# Uniformly most powerful test does not exists

I am having tough time understanding this concept The book says:

“We caution the reader that UMP tests for testing H0 : θ1 ≤ θ ≤ θ2 and H0′ : θ = θ0 for the one-parameter exponential family do not exist. An example will suffice.

Example 5. Let X1,X2,...,Xn be a sample from􏰂 N(0,σ2). Since the family of joint PDFs of X = (X1,...,Xn) has an MLR in T(X) = \$\sum_X_i^2, it follows that UMP tests exist for one-sided hypotheses σ ≥ σ0 and σ ≤ σ0.”

Does this means that if we have only one sample actually our z-test and t-test For inequality hypothesis does not exist?

The $$z$$-test and $$t$$-test still exist and they have all the nice power properties that are available. It's just that UMP tests don't make sense with a two-sided alternative.
If you have a two-sided test of equality of means, you can increase the power in one direction (eg, against an increase in mean) at the expense of lower power in the other direction (a decrease in mean). You can think of it in terms of partitioning the size $$\alpha$$ of the test: the rejection region has to have probability $$\alpha$$ in total, and it comes in two parts with the non-rejection region separating them. If you make the upper rejection region larger, by moving the threshold down towards equality, you need to make the lower rejection region smaller by moving its threshold down, away from equality.