\alpha=\beta=\theta>0\) are the sufficient statistic for \(\theta\), as no other values derived from the sample data could provide any additional information about \(\theta\) beyond what these observations provide.