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
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.
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.