 We then have

which is exactly in the form of the exponential family shown in the previous section, with the sufficient statistic being

Thus, we conclude that the test

is the UMP test of size formula_20 for testing formula_46 vs. formula_47

Finally, we note that in general, UMP tests do not exist for vector parameters or for two-sided tests (a test in which one hypothesis lies on both sides of the alternative).The reason is that in these situations, the most powerful test of a given size for one possible value of the parameter (e.g.for formula_48 where formula_49) is different from the most powerful test of the same size for a different value of the parameter (e.g.