Let $(X_1 ,Y_1),(X_2 ,Y_2),\ldots,(X_n ,Y_n)$ be a sample from the uniform distribution on a disc $X^2 + Y^2 \leq \theta$, where $\theta$ is unknown. That is, the joint density function of $(X,Y)$ is $f_{(X,Y)}(x,y,\theta)=\frac{1}{\pi \theta^2} \mathbb 1_{ [0,\theta ]} (\sqrt{x^2+y^2})$.
(1) Find a complete sufficient statistic of $\theta$ and its distribution
(2) Find the UMVUE of $\theta$.
What I did is to multiply the density function to find the likelihood function $L=\frac{1}{\pi^n \theta^{2n}} \mathbb 1_{ [0,\theta ]} (\prod_{i=1}^n \sqrt{x^2_i +y^2_i})$, then we know $L=e^{\ln(L)}$, then I find a complete sufficient statistic $T=\sum_{i=1}^n \ln(x^2_i +y^2_i)$; however, I don't know how to find its distribution.
For part (2), I think it's related to the first part. My idea is to see the expectation of $T$ and its relation with $\theta$.
The likelihood function should be $1/(\pi^n \theta^n)$ when all $\sqrt{x_i^2 + y_i^2} \le \theta$, not when their product $\le \theta$. In particular it depends only on the maximum of $\sqrt{x_i^2 + y_i^2}$. So that maximum is your sufficient statistic. To find the distribution of this statistic, calculate the probability that all $\sqrt{x_i^2 + y_i^2} \le r$, where $0 \le r \le \theta$.