Recall that a test is deemed UMP at a level if for any value of the parameter in the alternative hypothesis, the same test is UMP for the test of the simple hypothesis vs. . The Neyman-Pearson Lemma tells us that when testing that the Likelihood Ratio Test is uniformly most powerful among all level tests.… Continue reading Simple and Composite Null UMP Tests
A quick note on convergence in probability vs. convergence a.s.
Convergence in probability says , which controls how a single random variable is close to its putative limit . Convergence almost surely controls how the entire tail approaches the limit simultaneously. I.e. we have that for all . This is equivalent to saying that . This means that if you were simulating out the sequence convergence… Continue reading A quick note on convergence in probability vs. convergence a.s.
Probability Exercise: If it must happen once it happens infinitely often.
Exercise: Let be a sequence of independent events with for all . Show that . Solution: which by independence means that . Now since , we know that for any , which means that Now suppose f.o. Then for sufficiently large . Thus the set f.o., can be written as , which is a countable union of… Continue reading Probability Exercise: If it must happen once it happens infinitely often.