So, I want to know if a test is a UMP test.
for:
$ θ_0=3 $
$ θ_1<θ_0 $
using neyman-pearson lemma suppose I got something like this where our estimator for
$ θ_0 $ is the mean and for some distributions of the hypothesis parameters'(doesnt need to matter):
$ Λ(X1,...,Xn )=⋯=(\frac{θ_1}{θ_0} )^{\bar x * n} $
$=((\frac{θ_1}{3} )^{n})^{\bar x} \leq constant $
$\implies \bar x \leq \log_{(\frac{θ_1}{3} )^{n}}{constant} $
and hence infer that: 1. the critical region is what I found(suppose we have some value of significant level):for every mean that is less than(or equals) the expression we reject. and also infer that 2. the critical region depends on $ θ_1 $ and hence the test is not a UMP test. why the two arguments above are false? Thank you very much in advance!