# simple vs composite hypothesis doubt

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)

$H_0 :\theta<=\theta_0$ 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 $H_0 :\alpha=\alpha_0 , \beta>=\beta_0$

Simple or composite?

$H_0 :\theta=\theta_0$ vs $H_1 :\theta>\theta_0$ is a composite hypothesis since for $H_1$ you can have many different $\theta$s.
• What matters most are the properties of the null hypothesis, because it determines the sampling distribution of the test statistic used to evaluate the null. When the null is composite, the situation is tricky because the test statistic does not have a definite distribution. $H_0:\theta=\theta_0$ might or might not be composite, depending on what other parameters might be in play and how they affect the test statistic's distribution, but at face value most would consider this to be a simple hypothesis. The nature of $H_1$ plays no role in classifying hypotheses into simple or composite. – whuber Aug 20 '15 at 1:30