Sample maximum estimator

Question

Suppose $x_1, \cdots, x_n$ is a random sample from a uniform $(0, \theta)$ distribution. Suppose $\theta$ is unknown. Let $y_n= \max (x_1,\cdots, x_n)$. Based on the cdf of $y_n$, it is seen that $y_n$ converged in probability to $\theta$, indicating it is a consistent estimate of $\theta$.

I got the cdf of $y_n$ to be $$F\{y_n\} (t) = \left(\frac{t}{\theta}\right)^n $$

But I don't understand how the cdf helps us to see that $y_n$ is a consistent estimate of theta. Can someone explain?

please use mathjax for readability. Click edit on your question to see how they marked it up or google how to do it. — spaceisdarkgreen, Sep 26 '17 at 05:55

score 1 · Answer 1 · answered Sep 26 '17 at 05:52

1

HINT

We have $$P(|Y_n-\theta|\ge \epsilon) = P(Y_n\le \theta-\epsilon) = \left(\frac{\theta-\epsilon}{\theta}\right)^n.$$

answered Sep 26 '17 at 05:52

spaceisdarkgreen

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Sample maximum estimator

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