Estimating the side of a square with random points

Question

I have a square with side $\theta$ and origin in $(0,0)$. I simulate some random values inside the square's area. What is the maximum likelihhod estimator, confidence interval and a non-biased estimator for $\theta$?

Answer 1

Are you using the uniform distribution to generate your random numbers?

If so, then your random variable $X\sim U(0, \theta)$, which means that the likelihood is $L(\theta|\mathbf{x})=\prod_{i=1}^n f(x)=\prod_{i=1}^n \frac{1}{\theta}I_{(x_i\leq \theta)}=\frac{1}{\theta^n}I_{(x_{(n)}\leq\theta)}$, which means that the maximum likelihood estimator must be $x_{(n)}\equiv \max x$.

It is not unbiased (but consistent), but I don't want to spend time on that if this isn't answering your question.