Solved – simple vs composite hypothesis doubt

Question

So if we have

$H_0 :\theta=\theta_0$ vs $H_1 :\theta=\theta_1$

It is easy to see that this is a case of simple vs simple hypothesis (assuming that $\theta$ is the only unknown parameter of our distribution)

what about

$H_0 :\theta\leq\theta_0$ vs $H_1 :\theta>\theta_0$

Is this composite vs composite or simple vs composite?

Since it is somewhat equivalent to

$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$

Which I guess it's a simple vs composite hypothesis

And last, if we have two unkown parameters, is
$H_0 :\alpha=\alpha_0 , \beta\geq\beta_0$

Simple or composite?

Answer 1

$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$ is a composite hypothesis since for $H_1$ you can have many different $\theta$s.

You can check these links the explanations are pretty clear.

http://www.emathzone.com/tutorials/basic-statistics/simple-hypothesis-and-composite-hypothesis.html

http://isites.harvard.edu/fs/docs/icb.topic1383356.files/Lecture%2014%20-%20Intro%20to%20Hypothesis%20Testing%20-%204%20per%20page.pdf