3. Let Y be a singleobservation from a population with density function. f(y)= 2y/θ2 0 ≤ y...

3. Let Y be a singleobservation from a population with density function.

f(y)= 2y/θ2 0 ≤ y ≤ θ

f(y)= 0 elsewhere

(a) Find the MLE for θ.

(b) Is the MLE found in (a) also a MVUE of θ? Justify your answer.

(c) Show that Y/ θis a pivotal quantity for θ.

(d) Use the pivotal Y /θ to find a 100(1 − α)% upper confidence interval for θ, which is of form (−∞, θˆU ).

For the following questions, suppose for some θ0 > 0, we wish to test H0 :θ = θ0v.s. H1 :θ < θ0 based on a single observation Y .

(e) Show that a level-α test has the rejection region given by ’reject H0if Y <θ0’.

(f) What is the connection between the lower confidence interval found in (b) and the rejection region found in (e)?

(g) Derive an expression for type II error probability βof the rejection region in (e). It will be a function of α, θ0 and θ1 a particular value of θ that less than θ0.

(h) Suppose θ0 = 1 and one observe Y = 0.1. Calculate the P-value.

Expert Solution

orchestra answered 1 year ago

3. (Not Bayesian) Let Y have pdf fY (y, θ) = 2(θ − y)/θ2 if 0...

3. (Not Bayesian) Let Y have pdf fY (y, θ) = 2(θ − y)/θ2 if 0 < y < θ 0 otherwise . You are going to construct a confidence interval for θ based on a single observation Y . (a) Show that Y/θ is pivotal quantity. (b) Suppose Y = 6. Find (numerically) a 10% confidence interval for θ .

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