Let X be one observation from a Cauchy( Icirc cedil ) distribution (a) Show that this family does not have an MLR (b) Show that the test is most powerful of its size for testing H0 Icirc cedil 0 versus H1 Icirc cedil 1 Calculate the Type I and Type II Error probabilities (c) Prove or disprove The test in part (b) is UMP for testing H0 Icirc cedil 0 What can be said about UMP tests in general for the Cauchy location family f 1 z 3 otherwise (z) 10

Question: Let X be one observation from a Cauchy(Î¸) distribution. (a) Show that this family does not have an MLR. (b) Show that the test is

Let X be one observation from a Cauchy(Î¸) distribution.
(a) Show that this family does not have an MLR.
(b) Show that the test

is most powerful of its size for testing H0: Î¸ = 0 versus H1. Î¸ = 1. Calculate the Type I and Type II Error probabilities.
(c) Prove or disprove: The test in part (b) is UMP for testing H0 : Î¸ 0. What can be said about UMP tests in general for the Cauchy location family?

f 1 < z < 3 otherwise (z) 10

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