Let $ X_1, ... X_n $ a sample of independent random variables with uniform distribution $(0,$$ \theta $$ ) $ Find a $ $$ \widehat\theta $$ $ estimator for theta using the maximun estimator method more known as MLE
maximum estimator method more known as MLE of a uniform distribution
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0If you want to find the maximum likelihood estimate, you first need to derive the likelihood. Did you get that far? Here is a primer: http://en.wikipedia.org/wiki/Maximum_likelihood_estimator – 2011-07-05
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2You asked this question for the method of moments, but you wanted the MLE. I am assuming in that time you've come up with something... surely... what have you tried? What is your effort? I'll write something that will guide you, but I don't want to just write the solution. – 2011-07-05
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0The following video really helped me: https://www.youtube.com/watch?v=XaAtkCzdjLE – 2015-08-31
2 Answers
First note that $f\left({\bf x}|\theta\right)=\frac{1}{\theta}$ , for $0\leq x\leq\theta$ and $0$ elsewhere.
Let $x_{\left(1\right)}\leq x_{\left(2\right)}\leq\cdots\leq x_{\left(n\right)}$
be the order statistics. Then it is easy to see that the likelihood
function is given by
$$L\left(\theta|{\bf x}\right) = \prod^n_{i=1}\frac{1}{\theta}=\theta^{-n}\,\,\,\,\,(*)$$
for $0\leq x_{(1)}$ and $\theta \geq x_{(n)}$ and $0$ elsewhere.
Now taking the derivative of the log Likelihood wrt $\theta$ gives:
$$\frac{\text{d}\ln L\left(\theta|{\bf x}\right)}{\text{d}\theta}=-\frac{n}{\theta}<0.$$ So we can say that $L\left(\theta|{\bf x}\right)=\theta^{-n}$ is a decreasing function for $\theta\geq x_{\left(n\right)}.$ Using this information and (*) we see that $L\left(\theta|{\bf x}\right)$ is maximized at $\theta=x_{\left(n\right)}.$ Hence the maximum likelihood estimator for $\theta$ is given by $$ \hat{\theta}=x_{\left(n\right)}.$$
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0I think you forgot the d theta in the denominator. but good answer! :) – 2011-07-05
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0Thanks aengle...its fixed...:) – 2011-07-05
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2@Nana Very old question, but still. Isn't there a problem with endpoints of the given interval? If they were included you solution would be perfectly fine, but the are not. How do deal with it? – 2013-06-04
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0I have another queston, I got an unbiased $\frac{n+1}{n}X_(n)$, if given that $\theta$ is greater than 1, will the estimator be changed? – 2014-10-12
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3How is differentiating valid here?? – 2018-05-25
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1https://math.stackexchange.com/questions/649678/how-do-you-differentiate-the-likelihood-function-for-the-uniform-distribution-in?rq=1 – 2018-10-05
This example is worked out in detail here (pages 13-14).
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0your link is broken (at least for me...) :p – 2011-07-05
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0@aengle: Thanks, now it works. – 2011-07-05
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0link is dead again – 2017-02-13