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In my book: $\mathbf{X}=(X_1,\ldots,X_n)$ $f(\mathbf{x})$ is the joint density, where $f$ is either $f_0 \text{ or } f_1$.

Suppose we want to test $H_0: f=f_0$ or $H_1: f=f_1$. The test, whose test function is

$$\phi(\mathbf{X})=1\text{ if }\frac{f_1}{f_0}\geq k;$$

$$\phi(\mathbf{X})=0 \text{ otherwise,}$$

(for some $0<k<\infty$) is a most powerful test of $H_0$ versus $H_1$ at level $E_0(\phi(\mathbf{X}))$.

My question is how is $k$ defined? Can I interpret the lemma as if $\forall k \in (0,\infty)$ there will be a test function $\phi(\mathbf{X})$ such that it will determine the size for which the test with test function $\phi(\mathbf{X})$ is most powerful?

I'm just trying to understand which kind of relationship $k$ and the size of the test have between each other.

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    $\begingroup$ Please don't cross-post your questions here and on math.stackexchange. $\endgroup$ – Stefan Hansen Jul 30 '14 at 15:00
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The value of $k$ determines the size of the test. You have to find the expression for $E_0 \phi(X)$ in terms of $k$, equate it to the desired size, and solve for $k$.

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