Let X be one observation from a Cauchy(θ) 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: θ = 0 versus H1. θ = 1. Calculate the Type I and Type II Error probabilities.
(c) Prove or disprove: The test in part (b) is UMP for testing H0 : θ 0. What can be said about UMP tests in general for the Cauchy location family?

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