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From Wald. Likelihood ratio, and lagrange multiplier tests in econometrics by Robert f. Engle:

(Tests whose) size does not depend upon the particular value of $\theta \in$ null $\Theta_0$ are called similar tests.

I don't quite understand the definition. Isn't the size of a test the sup of false positive rate over null $\Theta_0$? So it doesn't depend on a single $\theta \in \Theta_0$, but on $\Theta_0$?

Following provides more context from the source. Thanks and regards!

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  • $\begingroup$ +1 You have indeed uncovered an abuse of notation: $\alpha_T$ is explicitly an (ill-defined) function of $\Theta_0$ while "$\theta$" enters as a bound variable and so does not even exist semantically. It is clear, though, that Engle is referring at the end to a modified definition of size, equal to $\alpha_T(\theta)=\Pr(y\in C_T\mid \theta)$, which is a function of $\theta$, and that he really wants to define $\alpha_T=\sup_{\theta\in\Theta_0}\alpha_T(\theta)$. $\endgroup$
    – whuber ♦
    Commented Nov 17, 2014 at 18:18

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As defined above in Engle's book, there is no supremum in the expression for the size, so it may indeed depend on particular parameter value $\theta \in \Theta_0.$ As far as I know, it is now more common to call this expression null rejection probability (NRP).

The (finite sample) size is then defined as the supremum over the null set $Sz_n=\sup_{\theta \in \Theta_0} Pr(y \in C_T \vert \theta)$ and, furthermore, the asymptotic size is defined as $AsySz = \limsup_{n \to \infty} Sz_n$, the notions you probably messed up the above notion of NRP with.

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    $\begingroup$ This is a good interpretation but it's not what's in the book: $\alpha_T$ is explicitly written to be a function of $\Theta_0$--but that function doesn't even make sense unless $\Pr(y\in C_T|\theta)$ does not vary for $\theta\in\Theta_0$. $\endgroup$
    – whuber ♦
    Commented Nov 17, 2014 at 18:18

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