In Larry Wasserman's book "All of statistics" on p.151 there is the definition of a one-sided test:
A test of the form
$H_0: \theta \le \theta_0~~$ versus $~~H_1: \theta \gt \theta_0$
or
$H_0: \theta \ge \theta_0~~$ versus $~~H_1: \theta \lt \theta_0$
is called a one-sided test
And in the probability course website there is exactly the same definition.
So this definition looks quite popular and widespread. I want to know how to properly name tests in which only one hypothesis is one-sided? For example, can we call the following tests one-sided (I think the first two of them are widespread enough):
$H_0: \theta = \theta_0~~$ versus $~~H_1: \theta \gt \theta_0$
$H_0: \theta = \theta_0~~$ versus $~~H_1: \theta \lt \theta_0$
$H_0: \theta \lt \theta_0~~$ versus $~~H_1: \theta = \theta_0$
$H_0: \theta \gt \theta_0~~$ versus $~~H_1: \theta = \theta_0$
Best Answer
$H_0: \theta = \theta_0$ versus $H_1: \theta \gt \theta_0$ is OK, but some authors might write $H_0: \theta \le \theta_0$ versus $H_1: \theta \gt \theta_0.$
$H_0: \theta = \theta_0$ versus $H_1: \theta \lt \theta_0$ is OK, but some authors might write $H_0: \theta \ge \theta_0$ versus $H_1: \theta \lt \theta_0.$
In each instance above, a test of $H_0$ would use $\theta = \theta_0$ to get the null distribution, used to compute the P-value of the test.
Both of the formulations below are wrong, because $H_0$ must always contain an $=$-sign, whether as $\theta = \theta_0,$ $\theta \le \theta_0,$ or as $\theta \ge \theta_0.$
$H_0: \theta < \theta_0$ versus $H_1: \theta = \theta_0.$
$H_0: \theta > \theta_0$ versus $H_1: \theta = \theta_0.$