Question: Let X 1 , . . . , X n be i.i.d. with the exponential distribution with parameter . Suppose that we wish to test

Let X1, . . . , Xn be i.i.d. with the exponential distribution with parameter θ. Suppose that we wish to test the hypotheses

H0 : θ ≥ θ0,

H1 : θ < θ0.

Let X = E-1 Xi i=1. Let δc be the test that rejects H0 if X ≥ c.

a. Show that π(θ|δc) is a decreasing function of θ.

b. Find c in order to make δc have size α0.

c. Let θ0 = 2, n = 1, and α0 = 0.1. Find the precise form of the test δc and sketch its power function.

X = E-1 Xi i=1

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