Problem 2: Suppose we have an observation X from the logistic distribution given by f(x) =...

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Problem 2: Suppose we have an observation X from the logistic distribution given by f(x) = exp (x-6) (1+exp (x-0))' with x E R and 0 E R. 1. Show that this family has MLR property [5] II. Using the above find a UMP size a test for testing Ho: 0 = 0 vs. H:0> 0 [5] III. Let a + a2 = a < 0.05 all positive constants, using the CDF of X derive a 100(1 - a)% Confidence interval for 0. [7] IV. Derive the shortest confidence interval in the class (X-c,X-d) with confidence coefficient (1 - a). [5] V. What is the relationship between above III and IV. [3] Problem 2: Suppose we have an observation X from the logistic distribution given by f(x) = exp (x-6) (1+exp (x-0))' with x E R and 0 E R. 1. Show that this family has MLR property [5] II. Using the above find a UMP size a test for testing Ho: 0 = 0 vs. H:0> 0 [5] III. Let a + a2 = a < 0.05 all positive constants, using the CDF of X derive a 100(1 - a)% Confidence interval for 0. [7] IV. Derive the shortest confidence interval in the class (X-c,X-d) with confidence coefficient (1 - a). [5] V. What is the relationship between above III and IV. [3]