pdf of a maximum order statistic with a uniform distribution

I have the following problem,

$$X_1,X_2,...,X_n$$~$$U[\theta,2\theta]$$ I am tasked with finding the pdf of the mle. First off, I know that $$f(x)=\frac{1}{\theta}$$ just by definition and after some calculations, I have found the following,

$$lik(\theta)=\frac{1}{\theta^n} \implies \hat{\theta}_{mle}=\frac{X_{(n)}}{2}$$

Where I'm struggling is that I now want to find the pdf of $$\hat{\theta}_{mle}$$ using the order statistic $$X_{(n)}$$ by using the formula of $$f_u(u)=n[F(u)]^{n-1}f(u)$$ where $$U=\hat{\theta}_{mle}=X_{(n)}/2$$.

Using some calculations I get the following, $$f(u)=\frac{1}{\hat{\theta}_{mle}}$$ $$F(u)=\frac{x-\hat{\theta}_{mle}}{\hat{\theta}_{mle}}$$ $$\implies f_u(u)=n[\frac{x-\hat{\theta}_{mle}}{\hat{\theta}_{mle}}]^{n-1} \frac{1}{\hat{\theta}_{mle}}$$ Are my assertions correct on this? The reason I ask is because when I take the expectation of that function I should get $$E(X_{(n)})=\frac{2n+1}{n+1} \theta$$

However, that's not the outcome that I am getting. Thanks

Given $$\theta$$,

• the probability all $$n$$ values in a sample are less than or equal to $$y$$ is $$\left(\frac{y-\theta}{\theta}\right)^n$$ for $$\theta \le y \le 2 \theta$$

• so $$\mathbb P(\hat{\theta}_{mle} \le x) = \left(\frac{2x-\theta}{\theta}\right)^n$$ for $$\frac\theta2 \le x \le \theta$$

• so the density of the distribution of the maximum likelihood estimator is $$\frac{2n}{\theta}\left(\frac{2x-\theta}{\theta}\right)^{n-1}$$ when $$\frac\theta2 \le x \le \theta$$; (your density is for $$X_{(n)}$$ when $$\theta \le x \le 2\theta$$ rather than for $$\hat{\theta}_{mle}$$ and subtracts the estimator rather than the actual parameter)

and that gives a mean of $$\mathbb E[\hat{\theta}_{mle}] = \frac{2n+1}{2n+2} \theta = \theta - \frac{1}{2n+2}\theta$$ and $$\mathbb E[X_{(n)}] =\mathbb E[2\hat{\theta}_{mle}]= \frac{2n+1}{n+1} \theta = 2\theta - \frac{1}{n+1}\theta$$ as you might expect

• Thank you for the input on this. I just have a quick question what exactly does $t$ here represent the order statistic $X_{(n)}$? May 9 '20 at 22:10
• @Warhawk1987 $t$ should have been $\theta$ May 9 '20 at 22:53
• Okay I get it now. Thank you. May 9 '20 at 23:08