I have a question about the MLE of the following distribution.
let $Y$ be a Uniform$(0,\theta)$ random variable, where $0<\theta<\infty$ and $\theta$ is to be estimated.
The first question asks me to write down the joing PDF of $Y_1…Y_n$ which I believe is the following: $$ \frac{1}{\theta^n}$$
The second question asks for the likelihood function which I think is:
$$Likelihood(y_1…y_n|\theta)= \frac{1}{\theta^n}$$
Now to find the ML Estimator of $\theta$, I would have to take the natural log and the derivative correct? After doing so I get the following:
$$-\frac{n}{\theta}$$
Unfortunately, setting this equation to $0$ does not yield anything valuable. How would I be able to find the ML Estimator and Estimate without using the derivative. The "answer" that I have in my notes says that I should "argue that making $\hat\theta$ equal to the largest observation maximizes the likelihood." Can someone please explain this argument in non-technical language?
Also, if we let $X$ denote the largest observation among $Y_1…Y_n$, how can we show that the PDF of $X$ is $$\frac{n}{\theta^n}x^{n-1}$$
How can one show that? Help would be greatly appreciated! Thanks so much!!
Best Answer
Since your question, "how would I find the PDF of $X$ if $X$ represents the largest observation" is a distinct question from finding the MLE of $\theta$, it warrants a separate answer.
For IID $X_1, X_2, \ldots, X_n \sim {\rm Uniform}(0,\theta)$, the last order statistic $$X_{(n)} = \max_i X_i$$ is the largest of the observed values in the sample. Consider its CDF: $$F_{X_{(n)}}(x) = \Pr[X_{(n)} \le x] = \Pr\left[ \bigcap_{i=1}^n X_i \le x \right].$$ This is because the largest of the observations is less than or equal to $x$ if and only if every observation is less than or equal to $x$. But since the observations are IID, it follows that $$F_{X_{(n)}}(x) = \prod_{i=1}^n \Pr[X_i \le x] = \begin{cases} 0 & x < 0 \\ (x/\theta)^n & 0 \le x \le \theta \\ 1 & x > \theta.\end{cases}$$ Consequently, the PDF of the last order statistic is $$f_{X_{(n)}}(x) = \frac{nx^{n-1}}{\theta^n}, \quad 0 \le x \le \theta.$$
As a further exercise, what is the PDF of the first order statistic (i.e., the minimum of the sample)?